ultra-slow, stopped, and compressed light in bose-einstein condensates

Ultra-slow, stopped, and compressed light inBose-Einstein condensates

A thesis presented

by

Zachary Dutton

to

The Department of Physics

in partial fulfillment of the requirements

for the degree of

Doctor of Philosophy

in the subject of

Physics

Harvard University

Cambridge, Massachusetts

January 2002

Thesis advisor Author

Lene Vestergaard Hau Zachary Dutton

Ultra-slow, stopped, and compressed light in Bose-Einstein

condensates

Abstract

We present theoretical analysis and experimental results of methods to achieve

and use ultra-slow light (USL), stopped light, and compressed light in sodium Bose-

Einstein condensates (BECs). We present applications of these methods to study and

harness both the coherent and superfluid properties of BECs.

A description of the propagation of probe and coupling laser fields in three-level Λ

configuration atoms is presented in a semi-classical description. This formalism is used

to derive how electromagnetically induced transparency (EIT) and USL arise. We

present novel theoretical results on the effect a fourth level, and effects of nonlinearities

associated with a strong probe. Experimental demonstration of ultra-slow light is

presented. A description which includes atomic motion in Bose-condensed samples

of alkali atoms is developed in a mean field description and coupled Gross-Pitaevskii

equations are derived. A numerical code which solves these equations is presented. An

analytic and numerical analysis reveal the limits on ultra-slow light and compressed

light imposed by the external atomic dynamics.

We then show that using USL and switching the coupling field off allows storage

of the coherent probe pulse information (amplitude and phase) in the atomic fields.

Switching the coupling beam back on writes the coherent information back onto the

Abstract iv

probe field. Experimental demonstration is presented. We present experimental data

and theoretical analysis showing how stopping light in a BEC creates an atom laser

with the highest reported phase space density flux to date. Alternatively, reviving the

probe pulse after significant BEC dynamics can be used to process the information

before it is written back onto the probe. Possible applications to quantum processing

are discussed.

We then present results on a light “roadblock”, whereby blocking part of the cou-

pling field spatially compresses probe pulses to sizes on the order of the condensate

healing length. The compressed probe creates large amplitude, short wavelength ex-

citations in the BEC which form solitons via quantum shock waves, and later multiple

vortices via the snake instability. Dynamics of multiple vortices in the BEC are ex-

plored. Further possibilities for studying superfluidity in two component condensates

are considered.

Contents

Title Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vCitations to Previously Published Work . . . . . . . . . . . . . . . . . . . ixAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

1 Overview 1

2 Dark states in three-level atoms 162.1 Semi-classical atomic evolution equations . . . . . . . . . . . . . . . . 182.2 Two-level example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3 The Dark/Absorbing basis . . . . . . . . . . . . . . . . . . . . . . . . 242.4 Coherent exchange and incoherent loss . . . . . . . . . . . . . . . . . 282.5 Dark states with small two-photon detunings and dephasing . . . . . 31

2.5.1 Two photon detuning (∆p = ∆c) . . . . . . . . . . . . . . . . 322.5.2 Dephasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.5.3 Effect on the excited level occupation . . . . . . . . . . . . . . 34

2.6 Effects of levels outside the three-level system . . . . . . . . . . . . . 342.6.1 Sodium level structure . . . . . . . . . . . . . . . . . . . . . . 352.6.2 Dephasing and AC Stark shifts due a fourth level . . . . . . . 38

3 Ultra-slow light via electromagnetically induced transparency 453.1 Classical light field propagation in three-level atoms . . . . . . . . . . 48

3.1.1 Propagation equations from Maxwell’s equations . . . . . . . . 493.1.2 Weak probe susceptibility . . . . . . . . . . . . . . . . . . . . 543.1.3 Group Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . 563.1.4 Illustration in the two-level case . . . . . . . . . . . . . . . . . 60

3.2 Ideal three-level USL . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.2.1 Coherent photon exchanges . . . . . . . . . . . . . . . . . . . 633.2.2 The self consistent three-level equations . . . . . . . . . . . . . 65

v

Contents vi

3.2.3 Weak probe limit . . . . . . . . . . . . . . . . . . . . . . . . . 653.2.4 Solution for the pulse in frequency space . . . . . . . . . . . . 663.2.5 Solution for the pulse in time . . . . . . . . . . . . . . . . . . 733.2.6 Spatial compression . . . . . . . . . . . . . . . . . . . . . . . . 753.2.7 Resulting atomic amplitude evolution . . . . . . . . . . . . . . 773.2.8 Comparison with numerical results . . . . . . . . . . . . . . . 80

3.3 Corrections to the ideal system . . . . . . . . . . . . . . . . . . . . . 823.3.1 Stronger probe regime and adiabatons . . . . . . . . . . . . . 823.3.2 Considerations in four-level systems . . . . . . . . . . . . . . . 90

3.4 Ultra-slow light experiments . . . . . . . . . . . . . . . . . . . . . . . 923.4.1 Observation of ultra-slow light . . . . . . . . . . . . . . . . . . 923.4.2 Measurement of dephasing . . . . . . . . . . . . . . . . . . . . 963.4.3 Experiments on the D1 line . . . . . . . . . . . . . . . . . . . 97

4 Ultra-slow light in Bose-Einstein condensates 1024.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.1.1 Second-quantized Hamiltonian . . . . . . . . . . . . . . . . . . 1054.1.2 Gross-Pitaevskii equations . . . . . . . . . . . . . . . . . . . . 1074.1.3 The initial ground state and the Thomas-Fermi approximation 1114.1.4 Correspondence with the previous formalism and light field

propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.1.5 Adiabatic elimination of level |3〉 and final equations . . . . . 1174.1.6 Experimental geometries and parameters . . . . . . . . . . . . 119

4.2 USL propagation in BECs with a weak probe . . . . . . . . . . . . . 1214.2.1 Equations in the weak probe limit . . . . . . . . . . . . . . . . 1224.2.2 Transverse dimension effects . . . . . . . . . . . . . . . . . . . 1274.2.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 1314.2.4 Limitations at low coupling intensities . . . . . . . . . . . . . 1334.2.5 Analogous limits for thermal clouds . . . . . . . . . . . . . . . 143

4.3 USL propagation in BECs with a strong probe . . . . . . . . . . . . . 1444.3.1 Dark state following and the dark state GP equation . . . . . 1444.3.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . 147

5 Stopping, storing and reviving light 1535.1 Fast switching of the coupling field . . . . . . . . . . . . . . . . . . . 157

5.1.1 Dark/Absorbing field basis . . . . . . . . . . . . . . . . . . . . 1595.1.2 Switching slowly compared to the natural linewidth . . . . . . 1625.1.3 Why are the ground state wave functions so robust? . . . . . . 1705.1.4 Storing an incompletely compressed pulse causes losses . . . . 1725.1.5 Switching fast compared to the natural linewidth . . . . . . . 1755.1.6 Orthogonal- and counter-propagating geometries . . . . . . . . 178

5.2 Stopped and revived light experiments . . . . . . . . . . . . . . . . . 180

Contents vii

5.3 Bypassing band-width requirements with partial switching . . . . . . 186

6 Coherent two component dynamics and pulse processing 1916.1 Using stopped light to make an atom laser . . . . . . . . . . . . . . . 193

6.1.1 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 1946.1.2 Theoretical description of output coupled |2〉 atoms . . . . . . 197

6.2 Coherent two component dynamics and pulse processing . . . . . . . 2016.2.1 General considerations for dynamics and revivals . . . . . . . 2026.2.2 Revivals with a momentum kick . . . . . . . . . . . . . . . . . 2056.2.3 Differently trapped |2〉 atoms (V2 = V1) . . . . . . . . . . . . . 2086.2.4 Equally trapped |2〉 case with U12 > U11 . . . . . . . . . . . . 2156.2.5 Dissipation: Inelastic loss processes . . . . . . . . . . . . . . . 2246.2.6 Long storage and processing: Trapped |2〉 case with U12 < U11 2256.2.7 Outlook: Applications beyond the mean field . . . . . . . . . . 229

7 Ultra-compressed light 2327.1 Motivation and basic picture . . . . . . . . . . . . . . . . . . . . . . . 2347.2 Light roadblock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

7.2.1 1D picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2417.2.2 Effects of the transverse propagation of the coupling beam . . 2457.2.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 248

7.3 More complicated spatial engineering . . . . . . . . . . . . . . . . . . 251

8 Using slow light to probe superfluid dynamics 2548.1 1D dynamics: Quantum shock waves and solitons . . . . . . . . . . . 258

8.1.1 Linearized hydrodynamics and sound waves . . . . . . . . . . 2598.1.2 Solitons via Quantum Shock waves . . . . . . . . . . . . . . . 2648.1.3 Sound and solitons in a Thomas-Fermi condensate . . . . . . . 2708.1.4 Solitons from defects created with the light roadblock . . . . . 277

8.2 2D dynamics: Vortex nucleation and dynamics . . . . . . . . . . . . . 2798.2.1 Vortices in Bose-condensed systems . . . . . . . . . . . . . . . 2798.2.2 The snake instability and vortex nucleation . . . . . . . . . . . 2848.2.3 Vortex dynamics and vortex-vortex collisions . . . . . . . . . . 2918.2.4 3D considerations . . . . . . . . . . . . . . . . . . . . . . . . . 296

8.3 Experimental observations . . . . . . . . . . . . . . . . . . . . . . . . 2988.3.1 Observation of soliton arrays . . . . . . . . . . . . . . . . . . . 2988.3.2 Observation of the snake instability, vortex nucleation, and vor-

tex dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 2998.4 Outlook: Possibilities for two component studies . . . . . . . . . . . . 304

8.4.1 One condensate moving through another . . . . . . . . . . . . 3048.4.2 Light propagation at the sound speed . . . . . . . . . . . . . . 306

Contents viii

A The Optical Bloch Equations versus amplitude equations. 307A.1 Outline of derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 308A.2 Comparison with amplitude equations . . . . . . . . . . . . . . . . . 312

B Adiabatic elimination 315B.1 The adiabatic elimination solution . . . . . . . . . . . . . . . . . . . . 315B.2 Comparison with the exact solution for time independent terms . . . 317B.3 Considerations for time-dependent terms . . . . . . . . . . . . . . . . 320B.4 Considerations for three-level systems . . . . . . . . . . . . . . . . . . 321B.5 Imaginary G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

C Matrix elements and light field couplings 323C.1 Matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323C.2 Light field couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . 324

D Numerical algorithm to solve Amplitude-Bloch Equations 326D.1 Grid representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327D.2 Atomic density matrix evolution in time . . . . . . . . . . . . . . . . 327D.3 Light propagation in space . . . . . . . . . . . . . . . . . . . . . . . . 329D.4 Self consistent procedure . . . . . . . . . . . . . . . . . . . . . . . . . 330D.5 Diagnostics and extensions . . . . . . . . . . . . . . . . . . . . . . . . 331

E Numerical algorithm to solve Maxwell-GP Equations 332E.1 Fundamental equations and grid representation . . . . . . . . . . . . 333E.2 Light propagation in space: Runge-Kutta . . . . . . . . . . . . . . . 337E.3 Atomic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

E.3.1 Split operator approach . . . . . . . . . . . . . . . . . . . . . 338E.3.2 1D: Crank-Nicolson . . . . . . . . . . . . . . . . . . . . . . . . 339E.3.3 2D: Alternating-Direction Implicit method . . . . . . . . . . . 343E.3.4 3D: Special treatment at the origin . . . . . . . . . . . . . . . 343E.3.5 Internal Dynamics: Central differencing propagation . . . . . 344

E.4 Typical numerical parameters . . . . . . . . . . . . . . . . . . . . . . 345E.5 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

Bibliography 347

Citations to Previously Published Work

Some of the experimental results on ultra-slow light near the end of Chapter 3 werepreviously published in

“Light speed reduction to 17 metres per second in an ultracold atomicgas”, L.V. Hau, S.E. Harris, Z. Dutton, and C.H. Behroozi, Nature 397,594 (1999).

The stopped light experiments in Chapter 5 were published in

“Observation of coherent optical information storage in an atomic mediumusing halted light pulses”, C. Liu, Z. Dutton, C.H. Behroozi, and L.V.Hau, Nature 490, 409 (2001).

The theoretical ideas and experimental results for the light “roadblock” in Chapter 7were published in

“Observation of quantum shock waves created with ultra-compressed slowlight pulses in a Bose-Einstein condensate”, Z. Dutton, M. Budde, C.Slowe, and L.V. Hau, Science 293, 663 (2001); Science Express, publishedonline 28 June 2001, 10.1126/science.1062527.

This paper also contains many of the results of single component superfluid studies,including soliton formation, the snake instability, and vortex formation, discussed inChapter 8.

Acknowledgments

I thought, when I arrived in Cambridge in 1996, that Harvard would be a great

and stimulating place to do my graduate studies. I also thought that Bose-Einstein

condensation, which had first been observed just in the previous year, would be an

exciting and fast-moving field in which to be involved.

My expectations were exceeded on both counts. My advisor, Lene Hau, was a lot

of the reason for this. Her willingness to go in new directions and push new limits

has made working in her group very rewarding. I will never forget the excitement

of collaborating with her on the ultra-slow light project and the ideas which have

developed from it. As my advisor, she has constantly pushed me to seek new and

interesting problems, giving me freedom to develop my own research program, while

at the same time infusing important and novel ideas which have made that program

much more exciting. Her incessant questions and attention to detail have proved to

be extremely valuable to me and her ability to understand the intricacies of both

experimental and theoretical physics have kept my research relevant to experimental

realities. I will look back fondly on her contributions to my education.

I have benefitted greatly from a close collaboration of experiment and theory in

my group. I would like to acknowledge discussions and collaboration with Chien

Liu, Cyrus Behroozi, and Brian Busch over the years, and thank them for their

great experimental work. They contributed greatly to the experiments presented in

Chapters 3 and 5. Michael Budde and Chris Slowe kept me very busy this past

year explaining their data. As payback, I kept them busy by coming up with new

theoretical ideas for them to implement in the lab, which they have done remarkably

successfully and quickly. The experiments presented in Chapters 6-8 of this thesis

Acknowledgments xi

were a result of these efforts. I have enjoyed discussions with them and their infusion

of good ideas. The same goes for the other members of the Hau group who are now

busy building the next generation of experiments, Matt Eisaman, Naomi Ginsberg,

Laurent Vernac, Lilian Childress, and Maria Kandyla.

The Rowland Institute is a truly unique environment in which to do science and I

enjoyed my four years of research (and great views of Boston!) there. On top of that,

I can not thank the Rowland Institute enough for its financial and logistical support

of so many aspects of my research. Mike Burns is a truly kind and patient man, with

a tremendously broad and healthy perspective on science, and I’ve really appreciated

his advice, his encouragement, and his ability to tell me something interesting or

useful about any subject. I also greatly appreciate all the members and staff of

Rowland for their friendship, their knowledge and their help. There are too many

of them to list here. In particular, I would like to acknowledge Garrett Eastman for

his assistance and Alan Stern for discussions and insight, particularly in regard to

numerical issues.

As I said before, Harvard has exceeded my expectations and has been a place

where I have constantly been in contact with a tremendous variety of great physicists,

including students, faculty, and the many speakers and visitors which pass through.

There are too many to mention. In particular, I thank my thesis committee members:

Lene Hau, Jene Golovchenko, Paul Martin, and Roy Glauber for their comments and

input. I particularly enjoyed discussions with Chris Greene, Dimitry Budker and

Raymond Chiao during their visits here. I also would like to acknowledge the Aspen

Center for Physics for three weeks of beautiful hospitality during the summer of

Acknowledgments xii

2001. I thank Jane Kelly and the staff there, and the organizers of the conference on

quantum gases, Gordon Baym, Murray Holland, Randy Hulet, and Jason Ho. While

there I enjoyed discussions with many people including (but certainly not limited

to) Janne Ruostekoski, Mamaet Oktel, Wolfgang von Klitzing, Mark Edwards, Brian

Kennedy, Sandy Fetter, David Feder, Gora Shlyapnikov and Erich Mueller.

I would like to thank the physics department of the University of California at

Berkeley for providing me with a great undergraduate education and an environment

which seeded my interest in pursuing physics further. In particular, Steve Kahn and

Dan Rokhsar had a big influence in developing my interests in this direction. Also

two really good classes I took from Eugene Commins were the first to express to me

the beauty of Bose-Einstein condensation, and are probably the reason I am working

on that subject today.

On a personal note, in these past five years, Somerville has transformed in my

mind from a strange suburb north of Cambridge where the graduate students live, to

a home that I love. There I have developed wonderful friendships with people from

many walks of life who have constantly reminded me of the many joys of life beyond

physics. I thank my roommates over the years for the “homey” environment in which

I live. And I thank and love Devra for making this past year and one-quarter that

much better! Finally, I couldn’t ask for a better family than my parents, John and

Jane, and my sisters, Amy and Sarah. Graduate school has occasional moments of

despair and doubt. Having a family that I knew I could count on without fail and

that were always sure to ask what and how I was doing (and a mom who could say

it all with a pop-up book) was invaluable.

Dedicated to my dad, John Dutton, the first to show me math was fun.

Chapter 1

Overview

In early 1999, our group published the first observation of ultra-slow light (USL),

which brought the group velocity of light pulses down to 17 m/s [1], more than seven

orders of magnitude slower than its vacuum value. Later we further reduced it to

50 cm/s and then carried the concept out it to its logical extreme by stopping it

completely [2]. Associated with the reduction of the group velocity is a compression

of the light pulses’ spatial length by the same factor [1, 3, 2]. Pulses which are ∼ 2 km

long outside the medium are compressed to a size ∼ 50 µm, completely localizing them

inside the medium. Our later work compressed light pulses to ∼ 2 µm [4].

The initial experiments were concerned primarily with the light propagation itself.

However, one of the fascinating possibilities in this system is to bring light speeds

down to the relevant speeds in the atomic medium, such as the recoil velocity of

the light induced momentum kicks (∼ 3 cm/s) and the sound speed (∼ 0.5 cm/s)

in a Bose-Einstein condensate (BEC) of cold alkali atoms [5]. Similarly, the spatial

lengths of the pulses have been made smaller than the length of the medium and were

1

Chapter 1: Overview 2

even brought to near the healing length of the BEC. Under these conditions, the light

manipulates the atomic medium and drives unique kinds of excitations. In particular,

propagating USL in a BEC can be used to create two component condensates with

non-trivial spatial structure, produce an atom laser, and create solitons and vortices,

allowing study of the condensate’s superfluid properties [4].

In short, this system allows study and use of light propagation in a new regime.

Before expounding upon these ideas and outlining how we present them in this thesis,

we briefly review work by others and by our group on both slow light and Bose

condensation.

A brief introduction and review

Ultra-slow light and beyond

The underlying mechanism of the ultra-slow propagation is electromagnetically

induced transparency (EIT) [6, 7] in which a coupling laser field dramatically alters

the optical properties of a medium of three-level atoms. EIT can occur in three-level

atoms with a Λ energy level structure, as diagrammed in Fig. 1.1(a). There are two

stable internal states, labelled |1〉 and |2〉. Each of these has an available dipole

transition to a common excited state |3〉 which radiatively decays. The presence of

the coupling field, resonant with the |2〉 → |3〉 transition, lowers the on-resonance

absorption cross section seen by probe on the |1〉 → |3〉 transition by orders of mag-

nitude, as was observed in 1991 at Stanford [8]. This reduction arises from coherent

population trapping [9], in which the majority of the atomic sample is driven into a


|1>|2>

|3>

couplingprobe

ba

coupling

atoms

probePMT

Figure 1.1: Schematic of an ultra-slow light experiment.(a) The basic three-level Λ energy level structure used in EIT and considered throughout this thesis. (b)A typical geometry used to observe ultra-slow light in three-level atoms.

dark state [10], a superposition of the ground states in which the quantum mechanical

amplitudes for the paths by which atoms can be excited into |3〉 cancel. When an

atom is in the dark state, no absorption into |3〉 occurs, and when it is only slightly

perturbed from the dark state, the process of absorption into |3〉 is replaced by coher-

ent driving between the ground states. Unlike absorption, this driving is completely

coherent and reversible, and prepares the atom in a phase coherent superposition of

|1〉 and |2〉.

EIT only occurs in a very narrow region of frequencies about the |1〉 → |3〉 reso-

nance. A rather small detuning (∼ kHz) will drive the phases out of the dark state

superposition, destroying the EIT. An example of a transmission profile for the ex-

perimental parameters in [1], as a function of probe frequency, is shown in Fig. 1.2(a).

The presence of the coupling field causes a narrow transparency peak near the atomic

resonance, corresponding to a sharp dip in the imaginary part of the probe suscep-

tibility. This is accompanied by a steep slope in the real part of the susceptibility,

and therefore in the index of refraction, as shown in Fig. 1.2(b). While the index of

refraction is always near unity, leading to a phase velocity nearly equal to the vac-


Figure 1.2: Transmission and index of refraction profiles in USL. Figure takenfrom [1] (a) Transmission profile of a cloud of three-level atoms for parameters in thefirst observation of USL. The high transmission associated with EIT only occurs in anarrow range of frequencies near the atomic resonance. (b) A refractive index profilefor the same parameters. The index of refraction does not significantly deviate fromunity, so the phase velocity is ≈ c, however, the steep linear slope at the resonancegives rise to ultra-slow group velocities. In the case plotted it gives a group velocityof Vg ≈ 10−7c.

uum value c = 3 × 108 m/s, the steep slope gives rise to an extremely slow group

velocity. The physical origin of the slow group velocity is coherent exchange between

the atoms’ ground state amplitudes and the associated exchange of photons between

the two light fields.

An experimental geometry we use to observe ultra-slow light is diagrammed in

Fig. 1.1(b). A coupling field resonant with the |2〉 → |3〉 transition illuminates the

sample of atoms, and a pulse of much weaker probe field resonant with the |1〉 → |3〉

transition is then injected. Group velocities of the order of Vg = c/160 were observed

in 1995 [11] with hot lead atoms. By using cold atoms, which have negligible Doppler

shifts compared to room temperature atoms, we were able to bring slow light into a

completely new regime, slowing it in the initial experiment [1] to Vg = 17 m/s ∼ 10−7c.

We have studied USL in cases where the light fields are orthogonally propagating

(as in Fig. 1.1(b)) and with the light beams co-propagating, where group velocities

Vg < 1 m/s were observed. Two other groups subsequently accomplished USL in room


temperature atoms [12]. Those experiments were restricted to using co-propagating

light beams to suppress Doppler broadening.

The EIT width, and therefore the group velocities, are directly proportional to

the coupling field intensity and thus can be easily controlled. However, there is a

lower limit determined primarily by physical effects related to the atomic dynamics

(the Doppler broadening being one such example). These effects cause increased

absorption of the probe pulse when the coupling intensity is turned down. They

effectively give the atoms a finite frequency width which limits how steep we can

make the slope seen in Fig. 1.2(b).

It is important to note that the losses from the probe pulse scale with the column

density of the atomic cloud, since these effects accumulate as the probe passes through

the large number of atoms. Motivated by this observation, we have developed two

techniques in recent years which go beyond the limits of the original USL experiments

[1], where the coupling field was kept on at a constant value.

First, varying the coupling intensity in time while the pulse is localized in the

atomic cloud dynamically changes the group velocity. Last year, by taking this process

to the extreme, we were the first to observe stopped light [2], in which switching off the

coupling beam intensity brought the group velocity to zero. This was subsequently

observed in a hot gas [13]. In the stopped light procedure, the probe pulse intensity

is driven to zero and the coherent information (both amplitude and phase) is written

into a phase coherent superposition of the ground states |1〉 and |2〉 with a spatial

shape reflecting the pulse shape. Switching the coupling beam on at an arbitrary

later time revives the probe pulse in its original form, at which point it continues to


propagate out of the cloud. The method thus provides a way of transferring coherent

information from light to atoms, storing it and then transferring it back to light.

Storage times of several milliseconds were seen in this manner. Furthermore, recent

theoretical work [14] has suggested that quantum correlations (and not just mean

fields) could be successfully transferred between light and atomic fields in this sort of

system.

Second, we have also explored spatial variation of the coupling intensity by using

an orthogonally propagating setup, which is possible with cold atoms. This led us

this year to the development of a light “roadblock” [4] in which the coupling field was

blocked in such a way that it only illuminated the front half of the cloud. The probe

light pulses, upon reaching the roadblock, were compressed from their 50 µm length

inside the medium to only a few microns, creating a single small density perturbation

in the BEC below the imaging system’s optical resolution (about 5 µm in our system),

and close to the size of the wavelength of light (589 nm).

Bose-Einstein condensation

A Bose-Einstein condensate (BEC) is a macroscopically large number of identical

bosons occupying a single quantum state. Bose-Einstein condensation occurs when

a system of bosons is cooled below a critical temperature and is nature’s most ex-

treme demonstration of the consequences of Bose statistics. The phenomenon was

first predicted, naturally, by Bose and Einstein in 1924 [15]. Liquid helium provided

an experimental system demonstrating these effects [16]. However, direct compar-

isons with the theory describing these Bose condensed systems at the time [17, 18]


was clouded by the strong atom-atom interactions. Rather sophisticated theoretical

analysis and numerical techniques often had to be employed for quantitative compar-

ison [19, 20]. Furthermore, there were only limited experimental tools available to

control the temperature and densities in this system.

Recent advances in atomic trapping and manipulation techniques [21, 22, 23, 24,

25] culminated in 1995 in the production of Bose-Einstein condensed (BEC) sam-

ples of alkali vapors [26, 5]. Typically, condensation occurs in these trapped atomic

samples containing ∼ 104 − 106 atoms at a little less than one-millionth of a degree

above absolute zero. In contrast with the liquid helium case, these systems allow a

high degree of control over the properties such as density and temperature, as well

as accurate theoretical modelling [27]. Furthermore, later experiments have demon-

strated that once the atoms condense into a single mode, the BEC can be put into a

variety of excited states and maintain its coherence [28]. Completely new possibilities

still continue to arise in these systems with the creation of multiple component BECs

[1, 29, 30, 2, 4] and BECs in microtraps [31].

Here we emphasize two distinct subjects of interest in these new systems. First,

their extremely low temperatures and bosonic nature have provided a way of produc-

ing coherent matter fields, analogous to laser fields in light, opening up the field of

atom optics [32]. For example, BECs have been used to make atom lasers (a coherent

moving sample of atoms) [33, 34], to observe four-wave atomic field mixing [35], and to

create quantum states such as number squeezed states [36]. Recent theoretical work

has addressed the possibility of using two component BECs to create spin squeezed

states [37] or even as elements in quantum computation schemes [38].


We created Bose condensates of Sodium atoms in 1997. We have since used our

BECs to create USL, and used USL to study novel properties of condensates. The

stopped light technique creates two component BECs with novel spatial structure.

We applied this technique to produce an atom laser with the highest phase space

density flux reported to date. Theoretically, we have also seen that it can lead to

interesting two component dynamics, such as quantum reflections of matter waves

off of a hard boundary. Our investigation of stopped and revived probe pulses has

also begun to lead to ideas for using BECs for quantum processing. By storing the

pulse for a sufficiently long time, the coherent evolution of the BEC can process the

information which was input on the pulse, and upon revival this evolution is written

onto the output probe pulse.

Secondly, Bose-condensed samples exhibit superfluidity, or dissipationless flow [16,

20]. Because they are the first superfluid systems in the weakly interacting regime,

it has been possible to accurately model the many-particle quantum state with the

Bogoliubov formalism [17, 5] and to use the mean field Gross-Pitaevskii equation

[18, 27] to calculate the dynamics. This has led to quantitative comparison between

theory and experiment. Originally, work was done studying macroscopic collective

excitations [40]. More recently work has focused on localized excitations related to

superfluidity and other issues related to quantum fluids. Experiments have observed

quantized vortices [41, 42, 43, 4, 44], solitons [45, 4], and a superfluid critical speed [46,

51]. The mechanisms by which quantized vortices nucleate, interact, and annihilate

continues to be a subject of interest. It has been conjectured that these processes play

an important role in the transition from short to long range order at the condensation


critical temperature [47, 48]. Also, interactions they undergo with sound waves are a

mechanism by which dissipation is introduced into Bose-condensed systems [49, 50].

We have seen that our light roadblock, by creating controlled density perturbations

in the BECs with a spatial width of only a few microns, can be used as a completely

new tool to study these issues. The size of the density defects is on the order of the

healing length, which is typically ∼ 0.4 µm in our experiments. This is the spatial scale

governing topological excitations such as solitons and vortices [16, 20]. By creating

these narrow defects, we observed quantum shock waves [4] whereby the nonlinearity

due to atom-atom interactions caused the density defects to shed off solitons. These

solitons then decay via the snake instability [52, 53, 54, 55, 43] into vortices. Our

experiment was the first to directly observe the evolution of the snake instability and

the subsequent nucleation of vortices in BECs.

The experiment was a first step in a continuing program of using ultra-compressed

light to study issues associated with superfluidity. We plan to look into the relatively

unexplored questions of both soliton and vortex nucleation and dynamics as well

as three dimensional vortex structure. Because ultra-compressed light creates large

numbers of vortices out of equilibrium, it also promises to allow observation and study

of vortex-vortex collisions and vortex-sound wave interactions, processes which have

not been observed in atomic BECs to date.

Outline of this thesis

This thesis presents a combination of theoretical and experimental work, with

the emphasis on theory. In Chapters 2-4 we develop a formalism which describes


ultra-slow light in Bose-Einstein condensates quite generally. It is first applied to the

initial ultra-slow light experiments, and then used to study and describe the recent

extensions discussed above. Lastly, we see how these extensions can and have been

used in continuing studies of Bose condensates. Figure 1.3 shows a logical flowchart,

which is expounded upon in the following.

USL and development of the formalism

In Chapter 2 we consider the evolution of a single atom with a Λ energy level

configuration (Fig. 1.1(a)) in the presence of the probe and coupling light fields. We

work in a semi-classical description, where the atoms are described quantum mechan-

ically while the light fields are treated as classical fields, and derive how the atomic

amplitudes evolve. This will allow us to see how, for given laser field amplitudes

Ωp and Ωc, the Λ atoms can be driven into a dark state which does not interact

with the light fields. We also derive a condition (the EIT adiabatic condition) under

which time-dependent Ωp,Ωc continuously drive the atoms into the time-dependent

dark state superposition. In this way, we see how light fields can be used to drive

atoms into coherent, uniformly phased superpositions of the two ground states |1〉

and |2〉. This chapter also introduces the concept of the dark/absorbing basis which

often is useful in our analysis. At the end of the chapter we present some novel results

extending the analysis to include the presence of a weakly coupled fourth level and

discuss the application of these results to real atoms, where many internal levels are

present.

In Chapter 3 we begin to treat the light fields Ωp,Ωc as dynamical variables by


Figure 1.3: Logical flowchart of the thesis. Chapters 2-4 are concerned withadding the three major pieces to our formalism, listed on the left hand side of eachof the top three boxes. In each of these chapters, we will apply the formalism toaspects of the USL experiments, as listed on the right hand sides of these boxes. InChapters 5 and 7 we apply the formalism to experiments where the coupling field isvaried in time and space and again present experimental results. In Chapter 6 weuse the stopped light technique to investigate BECs as coherent matter waves, andin Chapter 8 we look into ways of using the light roadblock to probe their superfluidproperties.


deriving equations for their amplitude propagation based on the Maxwell’s equations.

We now also consider many-atom clouds. We solve analytically the set of equations

governing the coupled system of the atomic internal state amplitudes and laser light

fields. We also present and use a novel numerical code, described in Appendix D,

which we developed to analyze this problem. Applying the analytic and numerical

techniques to the ultra-slow light experiments, we see how the remarkable transmis-

sion and index of refraction profiles (Fig 1.2) arise, how they lead to ultra-slow light

propagation and pulse compression, and also derive the leading order losses. Most

of the analytic results in the chapter rely on the assumption of the weak probe limit

(Ωp Ωc). We then go beyond this approximation by considering the nonlinear

corrections when Ωp ∼ Ωc, a regime which is important in many latter parts of the

thesis where we focus on using USL to couple to excitations in BECs. We also apply

our results for the effect of a fourth level to see how it affects light field propagation.

We conclude the chapter with a presentation of the first USL experiments, published

in [1], and later extensions of it.

Up to this point, the atoms had been assumed to be stationary and in the Chap-

ter 4 we add the effects of atomic motion. We develop a formal description of the

BEC dynamics based on a second-quantized Hamiltonian, which leads us to a set of

coupled Gross-Pitaevskii (GP) equations governing the evolution of the atomic mean

fields. This formalism describes the atomic internal evolution, external evolution and

light field propagation all self-consistently. This is the basis of another novel numer-

ical code, outlined in Appendix E, which can account for dynamics in one, two, and

under certain conditions, three, dimensions. It can used to analyze a wide variety of


experiments and is used extensively through the remainder of the thesis. We again

present experiments which corroborate some of the analytic and numerical results

and also see how, under certain conditions, the more complete description presented

here reduces to the results already seen in Chapter 3, where the atomic motion was

neglected. We then use the formalism to derive some of the limitations on group

velocities imposed by the atomic external dynamics as well as some of interesting

consequences this motion can have.

Variations on USL: Stopped, revived, and ultra-compressed

light

With our formalism completely in place, we next use it to consider more com-

plicated situations. In Chapter 5 we address the issue of dynamically changing the

coupling field intensity and find this dynamically changes the probe’s group velocity.

Taken to the logical extreme of completely switching the coupling field off leads to

stopped probe pulses. Remarkably, they can then be revived at much later times by

switching the coupling field back on, and the probe pulse then continues to propagate

out. The atomic coherence between the ground states, which is created during the

probe propagation, is found to be extremely robust when the coupling field changes

in time. Experimental demonstration of stopped and revived light pulses, published

in [2], is presented and compared with the theory. Also we consider how extensions

of switching technique can be used to bring group velocities to non-zero speeds much

lower than would be possible with conventional USL. This technique could be used

in our continuing superfluid studies.


Before discussing Chapter 6, we mention that in Chapter 7 we introduce the light

roadblock, whereby probe pulses can be compressed to sizes far smaller than possible

with conventional USL. The theoretical ideas, with supporting numerical simulations,

are presented and the fundamental limits of the technique are discussed. We present

experiments, published in [4], which demonstrate creation of sub-optical resolution

defects in a BEC. We also consider how more complicated spatial engineering of the

coupling field could be used to create novel two component localized excitations in

BECs.

Applications to studies of BECs

In Chapter 6 we turn to the question of storage times long enough for significant

atomic dynamics to occur. In particular we present experimental results of the appli-

cation of stopped light to the creation of an atom laser and a corroborating theoretical

analysis. We also address the question of using long storage times in BECs for coher-

ent processing the information of the probe pulse . Depending on parameters such as

the atomic magnetic moments and the scattering lengths of atom-atom interactions,

a wide variety of atomic dynamics can ensue during the storage time. When the time

scale of these dynamics is comparable to the storage time, non-trivial processing of

the revived probe pulse occurs. At the conclusion of the chapter, possible applications

to quantum processing and studies of BECs beyond mean field theory are considered.

In Chapter 8 we then consider the interesting question of using the techniques

discussed in Chapter 7 to study superfluidity in BECs. First we analyze the one com-

ponent dynamics, and see how the highly localized, sub-resolution density defects


created by the light roadblock decay into the topological defects known as solitons

[56]. The soliton formation occurs via quantum shock waves [4] which are the result

of the nonlinearity in density wave propagation. These solitons then break up into

vortices via the snake instability [52, 53, 55, 54, 43]. This idea is first discussed theo-

retically and our corroborating experimental results are then presented. In particular,

we present the first direct observation of the nucleation of vortices via the snake in-

stability. We obtain detailed images not possible with normal absorption imaging by

pumping a selected slice of the BEC for imaging. We also discuss the prospect of

observing the 3D topology of the vortices with this method.

We also consider further possible extensions of the technique to study vortex dy-

namics and interactions. We show how the light roadblock technique is unique in that

it nucleates multiple vortices circulating in both directions, effectively make a “gas” of

interacting particle-like objects [47]. Our numerical simulations indicate that vortex-

vortex scattering, annihilation events, and sound-wave vortex interactions should all

occur. These phenomena have not yet been observed in atomic BECs and a direct

observation could address some of the issues discussed in [49, 50, 16], which consider

the role vortices play in introducing friction into superfluid systems. We conclude the

chapter by considering similar experiments with two component condensates which

should be possible with extensions of the stopped light and roadblock techniques.

Chapter 2

Dark states in three-level atoms

In this chapter, we introduce the basic idea of dark states [6, 10] in three-level

atoms in a Λ configuration, as shown in Fig. 2.1. We will see how, when atoms

are in a particular superposition of the two ground states |1〉 and |2〉 known as the

“dark state”, a quantum mechanical interference between two processes suppresses

the absorption of photons into the excited state |3〉, thereby allowing atoms to remain

in a steady state which does not interact with the light fields. Furthermore, we will

see how this dark state superposition can be changed adiabatically, allowing atoms

to be coherently driven between the two ground states.

The existence of the dark state is the key ingredient for electromagnetically induced

transparency (EIT) [6, 7, 10] whereby an otherwise opaque medium of atoms can allow

the transmission of a probe pulse. Associated with the high transmission of EIT, is

ultra-slow light (USL) [1, 58, 59, 11], a reduction in the group velocity of the probe

pulse by many orders of magnitude. We discuss these effects in Chapter 3, when we

treat the light fields as dynamical variables and investigate how they are affected by

16

Chapter 2: Dark states in three-level atoms 17

p

c

p

p c

|1>|2>

|3>

Figure 2.1: Λ system energy level diagram. The basic Λ atomic energy levelstructure we will consider consists of two ground states, |1〉 and |2〉. Each is coupledwith near resonant lasers to a common excited level |3〉. The strength of the couplingsare given by the laser Rabi frequencies Ωp and Ωc respectively (these are defined laterin the text). The detunings are allowed to be non-zero but are generally much smallerthan the excited state decay rate, Γ. Atoms which spontaneously decay from |3〉 areassumed to exit the levels under consideration.

the atomic medium. In the present chapter, we simply take the light fields as given

quantities, and calculate the resulting evolution of the atomic internal states. For the

time being, we neglect the effects due to external potentials, atom-atom interactions,

and recoil momentum, effectively treating the atoms as stationary.

In Section 2.1, we begin with an idealized three-level atom in the presence of

two light fields. We derive the semi-classical equations of motion for the amplitudes

of the internal states within the electric dipole approximation and rotating wave

approximations. We briefly solve the equations for the simpler two-level case to

ease comparison with the three-level case later on. In Sections 2.3 and 2.4 we then

investigate the existence of the dark state and introduce a transformation of the

atomic system into the dark/absorbing basis. This allows us to see the adiabatic

requirements to keep atoms in the dark state in the case of time-dependent fields. In

Section 2.5 we then discuss the effects of imperfections in the system, such as small

losses from the ground states, and small uncontrollable frequency shifts. Because EIT


relies on a sensitive interference of two amplitudes, small effects can induce absorption

into |3〉. In Section 2.6 we specifically discuss the implementation of EIT in sodium

atoms and discuss the effect of levels outside the ideal three-level system.

2.1 Semi-classical atomic evolution equations

We now derive equations of motion for the internal atomic states in a semi-classical

formalism, so called because the atomic internal states are treated quantum mechan-

ically, while the light fields are classical. Assume we have a single atom at the origin

(spatial dependence and the effect of a cloud of many atoms will be added in Chap-

ter 3) with three internal states (corresponding to electronic states of the valence

electron in an alkali atom). Two are the stable ground states, labelled |1〉 and |2〉,

and |3〉 is an excited level decaying at a rate Γ ( = (2π)10 MHz in the case of sodium).

This is known as a Λ configuration (Fig. 2.1). We describe the amplitude of each in-

ternal state with a complex number, c0i (t), i ∈ 1, 2, 3. Initially (t = −∞), the atom

is in |1〉, so c01(−∞) = 1, c02(−∞) = c03(−∞) = 0.

The internal states are coupled to each other via two light fields : a probe beam,

near resonant with the |1〉 → |3〉 transition, and a coupling beam, near resonant with

the |2〉 → |3〉 transition (Fig. 2.1). They are written, respectively, as:

Ep(R, t) =1

2εpEp0(R, t)(e

ı(kpz−ωpt) + c.c.),

Ec(R, t) =1

2εcEc0(R, t)(e

ı(kcw−ωct) + c.c.), (2.1)

where c.c. denotes the complex conjugate, and εp,c are unit vectors denoting the beam


polarizations (We assume here that the polarization vectors are constant in space

and time, effectively neglecting refraction effects. This is justified and discussed in

Subsection 3.1.1.). The probe is assumed to propagate in the z direction, while the

coupling beam can propagate along w = z,−z or x. The near resonant condition here

implies that the laser detunings, ∆p,c ≡ ωp,c − (ω3 − ω1,2) Γ.

The light field-atom coupling is calculated within the electric dipole approxima-

tion, which assumes that the spatial extent of the electronic states (typically ∼ A)

are much smaller than the wavelength of the light fields (λ ≡ (2π)/kp,c ≈ 589 nm

for resonant fields in sodium. In practice |1〉 and |2〉 are split by hyperfine frequen-

cies, which are small compared to ωp,c so λp − λc ∼ 10−6λp). A careful treatment of

the various electric and magnetic contributions reveals that the electric dipole term

dominates the quadrupole electric and magnetic dipole terms by a factor of the fine

structure constant [60, 7], allowing us to keep only the electric dipole term. The single

atom Hamiltonian then becomes:

H = H(0) + H(dip);

where H(0) ≡∑

i=1,2,3

ωi|i〉〈i|,

H(dip) ≡ er · E, (2.2)

r is the electron position operator, −e is the electron’s charge, and the total electric

field is E = Ep + Ec. Because the light fields are varying slowly compared with

the electronic state spatial sizes, we can pull the electric fields out of the integrals

when we are calculating matrix elements for transitions between internal states. This

motivates the introduction of the Rabi frequencies, Ωijp,c ≡ eEp0,c0εp,c · rji/, where


rij ≡ 〈i|r|j〉 (note rij = r∗ji). In the definitions of the Rabi frequencies, we have not

included the quickly varying space and time phase dependence of the electric field.

Next we transform to the interaction picture [60], whereby we evolve H(dip) according

to the Heisenberg equations as determined by H(0). In the interaction picture the

Hamiltonian becomes

H(dip) = exp

(+i

H(0)t

)H(dip)exp

(− i

H(0)t

)

=1

2

∑i,j=1,2,3

(Ωji

p ei(kpz−ωpt) + Ωji

p e−i(kpz−ωpt)

+ Ωjic e

i(kcw−ωct) + Ωjic e

−i(kcw−ωct)) |i〉〈j|ei(ωi−ωj)t. (2.3)

We further make the rotating wave approximation (RWA). This approximation

ignores terms in the Hamiltonian which are varying in phase rapidly compared to other

frequencies in the problem, as these terms effectively time average to zero. Physically,

the RWA eliminates terms which violate energy conservation. The cases which we

will study involve Rabi frequencies on the order of (2π)1 − 10 MHz and Γ = (2π)10

MHz. In (2.3) we see that the electric fields contribute terms which rotate in phase

at the optical frequencies, ±ωp,c ∼ ±1015 Γ,Ωijp,c These terms are then multiplied

by the matrix elements, which rotate at the atomic energy splittings ωi − ωj. When

we are near resonance (∆p,c Γ,Ωijp,c), the RWA allows us to ignore terms involving

the product of an electric field and a dipole matrix element rotating with the same

sign. They are dominated by the terms where the two have opposite sign and the two

phase rotations nearly cancel. We also generally have a large hyperfine splitting (∼

GHz) between the ground states |1〉 and |2〉. With this assumption, matrix elements

involving the probe and the |2〉 ↔ |3〉 transition (as well as the coupling and |1〉 ↔ |3〉)


involve quickly varying phases, and we use the RWA to eliminate these terms as well.

Generally, we keep terms which are rotating up to frequencies two orders of magnitude

more than Γ,Ωji,p,c, and throw out terms which rotate faster. This means that we

allow intermediate states which violate energy conservation by up to two orders of

magnitude more than our transition matrix elements. When this is done, the only

Rabi frequencies which are retained are Ωp ≡ Ω13p ,Ω

∗p ≡ Ω31

p ,Ωc ≡ Ω23c ,Ω

∗c ≡ Ω32

c .

We now write down the Schrodinger equation in the interaction picture for the in-

ternal states, i ˙c0i =∑3

j=1 H(dip)ij c0j , where c0i = exp(iωit)c

0i are the interaction picture

amplitudes. We make a further transformation to eliminate the electric field rotation

frequencies, c1 ≡ c01, c3 ≡ c03ei(∆p)t, c2 ≡ c02ei(∆p−∆c)t to obtain our final equations of

motion:

c1

c2

c3

=

0 0 −iΩ∗p

2

0 i(∆p − ∆c) −iΩ∗c

2

−iΩp

2−iΩc

2i∆p − Γ

2

c1

c2

c3

. (2.4)

Here we have included a loss term Γ/2 from state |3〉, representing spontaneous emis-

sion. It is added phenomenologically here, though it can be derived quantum me-

chanically when one considers the joint density matrix of the atom and all modes

of the vacuum radiation field. In Appendix A, we discuss this and show how the

non-Hermiticity of the Hamiltonian which gives (2.4) arises because one traces over

the degrees of freedom of the radiation field. When one considers the full quantum

treatment, the spontaneous emission rate is found to be [60, 7]:


Γ =∑j

ω331,32

c3e2|r3j|23πε0

, (2.5)

where ε0 is the permittivity of free space, c is the speed of light in vacuum and the sum

j runs over all ground states to which |3〉 can decay (including |1〉, |2〉 and ground

states outside the Λ system under consideration). Note that ω31 ≈ ω32 (they are

different only by the hyperfine splitting, which is one part in 106).

In our description, an atom which spontaneously emits a photon is assumed to

decay to other levels and is therefore lost (i.e. an open system). In any atom used in

practice, some fraction returns to ground states |1〉 and |2〉. However, because EIT

suppresses spontaneous emission, modelling the system as open does not introduce

serious errors. Rather, as we will see, the dominant processes are coherent, reversible

transitions between the ground states. The quantity which determines the importance

of the incoherent processes is given by the fraction of the population undergoing

spontaneous emission integrated over time:

Ploss = Γ

∫ ∞

−∞dt |c3(t)|2, (2.6)

which we will show later (Chapter 3) is small in USL propagation. This is in contrast

to many quantum optics problems, notably a simple two-level atom, in which the

excited state spontaneously decaying into the original ground state is an essential

ingredient in the calculating the cross section. In such cases, the Optical Bloch

Equations (OBEs) are used. In that formalism, the expectation values of the full

atomic density matrix are calculated (rather than just the amplitudes ci), allowing

for inclusion of incoherent population transfers within the system. In Appendix A we


compare the two descriptions in detail and justify our use of amplitude equations for

the purposes of this thesis.

2.2 Two-level example

Before discussing the dark state, we note that in the absence of the coupling field

(Ωc = 0, c2 = 0), (2.4) reduces to a description of a two-level atom, pumped by a

single laser, Ωp. The solution in this case is obtained by adiabatic elimination of c3

[61], an approximation which we use extensively in this thesis, and is justified and

discussed in Appendix B. Generally it is applicable when there is a large damping

rate, in the present case Γ/2, driving one of the dynamical quantities into a quasi-

steady state. This damping rate must dominate the coupling between states, in the

present case the Rabi frequency Ωp. Adiabatic elimination involves eliminating one

equation by setting c3 = 0, giving:

c3 = −i Ωp

Γ − 2i∆p

c1. (2.7)

Plugging this result back into the remaining equation, one obtains the rate of popu-

lation exchange between |1〉 and |3〉:

P1 =d

dt|c1|2 = −RP1 +RP3,

P3 =d

dt|c3|2 = +RP1 − (Γ +R)P3;

where R ≡ |Ωp|2Γ2 + 4∆2

p

Γ, (2.8)


and Pi ≡ |ci|2 represents the probability density in |i〉. In the limit R Γ, we

have the quasi-steady state solution P3 ≈ (R/Γ)P1, and atoms, through spontaneous

emission, are pumped out the system at a rate R. If one keeps the light field on for

some time τ0 then Eq. (2.6) gives Ploss = Rτ0. As we will now see, the presence of

the coupling beam significantly reduces this loss.

2.3 The Dark/Absorbing basis

A remarkable feature of the three-level Λ system is the existence of a dark state

when we have a non-zero coupling field Ωc, a feature which we exploit in EIT and

USL. Consider the case where the laser beams are exactly on two-photon resonance

(∆p = ∆c ≡ ∆). Then one easily verifies a steady state occurs when:

c2 = −Ωp

Ωc

c1, c3 = 0. (2.9)

This is an eigenstate of the system with eigenenergy zero (choosing ω1 as the zero

energy reference). Because it contains no amplitude in the excited state, it does not

spontaneously decay. More significantly, one sees the two terms representing laser

driven excitations to the excited level (in the third equation of (2.4)) exactly cancel.

The essence of EIT and USL is to keep the vast majority of the atomic sample in this

state.

This observation motivates a transformation into a dark/absorbing basis [10],

which is illustrated in Fig. 2.2:


|D>|A>

|3>

NA

Figure 2.2: The Dark/Absorbing basis. The Λ system in the basis defined bytransformation M (Eq.(2.10)). The Dark state |D〉 is decoupled from the excitedlevel, while the absorbing state |A〉 is strongly coupled. Time dependence causes anon-adiabatic coupling ΩNA between |D〉 and |A〉.

cD

cA

c3

= M

c1

c2

c3

, M =

Ωc

Ω−Ωp

Ω0

Ω∗p

ΩΩ∗

c

Ω0

0 0 1

, (2.10)

where Ω ≡√|Ωp|2 + |Ωc|2. While |D〉 = (Ωc/Ω)|1〉−(Ωp/Ω)|2〉 is an eigenstate, |A〉 =

(Ωp/Ω)|1〉 + (Ωc/Ω)|2〉 and |3〉 are not. The other two eigenstates are superpositions

of |A〉 and |3〉, analogous to the two dressed states in a two-level atom, and depend

on Ω, ∆, and Γ. Because they contain a non-zero amplitude c3 they decay in time

and therefore have complex eigenfrequencies, (−iΓ ±√4Ω2 − Γ2)/4.

For the present section and Sections 2.4-2.5, let us assume the Rabi frequencies

Ωp,c are real and that we are on two-photon resonance (∆p = ∆c ≡ ∆). Then we apply

our transformation (2.10) to (2.4), allowing for the possibility of time-dependent light

fields, and obtain:


cD

cA

c3

=

0 −ΩNA

20

ΩNA

20 −iΩ

2

0 −iΩ2i∆ − Γ

2

cD

cA

c3

, (2.11)

where ΩNA ≡ 2(ΩpΩc − ΩpΩc)/Ω2 [62]. This term determines the rate of exchanges

between the dark and absorbing states due to non-adiabaticity (see Fig. 2.2).

First consider the case with non-time-dependent fields (ΩNA = 0). If we were

to start with a sample of atoms in an arbitrary superposition of |1〉 and |2〉 in the

presence of both light fields, the component in the dark state would not interact

with the light fields, while the absorbing component is coupled to the excited state

(with an effective Rabi frequency Ω) and is therefore damped in time. The equations

governing cA and c3 are identical to an open two-level atom system. Here adiabatic

elimination of c3 in (2.11) yields:

c3 = −i Ω

Γ − 2i∆cA. (2.12)

We see the cA, c3 system is analogous to the two-level system discussed above (see

Eq. 2.7). In near resonant case ∆ Γ the atomic population in |A〉(|cA|2) is pumped

into the excited state at a rateW ≡ Ω2/Γ and then lost through spontaneous emission

until cA = 0.

When we consider time-dependent Ωp,c Eq. (2.9) shows the dark eigenstate changes

with time. The degree to which the change is not adiabatic will introduce tran-

sitions between the dark and absorbing state at a rate ΩNA. For examples, if

one keeps Ωc constant and changes Ωp with a characteristic time scale, τ0, then

ΩNA ∼ 2(ΩpΩc/Ω2)(τ0)

−1. Since the absorbing state is constantly being depleted at


a rate W , the majority of the atoms will remain in the dark state if W ΩNA. In

this limit, the cA is strongly damped and can be adiabatically eliminated in a manner

similar to c3 (see Appendix B). Plugging (2.12) into the second equation of (2.11)

(again assuming ∆ Γ), then adiabatically eliminating cA (i.e. setting cA = 0)

gives:

cA ≈ ΩNA

WcD ∼ 2

ΩpΩc

Ω2

1

τ0WcD. (2.13)

Combined with (2.12) this provides a good estimate of the excited level amplitude

for time-dependent fields :

c3 ∼ −2iΩpΩc

Ω3τ0cD, (2.14)

showing that for a given τ0, using stronger fields stimulates occupation of the dark

state and actually reduces losses. The total loss rate can be calculated by plugging

(2.14) directly into (2.6). An alternative but equivalent formulation is to plug (2.13)

into the first equation of (2.11) to find a loss rate of the dark state amplitude:

˙cD = −Ω2NA

2WcD ∼ −2

Ω2pΩ

2c

Ω6τ 20

ΓcD, (2.15)

which can be kept small by using sufficiently large τ0 or Ω. When the loss rate (or

equivalently the amplitude c3) is significantly reduced from its value in the case with

no coupling field (Ωc = 0), we say we are under conditions of good EIT.


2.4 Coherent exchange and incoherent loss

Since one can change the dark state |D〉 by simply changing the light field inten-

sities, one can drive atoms between the ground state |1〉 and |2〉. Importantly, this

driving is coherent, in the sense that the phase relationship between c1 and c2 is well

defined and preserved in the process. By contrast, atoms which are stimulated into

|3〉 and then spontaneously emit photons lose coherence between their ground states.

The scaling of c3 with τ0 is important for the purpose of coherent transfer. From (2.6)

and (2.14) (or (2.15) directly) one sees that the total loss rate from the system will

be ∝ τ−20 . The total loss is then obtained by integrating over all time (∝ τ0) so the

total integrated loss is ∝ τ−10 . One can thus arbitrarily reduce the total loss during

the transfer by increasing the time scale τ0.

This can be seen directly by calculating the rates of loss versus the rate of coherent

transfer. Considering the case of exact two-photon resonance (∆p = ∆c) and near

one-photon resonance (∆p,c Γ), we simplify (2.4) by adiabatically eliminating c3.

Doing this and plugging in the dark/absorbing transformation M from (2.10) gives,

after some algebra:

P1 = −RA1 −RS,

P2 = −RA2 +RS;

where RS ≡ ΩpΩc

ΓcAcD, RA1 ≡

Ω2p

Γc2A, RA2 ≡ Ω2

c

Γc2A. (2.16)

This clearly distinguishes the coherent, reversible processes from the incoherent, lossy

ones. The total loss rate of the system RA ≡ RA1 + RA2 is the rate at which atoms


are absorb and spontaneously emit photons, while RS indicates the rate of stimulated

coherent exchange between the two ground states. The sign of RA is either positive

and negative depending on which direction the light fields must drive the atoms to

achieve the dark state (2.9).

When there is no time dependence in the light fields, cA = 0 and we have a

steady state. Introducing a slow change in one of the fields puts a small amplitude

in cA cD ≈ 1 (see Eq. (2.13)), which causes coherent exchange at a rate RA and a

much smaller amount of loss at a rate RA. The ratio between the rates of loss and

stimulated exchange is:

RA

RS

=cAcD

Ω2

ΩcΩp

∼ 2

Wτ0, (2.17)

where the last estimate uses (2.13) for cA. We then see that keeping the ratio small

requires:

τ0 W−1. (2.18)

This is known as the adiabatic condition for EIT. When it is violated, absorption

events become as common as stimulated exchange (RA ∼ RS).

In Fig. 2.3, we present verification of this with numerical integration of (2.4).

In each case, we start with all atoms in state |1〉, and the coupling field on at some

constant value, Ωc0. The probe beam, originally off, is then ramped up to a value equal

to the coupling field and back down, with a time constant τ0. Here (and throughout

this thesis) we will assume a time-dependent input probe of the form:


Figure 2.3: Coherent transfer between |1〉 and |2〉. Plots present the resultsof numerical integration of (2.4) in the presence of a c.w. coupling field of Ωc0 =(2π)2 MHz and a probe pulse of the form (2.19) with Ωp0 = (2π)2 MHz. ThusW−1 =Γ/Ω2 varies between 0.4µs when the probe field is off to 0.1µs when it is at its peakvalue. (a) Atomic amplitudes c1(t) (dashed curve) and c2(t) (solid curve) when τ0 =15 µs. (b) The amplitude cA(t). The analytic expression (2.13) is indistinguishablefrom the numerical result. (c,d) The same calculation for a probe pulse width τ0 =1.5 µs. In (d) the analytic result (dashed curve) is seen to deviate slightly from thenumerical result (solid curve). (e) Data points indicate the total fractional loss ofatoms (2.6) during the process for a series of calculations at different τ0. As expected,the integrated loss scales approximately linearly with τ−1

0 when τ−10 W . The arrows

indicate the two cases plotted in (a-d).

Ωp(in)(t) = Ωp0e

−(

t2

2τ20

), (2.19)

so τ0 is the (1/e) half width of the intensity and t = 0 denotes the time the peak probe

intensity is input. The atom is driven into the changing dark state superposition of

|1〉 and |2〉 (2.9) at all times, as shown in Fig. 2.3(a), and a small amplitude is in the

absorbing state (Fig. 2.3(b)). When the probe has reached its peak value (t = 0),

Ωp = Ωc0 and half of the probability is in |2〉. The population is then transferred back

purely into |1〉 as t→ ∞. In Fig. 2.3(c-d) we plot the results of a calculation with a

smaller τ0, and one sees a larger cA. The analytic estimate (2.13) is seen to be quite


accurate in both cases. The total loss of atoms integrated over all time is plotted for

various τ0 in Fig. 2.3(e), and we see it is indeed ∝ τ−10 for sufficiently long τ0. For

the shortest time scales plotted RA ∼ RS so the loss (∼ 30%) becomes comparable to

the number of atoms transferred back and forth (50%). In this regime the expression

(2.13) begins to lose validity and the losses no longer scale linearly with τ−10 .

The near resonant condition (∆ Γ) which we assumed in deriving Eqs. (2.16)

distinguishes EIT from the process of off-resonant Raman transition. In a Raman

transition, the two-photon resonance is still satisfied (∆p = ∆c ≡ ∆), but the beams

are far detuned from the excited level (∆ Γ). Rather than driving atoms into

the dark state, the two photon process cycles atoms between the two ground states

with an effective Rabi frequency Ω ≡ ΩpΩc/2∆. In this case absorption into |3〉 is

suppressed due to the large detuning, not the quantum interference of two terms. By

contrast, in EIT (∆ Γ) we get strong damping of cA, and all atoms are kept in the

dark state superposition. One photon detunings on the order of a natural linewidth

(∆ ∼ Γ) correspond to a crossover between the two regimes.

2.5 Dark states with small two-photon detunings

and dephasing

Thus far we have assumed a perfect three-level system without ground state de-

phasing or loss, and in perfect two-photon resonance ∆p = ∆c. In steady state, we

can then achieve a perfect dark state, and for adiabatically slowly varying light fields,

we can change the dark state superposition in time, and drive atoms between the two


ground states without introducing significant absorption.

However, the two-photon resonance condition is not exactly satisfied in general,

since atomic motion causes Doppler shifts, the external trapping potentials cause

Zeeman shifts, and interactions cause mean field energy shifts. Similarly, the presence

of levels outside the Λ system and atomic collisions introduce a finite loss rate for the

ground states.

As we now discuss, dephasing of the ground states (i.e. damping of c1,2) and

two-photon detuning introduces a finite lifetime of the dark state and thus causes

some non-zero absorption into |3〉 even in the steady state. An alternative viewpoint

is that they introduce a small imaginary part into the eigenenergy of the dark state.

2.5.1 Two photon detuning (∆p = ∆c)

Let us define a small two-photon detuning, ∆2 ≡ ∆p − ∆c, but assume now that

the light fields are time independent (ΩNA = 0). Recalculating (2.11) in this case

gives:

cD

cA

c3

=

−iΩp2

Ω2 ∆2 iΩpΩc

Ω2 ∆2 0

−iΩpΩc

Ω2 ∆2 iΩc2

Ω2 ∆2 −iΩ2

0 −iΩ2

−i∆ − Γ2

cD

cA

c3

. (2.20)

Then by the same adiabatic elimination procedure we applied in Section 2.3 we get:

cA = −2iΩpΩc

Ω2

∆2

WcD. (2.21)

Keeping cA cD gives a criterion on ∆2 to keep the atom primarily in the dark


state, which is analogous to the adiabatic condition (2.18). The similarity of the two

criteria is reasonable since the non-adiabatic loss due to fast time scales in the probe

pulse could also just be thought of as giving it a small width of frequency components

(with the width of the distribution of ∆2 being comparable to τ−10 ).

2.5.2 Dephasing

We now introduce a finite loss rate, γ into the ground state |2〉, so (2.4) becomes:

c1

c2

c3

=

0 0 −iΩ∗p

2

0 i∆2 − γ −iΩ∗c

2

−iΩp

2−iΩc

2i∆p − Γ

2

c1

c2

c3

. (2.22)

Such a rate can occur from processes which cause loss from the ground states. For

example, in the Section 2.6, we will see how to include the effect of a fourth level

by introducing an effective γ and in Chapter 4 we will see how it can account for

collisional losses. In general, if there is some loss rate Rloss of the population in |2〉,

we set γ = Rloss/2.

Mathematically, we see the roles played by γ and ∆2 are identical except the

contribution of the former is real (reducing the amplitude of c2) and the one of the

latter is imaginary (thus affecting the phase). If we have a small dephasing γ then

we have a generalized form of (2.21):

cA = 2ΩpΩc

Ω2

γ − i∆2

WcD. (2.23)

Just like ΩNA, both ∆2 and γ act to drive transitions between |D〉 and |A〉 and thus


cause cA to be non-zero.

2.5.3 Effect on the excited level occupation

The excited state amplitude can then be obtained from (2.12). When we are near

one-photon resonance we can show:

c3 = −iΩp

Γ

(2γ − 2i∆2

W

)c1, (2.24)

where we have used (2.9) (which approximately holds when we are in good EIT) to set

(Ωc/Ω)cD → c1. We see, upon comparison of (2.24) with (2.7), that c3 is suppressed

by a factor equal to the second ratio. Combining our results to this point, we see

that in order to obtain good EIT, the driving rate must dominate three separate time

scales:

W τ−10 ,∆2, γ. (2.25)

Losses from the other ground state |1〉 have a similar impact. Here and in Chap-

ter 3 we restrict the loss to |2〉 for simplicity.

2.6 Effects of levels outside the three-level system

Often light field coupling to a fourth level is the leading order contribution to the

loss rate γ discussed in Section 2.5. In practice, atoms have many internal sublevels,

and part of the challenge of achieving good EIT is choosing levels and polarizations

in a way that minimizes the light field coupling to levels outside the Λ system. In


this section, we calculate the effect of a small, but non-zero coupling to a fourth level,

|4〉 and point out several examples which occur in experiments in sodium. As we

will see, the effect can often be encapsulated in an effective loss γ and an AC Stark

shift. In Chapter 4, when we develop a more formal description which also accounts

for external motion, we will restrict ourselves to three internal levels. The results

presented here are a good guide for incorporating the effects of other levels in that

formalism phenomenologically.

2.6.1 Sodium level structure

Alkali vapors have provided the easiest system in which to cool and trap atoms,

and produce Bose-Einstein condensates [5, 26]. Our experiments have produced con-

densates and cold atom clouds in sodium [39] and so in the following we will use

examples from the sodium level structure. However, other alkalis have a similar hy-

perfine structure and so even specific examples discussed are applicable to them as

well.

In our original observation of ultra-slow light [1] we used the D2 line, shown

in Fig. 2.4(a). The available transitions are indicated along with their oscillator

strengths, which in each case are proportional to the square of the matrix element

|rij|2. The signs of the matrix elements, rij are not indicated in the figure but are

listed in Appendix C. In certain contexts, the relative signs of the matrix elements

can be important (as in double-Λ system analyzed below). They can be calculated

using the 9 − j symbols (see for example [63]).

The lower manifold was the 3S1/2 state for the valence electron, which is split due


Figure 2.4: Sodium level structure. (a) This part of figure taken from [23]. TheD2

line of sodium, relevant for the experiments in [1]. The atoms are originally trapped inthe |3S1/2, F = 1,mF = −1〉 state (circled). The numbers indicate the dimensionlessoscillator strengths fij, times 60 (chosen so all numbers are integers), which areproportional to |rij|2. Note that the total decay from each excited state sums to 60.(b) The D1 line of sodium, relevant to [2, 4] and all previously unpublished datapresented in this thesis. On this transition, one avoids coupling to extra levels in theF = 3 manifold. The splitting between the 3P1/2 and 3P3/2 lines is 17.19 cm−1.


to hyperfine interaction into the F = 1 and F = 2 manifolds, with a (2π)1.77 GHz

splitting. The upper manifold, 3P3/2, is an optical frequency (∼ (2π)5×1014 Hz) away,

and is split into four hyperfine states as shown. The atoms are originally trapped in

the internal state |1〉 ≡ |3S1/2, F = 1,mF = −1〉 (circled in the figure). The probe

beam was propagating along the z-axis (the axis of quantization) negatively circularly

polarized (εp = (x− iy)/√2), and was resonant with the |1〉 → |3〉 transition, where

|3〉 ≡ |3P3/2, F = 2,mF = −2〉. With this polarization, one sees that there is no

coupling from |1〉 to any other exited level. The coupling beam was linearly polarized

(εp = z) and propagating along the x axis. It was resonant with the |2〉 → |3〉

transition, where |2〉 ≡ |3S1/2, F = 2,mF = −2〉. These three states form the Λ

system we have discussed.

A coupling beam with that polarization can also drive the |2〉 → |4〉 transition,

where |4〉 ≡ |3P3/2, F = 3,mF = −2〉, but the detuning is ∆4 = −(2π)60 MHz as seen

in Fig. 2.4(a). For completeness, we also note that the probe can couple |2〉 to other

excited levels, and the coupling beam can do the same for |1〉 but these transitions

will be detuned by ≈ (2π)1.77 GHz and so will be much less important. In fact, these

are terms we decided to completely neglect in the RWA.

Before we calculate the effect of this fourth level, we note that in later experiments

[4], we circumvented coupling to |4〉 by carrying out experiments on the D1 line,

diagrammed in Fig. 2.4(b), where the |3〉 level is in the 3P1/2 manifold. Here we have

the same structure, but without the F = 0 and F = 3 manifolds. In this way, we

achieved a three-level Λ system, with the closest off-resonant transition to a fourth

level on the order of |∆4| = (2π)1.77 GHz away.


2.6.2 Dephasing and AC Stark shifts due a fourth level

To understand the effects of a fourth level, consider a system, diagrammed in

Fig. 2.5, where each beam drives transitions to an additional excited state |4〉. In

general there will be a difference in dipole moments of the various transitions, so we

define βc ≡ (εc · r42)/(εc · r32), βp ≡ (εp · r41)/(εp · r31). For the experiment in [1], one

sees from Fig. 2.4(b) that βp = 0, |βc| = 1. We assume |4〉 has the same decay rate Γ

as |3〉, which is true of all the 3P1/2 and 3P3/2 levels. We also assume, for algebraic

simplicity that the probe is on resonance (∆p = 0) while allowing the two-photon

detuning ∆2 ≡ ∆p −∆c to be non-zero. In all the results we will derive, it is only ∆2

which is important so long as both ∆p,∆c are much smaller than Γ. The equations

of motion governing this four-level system are:

c1

c2

c3

c4

=

0 0 −iΩ∗p

2−iβ∗p Ω∗

p

2

0 i∆2 −iΩ∗c

2−iβ∗c Ω∗

c

2

−iΩp

2−iΩc

2−Γ

20

−iβp Ωp

2−iβc Ωc

20 i∆4 − Γ

2

c1

c2

c3

c4

. (2.26)

where ∆4 ≡ (ω3−ω4). In all cases of interest, c4 is weakly coupled to the system, either

because of a very small matrix element (βp,c 1) or far detuning (∆4 Γ/2). We

now consider this system in three cases: the coupling only case (βc = 0, βp = 0), the

probe only case (βp = 0, βc = 0), and what we call the double-Λ case (βp = 0, βc = 0).

An Extra coupling beam transition

In the coupling only case, we now show how the presence of |4〉 causes an effective

dephasing γ4 and an AC Stark shift. The weak coupling to c4 allows us to adiabatically


p

c

p

2

|1>|2>

|3>

cc

pp

|4>

4

Figure 2.5: Four-level energy diagram. The four-level system considered. Each ofthe fields is coupled to the additional level |4〉. The text considers, (1) the couplingonly case (βp = 0), (2) the probe only case (βc = 0) and (3) the case where both arepresent, respectively. The coupling to the |4〉 is small either due to matrix elementsβp,c 1 or large detuning |∆4| Γ.

eliminate it [61] (also see Appendix B). Setting c4 = 0 in (2.26) and assuming βp = 0 :

c4 = −i βcΩc

Γ − 2i∆4

c2. (2.27)

Plugging this into the second equation of (2.26) gives:

c2 = −iΩ∗c

2c3 + (i(∆2 − ∆AC) − γ4)c2;

where ∆AC ≡ |βcΩc|2Γ2 + 4∆2

4

∆4, γ4 ≡ |βcΩc|2Γ2 + 4∆2

4

Γ

2. (2.28)

Both new terms have simple physical interpretations. The term ∆AC is an AC Stark

shift of |2〉 induced by the coupling between |2〉 and |4〉. In the limit, ∆4 Γ/2, it

reduces to ∆AC = |Ωc|2/4∆4, the usual result for AC Stark shifts [64]. This changes

the detuning of the coupling beam but the coupling frequency ωc can always be tuned

so ∆2 = ∆AC to exactly compensate this shift.

The other effect of the fourth level is seen to be an effective dephasing γ4 which is

mathematically equivalent to the γ discussed in the Section 2.5. The physical rate of


absorption on the |2〉 → |4〉 transition is R24 = 2γ4 supporting our earlier claim that

for a given loss rate Rloss from |2〉, we should set γ = Rloss/2.

If |4〉 is a far detuned level we have γ4 = Γ|βcΩc|2/8∆24. Plugging this into (2.24)

(when we have compensated for the AC Stark shift by setting ∆2 = ∆AC) we see that

the suppression factor of the excited level amplitude c3 depends only on ∆24 compared

to (Γ/2)2. The dependence on coupling intensity Ω2c cancels out since the rate of

preparation W and the rate of dephasing γ4 both scale with it linearly.

Dephasing due to bad polarizations

The results (2.28) holds equally well if |4〉 is nearly resonant with the coupling

field so long as the matrix element is small, i.e. βc 1. This allows us to apply

it to the case of a bad polarization. As mentioned in Subsection 2.6.1, when ex-

periments are carried out on the D1 transition, the nearest off-resonant transition is

∼ (2π)1.77 GHz away, and the γ4 due to this becomes completely negligible. In such

a case, often a small bad polarization can instead become the dominant dephasing

mechanism. Consider the D1 configuration discussed above and imagine that some

small fraction |β(pol)c |2 1 of the coupling intensity is positively circularly polar-

ized. Typical experimental values for bad polarizations are |β(pol)c |2 ∼ 10−4. We set

βc = β(pol)c in (2.28) to estimate the dephasing and get γ

(pol)4 = |β(pol)

c |2|Ωc|2/2Γ. Note

that again we get a cancellation in the dependence of c3 on the coupling intensity.

One can estimate the relative influence of bad polarizations versus far off-resonant

transitions by comparing the γ4 and γ(pol)4 for the two cases. One sees γ

(pol)4 will dom-

inate if |β(pol)c |2 > |βc|2Γ2/4∆2

4. For the D2 case discussed Γ2/4∆24 = 1/144 so γ4 is


likely to dominate while for the D1 case Γ2/4∆24 ∼ 10−5, meaning γ

(pol)4 is likely the

leading dephasing.

Bad polarizations could arise from bad polarizers, which is rather straightforward

to include once the fraction of bad polarization is known. However, any deviation

from the intended beam directions (with respect to the quantization axis) will also

lead to a fraction of light in the wrong polarizations. The inhomogeneities of the

trapping fields actually causes a slight inhomogeneity in the quantization direction

and so even perfectly aligned beams will have a small amount of bad polarization.

Calculating the matrix element of an arbitrarily directed beam with an arbitrary

polarization is tedious but straightforward. In Appendix C we discuss how this is

done.

An extra probe beam transition

If instead the probe is coupled to a fourth level (βp = 0, βc = 0), the situation is

quite similar. One adiabatically eliminates the fourth level and in this case obtains:

c4 = −i βpΩp

Γ − 2i∆4

c1. (2.29)

In most ultra-slow light experiments, the probe is kept substantially smaller than the

coupling beam (Ωp Ωc). At first site, a comparison of (2.29) and (2.27) would

seem to imply the effect of an extra probe transition is smaller. However, in this

limit, the dark state occupation of c2 is smaller than that of c1 by the ratio Ωp/Ωc, so

the population in |4〉 is the same order of magnitude as the previous case (if βc and

βp have the same magnitude). Thus again, it is the detuning and matrix element to


|4〉, not light field intensity, which is important in calculating its effect.

EIT in a double-Λ level system

An interesting case which arises in multi-level atoms is the one where the probe and

coupling are coupled to the same additional level |4〉 (βc = 0, βp = 0). This is called a

double-Λ system. We will show in this section how the two transitions contribute to

a combined AC Stark shift and dephasing that are not merely the sum of the results

for the two cases considered separately. Rather we will see that, depending on the

sign of the dipole moments, they can either constructively interfere, leading to four

times the loss that would be induced by a single transition, or destructively interfere,

giving a perfect four-level dark state. Thus, unlike the previous cases, the relative

signs of the rij are important.

An example of a double-Λ system which occurs experimentally is when the two

beams are co-propagating, as in [2]. In that experiment, state |2〉 was chosen to be

|F = 2,mF = +1〉 by making the probe beam positively circularly polarized (εp =

(x+iy)/√

2), the coupling beam was negatively circularly polarized (εc = (x−iy)/√2),

and both were tuned resonant with the excited level |3〉 = |F = 2,mF = 0〉. One sees

that in this configuration, both couple to the additional level |4〉 = |F = 1,mF = 0〉

with a detuning ∆4 = (2π)192 MHz. In that example βc = −βp (anti-symmetric

coupling) and constructive interference results (see Appendix C for the signs of the

matrix elements).

Consider the equations (2.26) in the general case. One can immediately under-

stand the role of the sign of the matrix elements βp,c by considering the dark state


(2.9), which depends on a specific phase and amplitude relationship between c1 and

c2. Adding the two additional terms due to |4〉 in the fourth equation of (2.26), one

sees that the dark state for |3〉 (2.9) is also a dark state for |4〉 (in the sense that there

is no light coupling with the ground states) if and only if βc = βp. Otherwise it is

impossible to choose c1 and c2 to simultaneously satisfy a dark state relationship for

|3〉 and |4〉. The following calculation considers the case of arbitrary βp,c and derives

the dephasing and Stark shift.

Adiabatically eliminating c4:

c4 = −iβpΩpc1 + βcΩcc2Γ − 2i∆4

, (2.30)

and plugging this back into (2.26) :

c2 = i∆2c2 − iΩ∗c

2c3 −

(β∗cβpΩ

∗cΩpc1 + |βcΩc|2c2

)(Γ + 2i∆4

8∆24

);

where ∆24 ≡ ∆2

4 +Γ2

4. (2.31)

The symbol ∆4 has been introduced for notational convenience and reduces to ∆4

in the limit ∆4 Γ. To cast this equation in the language of AC Stark shifts and

dephasings, in analogy to the lone coupling case, we use the fact that the |4〉 level is

a small perturbation and take the strategy of iterating with our known leading order

solution, the dark state relationship c1 = −(Ωc/Ωp)c2 + · · · . Plugging this into (2.31)

we obtain:


c2 = −iΩ∗c

2c3 + (i(∆2 − ∆

(2Λ)AC ) − γ(2Λ)

4 )c2;

where ∆(2Λ)AC ≡ β

∗c (βc − βp)|Ωc|2

4∆24

∆4, γ(2Λ)4 ≡ β

∗c (βc − βp)|Ωc|2

4∆24

Γ

2. (2.32)

So we see the symmetric case βc = βp leads to a cancellation of two Stark shifts so the

two-photon resonance is again on the bare resonance ∆2 = 0. Similarly, there is no

dephasing, since, as we conjectured above, the dark state relationship which cancels

absorption into |3〉 also does so for |4〉.

By contrast, in the anti-symmetric case βc = −βp, the AC Stark shift ∆(2Λ)AC is

seen to be twice as large as the lone coupling case (∆AC in Eq. (2.28)), and, more

importantly, c2 is forced out of the dark state at a rate γ(2Λ)4 = 2γ4, again twice as

high as the lone coupling case.

For this γ(2Λ)4 , we can determine c3 from (2.24) and c4 from (2.30):

c3 = −iβ∗c (βc − βp)(

Γ2

4∆24

)Ωp

Γc1,

c4 = −i(βp − βc)(

Γ2 + 2iΓ

4∆24

)Ωp

Γc1. (2.33)

The symmetric case then has no excited level occupation, while the anti-symmetric

case has twice as much amplitude in each of the excited states. In the next chapter,

we will return to this issue to see how it effects the absorption cross section seen by

the light fields.

Chapter 3

Ultra-slow light via

electromagnetically induced

transparency

In Chapter 2, we looked in detail at the way in which three-level atoms can be

driven into dark states which do not interact with the light fields. In this chapter,

we turn our attention to the question of the back action on the light fields. In this

way we will develop a self-consistent picture of the light propagation in the atomic

medium.

We will concentrate on the case where the coupling field is kept on at some constant

value and a weaker probe pulse (Ωp Ωc) is then injected into an atomic cloud with

density distribution n(R). A diagram of the geometry is presented in Fig. 3.1. In the

figure, the coupling beam is propagating in an orthogonal direction to the probe, as in

the experiment in [1], however, in general we will consider orthogonal, co-propagating,

45

Chapter 3: Ultra-slow light via electromagnetically induced transparency 46

Figure 3.1: Ultra-slow light (USL) experimental geometry. Schematic of theexperiment we consider in this chapter. The coupling field illuminates the atomcloud, propagating in either the x (orthogonal, as in the figure), z (co-propagating)or −z (counter-propagating) direction. The atoms are all originally in |1〉 and havea density distribution n(R). A probe of some characteristic temporal half-width τ0is then input along z. Information about the transmission and group velocity isobtained by recording the probe intensity as a function of time on a photo-multipliertube (PMT) located on the opposite side of the cloud. The pinhole images only thecentral part of the probe pulse, which has traversed near the center of the atom cloud,onto the PMT. The planes xin, zin refer to the planes where the input light fields aredefined.

and counter-propagating geometries. When the time scale of the probe pulse τ0

satisfies the adiabatic requirement and the dephasings and detunings are sufficiently

small (see Eq. (2.25)), the atoms are constantly driven into the dark state, just as in

the last chapter. When this happens, the probe is transmitted through what would

be an opaque medium in the absence of the coupling field. This transmission is the

basic idea of EIT [6] and was first observed in 1991 by Boller, et al. [8].

Although EIT dramatically reduces the light-atom interaction, there is some non-

zero interaction and so the light fields induce a polarization in the atomic medium.

This polarization acts back on the light fields, according to Maxwell’s equations,

and gives rise to a steep, linear dependence on frequency in the index of refraction.

This steep slope causes the reduction in the group velocity. When the probe is a

time-dependent pulse (rather than continuous wave), the reduced group velocity is

observable as a delay in the propagation time of the pulse. This effect first observed


in 1995 [11], where group velocities ∼ c/160 were observed. Our experiments in 1999

[1], and later others [12], have observed a group velocities ∼ c/107 or ∼ 10 m/s,

getting into the regime of ultra-slow light (USL). USL offers the prospect of bringing

group velocities down to the scale of other relevant velocities, such as recoil velocities,

and the sound speed in the atomic medium. In addition, the spatial length of the

pulse is reduced by the same factor as the group velocity, allowing the creation of

probe pulses with a length smaller than the atomic medium in which it is propagating

[1, 3, 2].

In Section 3.1 we present a semi-classical formalism, based on Maxwell’s equations,

which gives us equations governing the spatial propagation of the probe and coupling

fields in terms of the atomic field amplitudes ci. These propagation equations, com-

bined with our equations for the ci (2.4), will be the basis for analytic and numerical

results throughout this thesis. We use them to introduce the concepts of susceptibility

and group velocity, and apply them briefly to the two-level case for illustration. In

Section 3.2 we then use Fourier transform methods to solve the three-level problem in

the weak probe limit (|Ωp| |Ωc|), where the coupling field Ωc and the amplitude of

atoms in the initial state c1 are, to first order, unaffected by the probe propagation.

With this analysis, we will obtain the central results of EIT and USL propagation in

an atomic medium. Our analysis also allows us to obtain the evolution of the atomic

amplitudes ci as the pulse propagates through. To check the validity and limits of our

approximations, the analytic results are compared with direct numerical integration.

In Section 3.3 we consider effects which arise in experiments which go beyond the

simple linearized, three-level theory. First we look at how the coupling field is dy-


namically affected by the presence of a stronger probe (|Ωp| ∼ |Ωc|), which introduces

nonlinearities in the probe propagation. Then, as an extension of our results for the

four-level problem in Section 2.6.2, we see how a fourth level affects the probe prop-

agation. In Section 3.4 we then present the first ultra-slow light experiments, where

light was brought to 17 m/s, and our later experiments which pushed it to 50 cm/s.

These experiments demonstrate and corroborate various parts of the analytic results

obtained earlier in the chapter.

3.1 Classical light field propagation in three-level

atoms

Like Chapter 2, the formalism in this chapter is semi-classical, with the light fields

being treated classically, and the internal states of the atoms quantum mechanically.

However, we will now assume a many atom cloud, with an initial density distribution,

n(R) and allow the ci to be spatially dependent. The atomic amplitudes ci are de-

scribed by the time evolution equations (2.4) and the boundary conditions are tempo-

ral (i.e. given conditions at t = −∞): c1(R,−∞) = 1, c2(R,−∞) = c3(R,−∞) = 0.

Through Maxwell’s equations, we now treat the light fields, Ωp,c as complex dynam-

ical variables, which are also space- and time-dependent. We will see the dominant

contribution determining the light field dynamics is a first order equation for spa-

tial propagation, and thus our boundary conditions for Ωp,c will then be specified

as a function of time at the input planes z = zin and w = win, respectively (We

use w = x, z, or −z to denote the direction of the coupling beam propagation, i.e.


kc = wkc). We choose zin, win to be planes outside of the region occupied by the

atom cloud (where n(R) = 0, see Fig. 3.1).

In this section, our purpose is to use Maxwell’s equations to derive these light

propagation equations. We then apply them to find expressions for probe’s suscep-

tibility and group velocity and briefly illustrate the formalism by applying it to the

two-level problem (Fig. 3.1 with no coupling field).

3.1.1 Propagation equations from Maxwell’s equations

The following discussion follows Chapter 5 of [7]. Our strategy is to find the

polarization P in terms of the atomic amplitudes ci and then use the macroscopic

Maxwell equations to find the propagation of Ωp,c in terms of P. Using Maxwell’s

equation with no free charges or current (see Ch. 6 of [65]), we can derive an equation

of the electric field propagation in the presence of the macroscopic polarization P:

−∇2E +1

c2∂2E

∂t2= −µ0

∂2P

∂t2+

1

ε0∇(∇ ·P), (3.1)

where ε0 and µ0 are, respectively, the permittivity and permeability of free space.

The electric field E, consisting of probe and coupling (see Eq.(2.1)), contains terms

at four distinct frequencies: ωp,−ωp, ωc,−ωc.

Turning our attention to the polarization P, we find the collective effect of the

atomic cloud by simply summing up the contributions of individual atoms. The po-

larization can then be written as the sum over the individual electric dipole moments

of the individual atoms:


P(R, t) = −e 1

dV

∑R(m)∈dV

〈r (R(m), t)〉, (3.2)

where the sum is over all atoms m with positions R(m) in a small volume dV about

the position R. The quantity 〈r (R)〉 is the quantum mechanical expectation value

of the operator r:

〈r(R, t)〉 =∑i,j

c0j(R, t)c0∗i (R, t)rij. (3.3)

Parity considerations generally preclude any diagonal terms (rii = 0) and, in our Λ

system, also imply r12 = 0. The sum over the atoms gives us a factor of the density

n(R). Performing the sum and using (3.3) our expression for the polarization (3.2)

can then be written in terms of the ci:

P(R, t) = −e n(R)[(r13c3(R, t)c

∗1(R, t)e

i(kpz−ωpt) + c.c.)

+(r23c3(R, t)c

∗2(R, t)e

i(kcz−ωct) + c.c.)], (3.4)

where the slowly varying amplitudes are defined as c1 ≡ c01eiω1t, c3 ≡ c03ei(−kpz+ωpt)eiω1t,

c2 ≡ c02ei(−kpz+kcw+(ωp−ωc)t)eiω1t. This is the same time-dependent transformation as

just before (2.4). However, now that we are including spatial dependence in the ci, we

have also introduced a transformation which accounts for the quickly varying spatial

phases they acquire from coupling by the light fields (see Eq. (2.1)). Physically, the

spatial phase factor gives the |3〉 atoms a momentum kick from a probe photon ab-

sorption and |2〉 atoms a two-photon momentum kick from a probe photon absorption

and coupling photon stimulated emission. While we do not account for atomic motion


at present, these momentum kicks will have very important consequences when we

do account for it in Chapter 4.

Note that P in (3.4) has terms at the same four frequencies as E. The RWA

allows us to decouple parts of (3.1) at each frequency, effectively splitting (3.1) into

two simpler equations and their complex conjugates. We multiply these two equations

by εp,c, respectively, and write them as:

[(∂

∂z+

1

c

∂

∂t

)(− ∂∂z

+1

c

∂

∂t

)−∇2

⊥z

]Ep0(R, t)e

i(kpz−ωpt)

= e n(R) εp ·[µ0∂2

∂t2− 1

ε0∇(∇·)

]r13c3(R, t)c

∗1(R, t)e

i(kpz−ωpt),[(∂

∂w+

1

c

∂

∂t

)(− ∂∂w

+1

c

∂

∂t

)−∇2

⊥w

]Ec0(R, t)e

i(kcw−ωct)

= e n(R) εc ·[µ0∂2

∂t2− 1

ε0∇(∇·)

]r23c3(R, t)c

∗2(R, t)e

i(kcw−ωct), (3.5)

where ∇⊥j is the gradient in the two directions transverse to j.

We now make a number of assumptions regarding the slowly varying envelope

(SVE) shapes of the light fields to invoke some approximations. These will allow

us to throw out several terms, and simplify the remaining ones. First, we assume

that all time variation of the ci and Ep0,c0 are on time scales slow compared to the

optical frequencies ωp,c a condition which is satisfied by many order of magnitude in

experiments (generally ∼ 107 Hz versus ∼ 1014 Hz). A much more restrictive assump-

tion is that the same quantities vary slowly in space compared with the wavelength

λ = (2π)/kp,c (589 nm for sodium). We will see later in this chapter that we compress

pulses to lengths ≥ 20µm for typical parameters in a BEC, so the SVE inequality is

only satisfied by a factor of ∼ 30. Furthermore, in the light roadblock (Chapter 7)


experiments the pulse is compressed to only 2 µm, and solitons and vortices in BECs

(which we will study in Chapter 8) cause gradients in the atomic wavefunction with

a spatial scale on the order of the healing length, typically 0.4 µm. The consequences

of the breakdown of the SVE inequality is an interesting issue to consider, but goes

beyond the scope of this thesis.

Proceeding under the assumption of SVE in space and time, we denote the char-

acteristic length scale of the Ep0,c0 and ci by L and time scale by T , and then consider

terms of leading order in the dimensionless parameters (Tωp,c) and (Lkp,c). On the

right hand side (RHS) of the first equation of (3.5), the zeroth order term in (Tωp) of

(∂2/∂t2)c3c∗1e

i(kpz−ωpt) is −ω2pc3c

∗1e

i(kpz−ωpt). The derivatives of the ci contribute higher

order terms. Similarly, on the left-hand side (LHS) we will let (∂/∂z− (1/c)∂/∂t) →

2ikp. In the other term in that factorization, (∂/∂z + (1/c)∂/∂t), the quickly vary-

ing terms cancel and we are left with a leading order term O(Lkp,c), which we keep

exactly. The analogous result holds for the second equation.

Using the same logic for the terms involving transverse gradients ∇⊥z,w, we see

they contribute terms O(Lkp,c)2 on the RHS and O(Lkp,c) on the LHS. On both sides,

these are of higher order than terms we already have kept and so we neglect them.

(On the RHS we must use the fact that the polarizations εp,c are perpendicular to

kp,c). The perpendicular gradients which we are neglecting would lead to terms which

give refraction.

Neglecting terms and using substitutions according to these considerations, plug-

ging in our definitions for the Rabi frequencies and (2.5), and some straightforward

algebra then yields:


(∂

∂z+

1

c

∂

∂t

)Ωp(R, t) = −if13σ0

Γ

2n(R) c3(R, t)c

∗1(R, t),(

∂

∂w+

1

c

∂

∂t

)Ωc(R, t) = −if23σ0

Γ

2n(R) c3(R, t)c

∗2(R, t),

where σ0 ≡ 3

2πλ2. (3.6)

Here σ0 represents the cross section for absorption of resonant light in a two-level

system, with full oscillator strength. The oscillator strengths fi3 are the fraction

of atoms in |3〉 which spontaneously decay into |j〉. They serve as dimensionless

constants which parameterize the strength of a particular relevant transition line. To

obtain them, divide the numbers in Fig. 2.4 by 60.

When the RHS of (3.6) vanishes, these equations describe free space propagation.

In the presence of atoms, the RHS describes the polarization in terms of the ci. The

products of the amplitudes c3c∗i are complex dynamical numbers, and it is clear from

the form of the equations that the real parts of the products induce phase shifts in

the light fields, while the imaginary parts cause absorption. In (2.4) we allowed loss

of atoms from the system via spontaneous emission. Here the absorption present in

(3.6) corresponds to loss of photons which are spontaneously emitted. These photons

are scattered into a random direction and so are no longer in the coherent probe or

coupling fields. A quantum treatment of the light fields (see Appendix A) can be

used to derive this incoherent scattered field. In the present formalism we ignore

the scattered field, and therefore multiple scattering events. This issue has been

discussed in the literature [66]. We also note that, due to the SVE approximation,

these equations do not account for refraction or other effects related to the finite size


of the wavelength of light.

These equations in no way depend on linearity with respect to the probe field.

Rather, in (3.6) the atomic amplitudes (and therefore the polarizations) can have an

arbitrary dependence on the light fields and for this reason have quite a broad range

of validity. Therefore (3.6) are the basic equations we assume for both analytical and

numerical calculation of light field propagation throughout this thesis. Combined

with the amplitude evolution equations (2.4) they form a closed set of equations for

the atomic internal levels and light field propagation, a problem which we will solve

in Section 3.2.

3.1.2 Weak probe susceptibility

It is useful in many cases to consider a linear regime, where the probe beam

is sufficiently weak that it does not significantly effect the medium in which it is

propagating. This often allows us to derive analytic results which would otherwise be

impossible. In EIT and USL, we often consider a dressed atom picture in which the

medium consists of the atoms plus the strong coupling field and derive the resulting

susceptibility seen by the probe. Before solving the specific case of a three-level atom,

we will briefly define here a probe susceptibility, which can be applied to two-, three-,

and four-level atomic systems, whenever the probe is sufficiently weak.

A susceptibility χ relates an electric field of interest Ep to the polarization it

induces Pp via Pp = χEp [65]. In general, a χ defined in this way will not be

independent of the electric field Ep. The linear susceptibility regime, when it is most

useful, occurs when it is independent of it. In our case, the probe Ωp is the electric


field of interest and the polarization is proportional to the RHS of the first equation

of (3.6). So when this RHS scales as Ωp we have a linear susceptibility for the probe.

We will see later that in cases where linear susceptibility holds, c1 is a constant,

unaffected by the probe. In these cases, it is c3 which contains the linear dependence

on Ωp, and all other terms in the first equation of (3.6) are independent of Ωp.

We will generally find it easiest to solve our equations in frequency space using

Fourier transformations (FT). Throughout the thesis we will use the convention:

f(δ) =1√2π

∫ +∞

−∞dt f(t)eiδt (3.7)

Note that since Ωp has been defined as a slowly varying quantity (with the central

probe frequency ∆p transformed away), the FT variable δ represents the deviation of

the probe field frequency from ∆p (which arise from time dependence). If the probe

has a characteristic time scale τ0, then the FT quantity Ωp will thus contain a range

of frequencies δ with a width ∼ τ−10 .

Assuming the RHS of the first equation in (3.6) is proportional to Ωp we FT this

equation and use the definition of (2.5) to get:

(∂

∂z− iδc

)Ωp =

i

2kpχ Ωp, (3.8)

where we define the susceptibility in the frequency domain χ by the relation :

χ Ωp = −ρΓc3c∗1,

where ρ ≡ 3f13nλ3

4π2. (3.9)


(In order to ease notation later we have made χ a dimensionless quantity by dividing

by ε0 compared with the usual definition of χ (in MKS units) given above.) The

quantity χ is a function of the probe central frequency plus the additional frequency

introduced by time dependence ∆p + δ. The front factor ρ is a dimensionless param-

eterization of the density (in terms of the wavelength cubed). For typical condensate

densities (n ∼ 1014 cm−3) and an oscillator strength f13 = 1/2 (as was the case in

[1, 4]), ρ ≈ 0.7. Note that the FT we performed to obtain Eqs. (3.8)-(3.9), and

therefore the definition of χ, requires c3 ∝ Ωp to be valid.

3.1.3 Group Velocity

We now clarify how group velocity and pulse distortion arise in our susceptibility

description. We will start by investigating the effect of a χ which is independent of

frequency, then, one by one, add various physical effects which arise from frequency

dependence.

For this purpose, we will start with a slightly more general equation than (3.8) :

(∂

∂z− iδc

)Ωp =

i

2kp

(1 +

δ

ωp

)χ Ωp, (3.10)

Here we have added an additional term (1 + δ/ωp) after kp. This is necessary because

kp represents the probe wave vector for its central frequency ωp, and this term adds

the correction due to the true frequency ωp+δ. This correction is generally very small

and in fact, in the course of making the SVE approximation (in Subsection 3.1.1), we

suppressed it altogether. This is why it does not appear in (3.8). However, we keep

it here as it clarifies our discussion of group velocity.


With no polarization (χ = 0), the RHS of (3.10) vanishes. If one were to solve

this trivial problem and invert the FT, one finds the pulse propagates at the vacuum

velocity c (the problem becomes equivalent to the first equation of (3.6) with the RHS

equal to zero).

Now consider a χ = 0 which has almost no dependence on frequency. Then (3.10)

can be written as :

∂

∂zΩp =

i

2kp

[χ+ (2 + χ)

δ

ωp

]Ωp. (3.11)

From now on we will split χ into real and imaginary parts: χ = χR + iχI . From

the first term, we see the imaginary part χI will cause attenuation, thus reducing the

probe transmission through the medium, while the real part χR will cause a phase

shift. Both of these apply to all the frequency components of Ωp equally, and thus

will have the same effect on a both a pulsed or a c.w. probe. In addition, we can

see that (3.11), compared to the χ = 0 case, the front factor of the term linear in

δ changes 2 → 2 + χ. As a result the velocity, which was the vacuum velocity c,

becomes Vph = c/(1+χR/2). This is called the phase velocity and is the characterizes

the speed at which the probe photons travel in the medium. Therefore, one would see

a slight change in the propagation time of a pulse due to a non-zero χ, although this

change arises from a term that we have suppressed in our SVE approximation, and

so is always quite small for our parameters. This will be seen later when we explicitly

solve the three-level problem. From this term we also note that a non-zero χI can

also cause a slight frequency dependence in the attenuation.

Now assume there is some frequency dependence in χ. If the probe pulse width τ0


is sufficiently large, then δ will be small and we can expand χ about ∆p and rewrite

(3.11) as:

∂

∂zΩp =

i

2kp

[χ+ δ

((2 + χ)

1

ωp+dχ

d∆p

+ · · ·)]

Ωp, (3.12)

where χ and its derivative are evaluated at ∆p, i.e. at δ = 0. Because the new

term has the same dependence on δ as the vacuum speed term, we see it can be

combined it with the terms just discussed. In this way, all the real parts with a linear

dependence on δ give a renormalized velocity:

Vg =c

1 + χR

2+ ωp

2

(dχR

d∆p

) . (3.13)

The quantity Vg is called the group velocity [65] and is the speed at which a pulse

will propagate.

Without the last term in the denominator Vg would reduce to the phase velocity

already discussed. For our experimental parameters, we always have Vph ≈ c, while

the last term in denominator, which is proportional to the slope of χR, is typically

107! Thus Vg is completely dominated by this term. This can be seen graphically in

Fig. 1.2(b) in the introduction, which shows the index of refraction√

1 + χR. The

index is never far from one, meaning that the phase velocity Vph ≈ c, but the steep

slope on resonance (resulting from the large slope in χR), makes Vg is very small. The

group velocity arises from different frequency components of the probe interfering in

a specific way such that the pulse (or group of photons), as opposed to the individual

photons, travel at a slow velocity. Whenever we invert the solution to an equation of

the form (3.12)) back to the time domain, we see the first order term in δ shifts the


probe in time by an amount we call the delay τd, which is inversely proportional to

Vg.

Lastly, we consider what happens when there is even more frequency dependence

in χ. Suppose we must keep the second order term in the expansion :

∂

∂zΩp =

i

2kp

[χ+ δ

((2 + χ)

1

ωp+ δ

dχ

d∆p

+δ

2

d2χ

d∆2p

+ · · ·)]

Ωp, (3.14)

If there is a substantial real part at second order in δ, different frequency components

of Ωp will then propagate at different group velocities and the pulse shape distorts as it

propagates. Again, imaginary parts at both first and second order cause attenuation,

but in a frequency dependent way that will change the pulse shape as well.

Thus, we see that, generally, frequency dependent χ can distort the probe, but in

the special case that the real first order part dominates, this distortion is coordinated

in way that merely causes a time shift, no matter what the pulse shape is. As we

will see, USL propagation works in three-level atoms because there is a particular

frequency which has (1) a nearly vanishing imaginary susceptibility (at zeroth order),

allowing transmission; (2) a large, real linear (first order) part, causing a slow velocity

Vg; and (3), no substantial curvature (second order parts) or imaginary parts at first

order, allowing the pulse to preserve its shape. By contrast, even though steep slopes

in χ occur at certain frequencies in two-level media, these never occur at frequencies

which also satisfy conditions (1) and (3).


3.1.4 Illustration in the two-level case

To apply this to a simple concrete example, and to ease comparison with the three-

level case later, we briefly consider near resonant propagation in a two-level system.

Consider (3.6) and (2.4), which form a closed set of equations, and the experiment

diagrammed in Fig. 3.1. The two-level problem is described by taking Ωc = 0 and

c2 = 0, leaving us with three non trivial equations and three dynamical variables:

c1, c3, and Ωp.

Initially all the atoms are in |1〉 (c1(R, t = −∞) = 1) and for sufficiently short

and weak probe pulses c1(R, t) ≈ 1 always. This approximation, in effect, ignores

two common physical effects. The first is hole burning, whereby after a sufficiently

long time the population P1 = |c1|2 becomes depleted as it absorbs photons and

spontaneously emit into other levels. Then c1 ≈ 1 becomes a bad approximation. It

also ignores power broadening [60] whereby strong light intensities causes significant

changes in the inversion of the populations of |1〉 and |3〉, even at short times, and

again c1 ≈ 1 no longer holds. Mathematically, ignoring these effects amounts to a

linear probe approximation, and allows us to disregard the equation for c1. We FT

the equation for c3(z, δ) and obtain:

c3 = −i Ωp

Γ − 2i(∆p + δ)

=2(∆p + δ) − iΓΓ2 + 4(∆p + δ)2

Ωp (3.15)

Using (3.9) we then get the susceptibility:


- 20 - 10 0 10 20

- 0.2

0

0.2

0.4

0.6

- 200 - 100 0 100 200

0.2

0.4

0.6

0.8

1

Tra

nsm

issi

on

p (MHz) p (MHz)

a b

(2-lev)

~

Figure 3.2: Susceptibility in two-level atoms. (a) Real χR(2−lev) (solid curve)

and imaginary χI(2−lev) (dashed curve) parts of the susceptibility at the center of a

cloud of two-level sodium atoms (3.16) with central density n = 8.3 × 1013 cm−3.The Lorentzian width is determined by the natural linewidth Γ = (2π)10 MHz.The oscillator strength in the case plotted is f13 = 1/2. (b) The correspondingtransmission profile for an atomic cloud with D0 = 408. Because of the large D0,the frequency region of little or no transmission is much broader than the frequencywidth of the Lorentzian in χI(2−lev).

χ(2−lev) = ρ−(

∆p+δ

Γ/2

)+ i

1 +(

∆p+δ

Γ/2

)2 , (3.16)

which is a Lorentzian profile centered at resonance (∆p +δ = 0), with a width Γ. It is

plotted in Fig 3.2(a). The real part χ has a negative slope on resonance, corresponding

to a region of anomalous dispersion. Solving (3.10) to zeroth order in δ we get:

Ωp(z, δ) = Ωp(zin, δ) exp

−D

2

1 − i

(∆p+δ

Γ/2

)1 +

(∆p+δ

Γ/2

)2

. (3.17)

where we have defined a dimensionless optical density,

D(x, y, z) ≡ f13 σ0

∫ z

zin

dz′ n(x, y, z′). (3.18)

In the case of a c.w. probe (δ → 0), the intensity transmission through the cen-

tral column (x = y = 0) of the cloud is T(2−lev) = exp[−D0/(1 + 4∆2p/Γ

2)], where


D0 ≡ D(0, 0, zout). Typical condensates in our experiments have D0 ∼ 400, so res-

onant (∆p = 0) light is completely absorbed. The transmission profile is plotted in

Fig. 3.2(b). As we see in the wings of Fig. 3.2(b), significant transmission occurs

only at very large detunings (specifically when ∆p ≥ Γ√D0/2). Thus condition (1)

in Subsection 3.1.3 is not satisfied except at these large detunings.

If one puts in a time-dependent probe at this detuning (∆p = Γ√D0/2) and

considers the linear terms in δ, there would be a reduced group velocity (Vg ∼ 10−5c

according to (3.13)) for the parameters in Fig. 3.2), but this is still substantially

larger than the results we will derive next for the three-level case. More significantly,

the slope of χR(2−lev) is not constant at this ∆p, so the second order terms become

important and thus condition (3) is not satisfied. When one inverts (3.17) this results

in significant distortion of the input pulse shape.

3.2 Ideal three-level USL

We now address the three-level problem. We will primarily focus on the case where

the probe is much weaker than the coupling field (Ωp Ωc), allowing us to consider

terms up to linear order in Ωp and obtain analytic solutions. This will also allow

us to make the connection with the above discussion of susceptibility χ and group

velocity, however, we will return to our original equations for the atomic and light

field evolution in Subsection 3.2.2 and reintroduce the weak probe approximation in

this specific case in Subsection 3.2.3.

In Chapter 2, we saw how the existence of dark states suppressed the occupation

of |3〉. Since this suppression originated from less absorption of probe photons, we


expect, and show here, that on the two-photon resonance, where the dark state exists,

we get a large dip in χI , allowing high transmission. Just as we saw in Section 2.4

that absorptive events into |3〉 were replaced by coherent exchange between |1〉 and

|2〉, here will see in (Subsection 3.2.1) that absorption of probe photons are replaced

by coherent exchange of energy between the probe and coupling fields. The dip in χI

is accompanied by a steep, linear slope in χR. This leads to a steep linear refractive

index and therefore ultra-slow group velocities. The fact that χR is linear at the point

where the transmission occurs prevents distortion, so pulses with arbitrary shape to

can maintain their input temporal profile during propagation, even through large

optical densities D0. Our analysis will also give the leading order corrections which

introduce finite absorption and pulse spreading. We will also present some numerical

calculations which corroborate the analytic results.

3.2.1 Coherent photon exchanges

To understand the process of EIT, we can perform a calculation analogous to

Section 2.4, where we saw how the dark state allows coherent exchange to domi-

nate absorption losses. We begin with our propagation equations (3.6), adiabatically

eliminate the excited state amplitude c3, and plug in the dark/absorbing basis trans-

formation (2.10) to get:

(∂

∂z+

1

c

∂

∂t

)Ωp = −1

2f13σ0 n cA

(Ωp

ΩcA +

Ωc

ΩcD

),(

∂

∂w+

1

c

∂

∂t

)Ωc = −1

2f23σ0 n cA

(Ωc

ΩcA − Ωp

ΩcD

). (3.19)

Using our definitions of the Rabi frequencies, the relationship between intensity and


electric field I = (1/2)ε0|E|2, and the assumption that one-photon contains an energy

ωp,c, we find the number of photons per unit volume in the probe and coupling fields,

respectively, are:

Np,c ≡ |Ωp,c|2f13,23σ0Γc

. (3.20)

Using (3.19) we then find:

(c∂

∂z+∂

∂t

)Np = n(−RA1 −RS),(

c∂

∂w+∂

∂t

)Nc = n(−RA2 +RS). (3.21)

where RS, RA1, and RA2 were defined in (2.16). By comparing (3.21) and (2.16) we

see that a loss of an atom in |1〉 (|2〉) is accompanied by a loss of a probe (coupling)

photon, and exchange between |1〉 and |2〉 is accompanied by exchange of photons

between the two light fields.

This physical picture clarifies the role played by the coupling beam in the slow

propagation of the probe. Whereas in a two-level medium the |3〉 state absorbs the

incoming probe field and spontaneous emission scatters it into all directions, in a

three-level system the probe energy flows into the coherent coupling field. As the

probe propagates out of the medium, the flow reverses and energy in the coupling

field restores the probe pulse to its original input energy upon exit. The physical

process of this energy exchange occurs over a finite time, which causes the large delay

seen in the exiting probe pulse. In the analytic solution we present next, the coupling

beam is assumed to be a constant, which obscures this physical energy flow we just

outlined.


3.2.2 The self consistent three-level equations

We now obtain a solution for the full three-level problem diagrammed in Fig. 3.1.

We start with a cloud of atoms in |1〉, illuminated with a coupling field. We then inject

a much weaker probe pulse of width τ0 along z and look at the resulting evolution.

In particular we will derive the resulting output at zout (at the PMT). In doing this,

we will also obtain solutions for the atomic amplitudes ci in the cloud as a function

of space and time.

We start with the full set of five equations governing the evolution of the three-

level atoms (2.22) and light fields’ propagation (3.6). Note that in using (2.22) (as

opposed to (2.4)), we are including the dephasing term γ, as it does not significantly

complicate our solution and we will see it is often the leading order cause of pulse

attenuation. Three of the boundary conditions express that all the atoms begin

in state |1〉: c1(R,−∞) = 1, c2(R,−∞) = c3(R,−∞) = 0. The two boundary

conditions for the light fields are defined at the planes zin, win outside the cloud. The

probe we take to be a Gaussian, Ωp(z = zin, t) = Ω(in)p (t) (defined in (2.19)) and the

coupling beam is taken to be constant : Ωc(w = win, t) = Ωc0.

3.2.3 Weak probe limit

In order to solve the problem analytically, we assume the weak probe limit Ωp0

MinΩc0,Γ, allowing us to retain only terms up to first order in |Ωp|. Our solution

will then yield a linear susceptibility. From our atomic evolution equations (2.22) we

see that:


c2, c3 = O(|Ωp|) + · · · , (3.22)

which in turn implies that:

c1 = 1 + O(|Ωp|2) + · · · ≈ 1,

Ωc = Ωc0(1 + O(|Ωp|2) + · · · ) ≈ Ωc0, (3.23)

to the order we are considering. This reduces the number of equation and variables

from five to three. Physically, we are taking the coupling field and the reservoir of |1〉

atoms as constants of the medium which are unaffected by the probe propagation.

Also, by eliminating the (possibly orthogonally propagating) coupling beam, we

have completely decoupled each column of atoms along z (at a particular x, y) from

columns at other x, y. We then have essentially an independent one dimensional

problem of the probe propagation. The solution will still depend on the x, y through

the density n(x, y, z), but, due to this decoupling, the total solution will just be a

linear sum of the solutions at each x, y. In the following, then, we will drop the explicit

dependence and simply write the z dependence, e.g. n(z), with the understanding

that we are solving the problem at some specified x, y.

3.2.4 Solution for the pulse in frequency space

To proceed we then will find it easiest to transform to frequency space, using the

convention (3.7). Making the substitutions in (3.23) in our remaining equations and

transforming:


[−i(δ + ∆2) + γ] c2 = − i2

Ωc0c3,[−i(δ + ∆p) +

Γ

2

]c3 = − i

2Ωp − i

2Ωc0c2,(

∂

∂z− iδc

)Ωp = −if13σ0n(z)

Γ

2c3. (3.24)

We algebraically eliminate c2 by plugging the first equation into the second and then

find c3 in terms of Ωp:

c3 = −Ωp

Γα(Ωc, δ,∆2,∆p),

where α(Ωc0, δ,∆2,∆p) ≡ 2Γ [(δ + ∆2) + iγ]

Ω2c0 − 2 [2(δ + ∆p) + iΓ] [(δ + ∆2) + iγ]

. (3.25)

Using (3.9) and (3.25) we find the three-level susceptibility:

χ(3−lev) = ρα. (3.26)

The complex parameter α parameterizes the degree to which the coupling field alters

the susceptibility. In the two-level case (Ωc0 = 0)), (3.26) reduces to (3.16) and

on-resonance (∆p + δ = 0) we have α = i.

The three-level susceptibility (3.26) is plotted as a function of probe frequency

in Fig. 3.3(a) for the case where the coupling beam is on resonance (∆c = 0) and

Ωc0 = (2 pi)8 MHz. By comparing with the two-level case (Fig. 3.2 (a)), we see

that outside a small range of frequencies near resonance, we recover the two-level

Lorentzian result. Near resonance, by contrast, there is a sharp dip in χI(3−lev) and it

nearly reaches zero. From (3.25) we see the presence of dephasing (γ = 0) prevents


Figure 3.3: Susceptibility of Λ configuration atoms in the presence of a cou-pling beam. (a) Real part of the susceptibility χR(3−lev) (solid curve) and imaginary

part χI(3−lev) (dashed curve) in a three-level atom (3.26) at the center of an atomiccloud with the same parameters as Fig. 3.2. The coupling beam has a Rabi frequencyΩc0 = (2π)8 MHz and is on resonance (∆c = 0). The dephasing rate is γ = (2π)1!kHz.The oscillator strengths are f13 = 1/2 and f23 = 1/3. Exactly on resonance χI(3−lev)

nearly vanishes and χR(3−lev) has a steep linear slope. (b) The same plot with the

coupling beam chosen to be off resonance, ∆c = (2π)4 MHz, shows that the trans-parency dip shifts from the center of the Lorentzian to the two-photon resonance andintroduces some curvature to the slope of χR(3−lev) at the transparency dip. (c) The

intensity transmission profiles, according to (3.27), for the two cases plotted in (a)(thick curve) and (b) (thin curve). The optical density of the cloud is D0 = 407.This large D0 causes even the very small on-resonance value of χI(3−lev) to reduce the

transmission there to about 60%. It also causes the transmission width to be muchsmaller than the EIT width.

it from reaching exactly zero. The two peaks in the curve are always spaced by Ωc0,

and thus the width of the profile can be dramatically altered by simply changing the

coupling field strength Ωc0. Associated with the sharp dip in χI(3−lev) is a steep linear

slope in χR(3−lev) ( i.e., dχR(3−lev)/d∆p is large). It is this steep slope which gives rise to

the slow group velocity. The Kramers-Kronig relations [65], which are a consequence

of causality, actually require the dip in χI(3−lev) to be accompanied by a sharp feature

in χR(3−lev). Note that the presence of the coupling field reverses the direction of the

slope from the two-level case, and we go from anomalous to regular dispersion.

Setting ∆c = 0 simply shifts the position of the lowest point in the absorption

spectrum to the two-photon resonance ∆p + δ = ∆c, while the Lorentzian envelope


stays centered on ∆p + δ = 0. An example is shown in Fig. 3.3(b). Note also that

a significant curvature is introduced in the slope of χR(3−lev) in this case. This occurs

when the one-photon detunings become comparable to the natural linewidth ∆p,c ∼ Γ.

We can solve the remaining equation of (3.24), a first order differential equation

for the probe propagation, and plug in our boundary condition in frequency space,

Ωinp (δ) = τ0Ωp0exp(−δ2τ 2

0 /2), to get:

Ωp(z, δ) = τ0Ωp0 exp

[iδ

(z − zin)

c− 1

2δ2τ 2

0 + i1

2D(z)α

]. (3.27)

This is the solution to general equation (3.8) in the special case of a three-level atom

with a non-zero coupling field Ωc0.

The integration over z in our solution causes the large optical density D(z) to

enter the problem. This represents the cumulative effect of the probe seeing many

atoms as it propagates through the cloud. This is evidenced by Fig. 3.3(c) where we

plot the transmission profiles for the same parameters as Fig. 3.3(a-b). The small

dephasing γ is seen to reduce the on-resonance transmission to about 60% and the

width of the transmission profile is significantly reduced from the characteristic width

of the EIT profile (i.e. the dip in χI(3−lev) which is ∼ Ω2/Γ). We return to this point

later.

In Chapter 2 we found EIT required choosing a sufficiently largeW ≡ Ω2/Γ. In the

weak probe limit, Ω2 ≈ Ω2c0 in which case W ≈ Ω2

c0/Γ ≡ Wc. In an USL experiment

we are interested in the EIT regime (near one and two-photon resonance), and so

choose our pulses widths and detunings such that


Wc τ−10 ,∆2, γ

and Γ ∆p,∆c, τ−10 , γ. (3.28)

When we assume these, we can perform a first order binomial expansion (in W−1c ) of

the denominator of α in (3.25) and find:

α ≈ 2iγ + 2(δ + ∆2)

Wc

+4i(δ + ∆2)

2 − 2iγ2 − 4(δ + ∆2)γ

Wc2 + O

(γ, δ,∆2

Wc

)3

=

(2iγ + 2∆2

Wc

+4i∆2

2 − 4∆2γ − 2iγ2

Wc2

)+ δ

(2

Wc

+8i∆2 − 4γ

W 2c

)

+δ24i

W 2c

+ O(γ, δ,∆2

Wc

)3

, (3.29)

where in the second equality we have split the terms into terms of increasing order

in δ. This expansion of α is useful for two reasons. First, when it is valid we can

perform the inverse FT of (3.27) in closed form, which we will do below. Second, it

allows to pick out terms which are zeroth, first, and second order terms in δ, thus

making the correspondence with (3.14) and discussion in Subsection 3.1.3 clear.

Comparing the first zeroth order term in (3.29) with the on-resonance two-level

case α = i, we see that, on two-photon resonance (∆2 = 0), the imaginary part of

α and therefore χI(3−lev) is reduced by a factor 2γW−1c 1. Thus we get a much

higher transmission, though there is a finite absorption coefficient due to due to

dephasing. For example, this term is responsible for the finite attenuation on two-

photon resonance in Fig. 3.3(c). There is also a real part in this term proportional to

∆2, contributing a small uniform phase shift. In, the second zeroth term (∝ W−2c ), we


see the leading order cause of attenuation due to two-photon detuning (∝ ∆22W

−2c ),

thus giving a parabolic dependence on ∆2 for the attenuation of the pulse near the

two-photon resonance. The last two terms contribute small corrections which are

always dominated by the terms we just discussed.

Now we consider the terms linear in δ, which are responsible for the group velocity.

The first term (∝ W−1c ) dominates. Using the coefficient for δ in the first term in

(3.29) and (3.26) we see the group velocity (3.13) becomes:

Vg =Wc

f13σ0n. (3.30)

One can easily verify that the 1 + χ/2 ≈ 1 in (3.13) is completely unimportant for

typical parameters and are dominated by this slope term. The other linear terms

(∝ W−2c ) in (3.29) are also dominated by the leading term just considered.

Now we now look at the term at second order in δ in (3.29). We see that there

is an imaginary term ∝ W−2c . This term represents absorption of high frequency

components in a very narrow temporal pulse. It is the analogue of the absorption

due to ∆2 only it absorbs components of Ωp in a frequency dependent way. It results

in a narrowing of Ωp in frequency domain and thus in a spreading of the pulse Ωp in

the time domain. This is an example of pulse distortion from a frequency dependent

χ we discussed more generally in Subsection 3.1.3.

When the binomial expansion (3.29) is not valid, we cannot obtain an analytic

inversion of the FT. This happens, for example, in the case of a one-photon detuning

comparable to Γ, such as the one plotted in Fig. 3.3(b). We saw in the figure that this

introduces a curvature in the real part of χR(3−lev), and we will see later numerically


a

(MHz)c0

c/V

(10

)g

9

I(0)

- 3 - 2 - 1 0 1 2 3

- 0.2

0

0.2

0.4

0.6

0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0 2 4 6 80

0.2

0.4

0.6

0.8

1

b

c

p (MHz)

~

~

Figure 3.4: Three-level susceptibility at low Wc. (a) The χR3−lev and χI(3−lev) for

the much lower coupling Rabi frequency Ωc0 = (2π)1 MHz (Wc = (2π)0.1 MHz) withother parameters the same as in Fig. 3.3. (b) χI(3−lev)(0) as a function Ωc0, based on

(3.26) and (3.25). At Ωc0 = 0 the behavior goes smoothly into the two-level case. (c)The calculated group velocity reduction factor c/Vg, based on (3.13), is seen to reachmaximum and then decrease at low Ωc0.

that this causes pulse distortion.

The assumption on Wc (3.28) gives us a minimum coupling intensity to be in

the EIT regime. When these inequalities do not hold, then the behavior goes over

smoothly into the two-level case. In Fig. 3.4(a) we plot χ(3−lev) with a much smaller

coupling intensity, Ωc0 = (2π)1 MHz. As seen there, the EIT feature in both the real

and imaginary part becomes correspondingly narrow. WhenWc becomes comparable

to γ, the on-resonance imaginary susceptibility χI(3−lev)(0) begins to rise and absorp-

tion becomes significant. In Fig. 3.4(b) we plot χI(3−lev)(0) (according to (3.25)) as a

function of Wc. As Wc → 0 it becomes equal to the two-level result (Fig. 3.2(a)).

As this occurs, the slope of χR(3−lev) also begins to decrease. We plot c/Vg at

the center of cloud, calculated from the slope of χR(3−lev) and (3.13). We see from

the plot that the group velocity decreases as we turn down Wc all the way to 10−9

until a critical point, at about Wc ∼ 2γ, where lowering it further then leads to a


group velocity increase. Thus a given γ introduces a lower limit on Vg. At lower Wc

still, the sign of the slope eventually reverses and goes into the anomalous (negative

slope) dispersion as in two-level atoms. Before we reach this value, the value of the

absorption coefficient (via χI(3−lev)) has already become indistinguishable from the

two-level case and the probe pulses are completely ignored after a very small optical

density.

3.2.5 Solution for the pulse in time

We have discussed the probe pulse propagation in frequency space. When we are

in EIT conditions and the expansion (3.29) is valid, it is simple to take the inverse

FT of (3.27) and obtain a closed form solution for the Ωp:

Ωp(z, t) =τ0√

τ 20 + 4DWc

−2Ωp0 exp

[−D γWc

]

× exp

[−2D∆2

2

Wc2 + iD

∆2

Wc

]exp

[−(t−DWc

−1 − (z−zin)c

)2

2(τ 20 + 4DWc

−2)

], (3.31)

where D = D(z). This is our central result for the USL probe pulse propagation

solution and will be used as a basis for comparison with results throughout this thesis.

Fig. 3.5(a) shows this expression at the PMT (z = zout) for typical experimental

parameters along with the “reference” probe input at z = zin (or equivalently, the

probe output at z = zout when there are no atoms present, D0 = 0).

In the case plotted, the group velocity at the center of the cloud is Vg = 6 m/s

= 2 × 10−8c (see Eq. 3.30). This manifests itself in (3.31) as the shift in time in the

last exponential, we see that we get a large delay τd ≡ D0Wc−1 (recall D0 ≡ D(zout.


- 5 0 5 10 15 20

0.2

0.4

0.6

0.8

1

t s)

p|

p

0

- 40 - 20 0 20 40

0.2

0.4

0.6

0.8

1

z m)

a b

Figure 3.5: Ultra-slow probe propagation and compression in an EITmedium. (a) Weak probe solution for the probe pulse (3.31) as a function of time,at the output z = zout (solid curve) for the same cloud and coupling parameters asFig. 3.3. The input probe pulse (dashed curve), or reference pulse, which is inputat z = zin has a width τ0 = 1.5 µs and peak Rabi frequency Ωp0 = (2π)2 MHz.There is an 11 µs delay, corresponding to Vg = 6 m/s at the cloud center. Since τ0is comparable to τsp = 1.0 µs, there is some visible spreading and attenuation. (b)The solution as a function of space at the times t = 0.0, 2.3, 5.7, and 10.3 µs (solidcurves from left to right). For reference, the dashed line shows the atomic density inarbitrary units.

This dominates the (z − zin)/c by many orders of magnitude and so we neglect this

term from now on (note this term represents the 1 in the denominator of (3.13).

Next we see that in the case γ = 0, the pulse is partially absorbed. The intensity is

attenuated by (1/e) after propagating a distance such that τd = γ−1, so the dephasing

sets a time scale for the largest attainable pulse delays with significant transmission,

τ(max)d ∼ γ−1.

The attenuation due to the two-photon detuning ∆2 is expressed in the next

exponential. We see the attenuation scales as ∆22 and gives rise to a transmission full

width of ∆trans = Wc/√D0 (see Fig. 3.3(c)). In Eq. (3.31) we also see ∆2 causes a

small phase shift of the pulse.

The front factor and the denominator of the last exponential reflect the absorption


of the high frequency components of the pulse. We see its temporal width increases

as it propagates from τ0 at the input to τ ≡√τ 20 + τ 2

sp at the output, where τsp ≡

2W−1√D0. The input pulse widths τ0 must be wider than the characteristic time

scale τsp to prevent severe attenuation and spreading. This condition is the real time

analogue of the frequency width condition on ∆2. Note that τd = τsp√D0. Therefore

in the limit D0 1 we can use pulses with τ0 < τd/2 and get good pulse separation,

in the sense that the reference and output pulses are completely separated in time

(as in Fig. 3.5).

Integrating the intensity of the pulse over time we find the integrated transmission

of the output pulse at zout, relative to the energy in the input (reference) pulse:

T =

∫dt |Ωp(zout, t)|2∫dt |Ω(in)

p (t)|2

=τ0τ

exp

[−2D0

γ

Wc

]exp

[−4D0

∆22

Wc2

]. (3.32)

So there are three distinct contributions to the loss of transmission: dephasing γ,

detuning ∆2, and finite pulse widths (applicable when τsp is comparable to τ0).

3.2.6 Spatial compression

It is interesting to consider the spatial dynamics of the probe as it propagates

through the cloud. Fig. 3.5(b) shows a series of snapshots of |Ωp| as the pulse prop-

agates through the cloud, based on (3.31). Outside the medium there is no visible

spatial structure to the probe, however, the front edge is eaten away as it enters the

cloud. It is completely contained when the peak has reached the cloud center. It is


then restored as it exits out the far edge.

To estimate the length in the center of the cloud, assume for a moment that

the density is slowly varying over a range of z of interest. In such a range Vg is

approximately constant and D becomes a linear function of z. Then the spatial (1/e)

length of the probe Lp is found from (3.31) to be [58, 3]:

Lp = 2τ0Vg =2Ωc0

2τ0f13σ0nΓ

, (3.33)

which is a reduction by Vg/c from the spatial length of a pulse in vacuum, 2τ0c. For

the parameters in Fig. 3.5, Lp = 18 µm at the cloud center. We note that in this

case, Lp is smaller than the cloud length so the pulse is compressed completely inside

the cloud for a time. This will be the case whenever τd > 2τ0. The time between the

end of the reference pulse and the beginning of the output pulse corresponds to the

time interval that the entire probe is contained in the medium.

From these arguments we see all but a tiny fraction (Vg/c) of the probe length and

energy have vanished when it is in the medium. To answer where the energy has gone,

we must return to our discussion in Subsection 3.2.1, where we saw time dependence

in the light fields causes a small amplitude in cA, which leads to coherent flow of

energy into between the light fields. In the case of USL, the probe energy is eaten

away as it enters and flows into the coupling field. The probe is then reconstructed

by taking energy back when it exits and is recorded by the PMT. Because the process

is completely coherent, it is reversible, and the probe is reconstructed perfectly in the

ideal limit (i.e. in the absence of absorption of probe photons T = 1).

The fundamental limit on Lp is determined by the smallest possible pulse temporal


width (τ0 ∼ τsp). This would give a pulse at the center of the cloud of:

L(min)p = 2Vgτsp =

4√D0

f13σ0n, (3.34)

which is 13 µm for the parameters we have been considering. Note the optical density

is important in determining this limit, because of the role it plays in determining the

bandwidth.

3.2.7 Resulting atomic amplitude evolution

We can use our solution for the probe field Ωp (3.27) to derive expressions for the

atomic amplitudes c2 and c3 (c1 is assumed to be unity). Doing this we will recover

the results obtained with our dark/absorbing state picture developed in Section 2.3.

Eq. (3.25) gives us the excited state amplitude c3. By invoking the approximations

and expansion in (3.29), plugging in (3.27), and taking the inverse FT, we get

c3(z, t) =2Ωp(z, t)

Ω2c0

[i

(t− τd)τ 20 + 4DW−2

c

− iγ − ∆2

], (3.35)

where Ωp is given by (3.31). This result is plotted in the center of the cloud (z = 0)

in Fig. 3.6(a), for the same parameters as Fig. 3.5. The first term is the result of

coherent exchanges between the ground states during the propagation and dominates

when τ−10 γ,∆2, as is often the case in practice. When this term dominates

c3 is purely imaginary and negative as the pulse arrives, then reverses sign as it

leaves. Note that this term changed from real to imaginary upon the inverse Fourier

transformation from the frequency to the time domain. The peak magnitude of c3

scales as ∼ τ−10 , as expected from our adiabaticity arguments in Sections 2.3 and


Figure 3.6: Atomic amplitude evolution in USL. (a) The purely imaginaryamplitude c3 (solid curve) (3.35) at the cloud center (z = 0). For reference, the(purely real) Ωp at z = 0 is plotted (dashed curve) in arbitrary units. The sign of c3is determined by whether the pulse is entering or leaving the region (which determinesthe direction of flow of atomic probability between |1〉 and |2〉). (b) The purely realc2 (solid curve) again compared with Ωp as in (a).

2.4. In this limit, our leading order losses are from non-adiabaticity and we can use

(2.6) to calculate the total fractional loss of the atomic population during the pulse

traversal:

Ploss = 2√π

(Ωp0

2

Ωc02

)1

Wcτ0, (3.36)

which is 5.7× 10−3 for the parameters in the Fig. 3.6. Note the last ratio is generally

always small in cases of good EIT (see Eq. 3.28). Dephasing and detuning will add a

small additional amplitude in the excited level and thus cause a small additional loss.

Substituting (3.25) and (3.27) back into the first equation of (3.24) and performing

the inverse FT we have:

c2(z, t) = − Ωp

Ωc0

[1 +

2

Wc

(t− τd

τ 20 + 4DW−2

c

− γ + i∆2

)]. (3.37)

In the limit that the term in brackets is unity, the internal atomic amplitudes follow

the dark state (2.9) as the time and space dependent probe passes through. Any


deviations from unity represent a non-zero amplitude cA.

In Chapter 2 we used adiabatic elimination arguments to derive the amplitudes cA

and c3 in the presence of non-adiabatic coupling, dephasing and two photon detuning.

We find the results derived here, which use the more rigorous Fourier analysis and

treat the light fields dynamically corroborate our earlier results. We do this by con-

verting (3.35) and (3.37) into the basis (2.10) and using Ω ≈ Ωc. Doing this, we see

the relation (2.12) between cA and c3, as well as relations (2.23-2.24) for the effects

of γ and ∆2, all hold.

We conclude from this calculation that USL can be used as a tool to drive atomic

samples into complicated time and space dependent superpositions of the ground

states. Since for typical parameters, the pulse can indeed be compressed completely

within the atomic cloud, this can lead to local excitations. However, there is a

limit on the length scale of these excitations (3.34). Another important conclusion

is that the dark state consists of a fractional population in |2〉 only on the order

of the (Ωp0/Ωc0)2 (as is evident in (3.37)), meaning we must enter the strong probe

regime in order for to induce a significant change in the atoms’ internal state. For

this reason, in Subsection 3.3.1 we will address the behavior when the weak probe

limit is violated. A method of circumventing both the weak probe and pulse length

restrictions, and therefore create short wavelength, large amplitude excitations, is to

use a light roadblock, which is the subject of Chapter 7.


3.2.8 Comparison with numerical results

In order to check the approximations we have made and explore beyond them, we

have developed a code which propagates the self consistent three-level equations (see

Subsection 3.2.2) numerically. In particular, it allowed us to check the validity of the

weak probe approximation (3.23) and the expansion of (3.29) α, valid for small probe

frequency widths. Both of these approximations were necessary in order to obtain

the closed form solution (3.31). Here we outline the assumptions which go into the

code and compare its results with our analytic solution for a few simple cases. The

numerical method is described in detail in Appendix D.

In actuality, the code propagates the Optical-Bloch Equations (OBEs) described

in Appendix A. They allow for feeding of spontaneous emission back into the system.

In the case of an open system (all spontaneous emission is to levels outside of the

system), the OBEs formally reduce to the amplitude equations (2.22) which we solved

above. Only one spatial dimension (z) is considered and the coupling beam is assumed

to be co-propagating with the probe.

In Fig. 3.7 we compare the results of numerical simulations and the analytic result

(3.31) for the same parameters. The dotted line is the reference (input) pulse. The

thick dashed line shows the analytic prediction with the delay τd in the ideal case

(γ = ∆p = 0) with no non-adiabatic losses or spreading (i.e. τsp is artificially set

to zero). The thin solid line then shows the analytic solution with the corrections

due to τsp included. The numerical result with these parameters is indistinguishable

from that curve, showing that (3.31) correctly predicts this spreading. The thick

solid line shows a case with a small detuning (∆2 = (2π)50 kHz) and dephasing


ts)

pp0

|

- 5 0 5 10 15 20 25

0.2

0.4

0.6

0.8

1

Figure 3.7: Comparison of numerical and analytic results for Ωp. We compareseveral cases of numerical propagation of Eqs. (2.4) with the analytic result (3.31).The dotted curve shows the reference pulse. The input probe, cloud, and couplingparameters are the same as in Fig. 3.5. The dashed curve shows the analytic resultwith τsp set to zero. The thin solid curve then shows the analytic result with γ = ∆2 =0 with the correct τsp = 1.0 µs, which is indistinguishable from the numerical resultwith γ = ∆2 = 0. The thick solid curve shows the analytic result with γ = (2π)1 kHz,and ∆2 = (2π)50 kHz, which is again indistinguishable from the numerical result forthese parameters. The only case where significant deviation from the analytic resultwas seen was when we made the one-photon detuning comparable to Γ. For example,our analytic solution cannot describe the case ∆c = ∆p = (2π)4 MHz (thin dashedcurve), because the expansion (3.29) is no longer valid. There we see visible distortionand spreading due to the nonlinear dispersion (see Fig. 3.3(b)).

(γ = (2π)1 kHz). The numerical result is again indistinguishable from the analytic

formula, showing the attenuation due to these effects is accurately described with

(3.31).

To check the expansion (3.29), which we used to derive (3.31), we varied γ and

the detunings ∆p,c over a wide range. Increasing γ, we saw that the transmission goes

to zero before there is any visible correction to the (3.31). The same thing happens

when we increased ∆2 = ∆p−∆c. In contrast, if we vary ∆p and ∆c together, keeping


∆2 = 0, we begin to see deviations from (3.31) when ∆p,∆c become comparable to

Γ. The thin dashed line in Fig. 3.7 shows a case where ∆p = ∆c = (2π)4 MHz (thin

dashed curve). As seen there, distortion and spreading occur due to χR no longer

being exactly linear at the point of transparency (see Fig. 3.3(b)).

The validity of and corrections to the weak probe approximation are discussed

below in Subsection 3.3.1.

3.3 Corrections to the ideal system

We have seen how USL allows probe pulses to propagate through an atomic cloud

with good transmission, significant delays and little distortion. But we have made

a number of assumptions in our model so far, and here we will explore two sources

of corrections to the model which are often relevant in experiments: nonlinear sus-

ceptibility (i.e. beyond the weak probe approximation), and the presence of a fourth

level.

3.3.1 Stronger probe regime and adiabatons

Slow light manipulates atoms by creating superpositions of the two ground states.

However, in the weak probe limit only a small fraction of atoms ∼ (Ωp0/Ωc0)2 ever

populate |2〉. The ultimate limit on the number of atoms we can coherently transfer is

thus dependent on two things: first, how high can we make (Ωp0/Ωc0)2 before the weak

probe approximation breaks down, and second, what happens when this breakdown

occurs. In this section we explore analytically the evolutions of c1 and Ωc (hitherto

considered constant) to find out the degree to which the probe changes them. The


deviations of Ωc from its input value Ωc0 are called adiabatons [59]. Sufficiently large

adiabatons act back and effect the probe propagation, giving rise to nonlinear effects.

We will then present some numerical results which show how they affect the pulse

delays, transmission, and shapes.

Under conditions of EIT, a negligible portion of the atoms occupy the excited

level and, in fact, a negligible fraction (given by 3.36)) are lost. We can then say

|c1|2 + |c2|2 ≈ 1 and conclude, using (3.37), that:

c1 ≈√

1 − |Ωp|2|Ωc0|2 ≈ 1 − |Ωp|2

2|Ωc0|2 , (3.38)

where the second equality is the result to second order in |Ωp| (also, one can easily

verify that c1 is purely real for ∆2 = 0). We expect this to slightly reduce the

susceptibility (3.26) by a factor |c1|2. The dependence of the susceptibility on |Ωp|2

thus introduces nonlinearity. We expect this lower susceptibility to cause a slightly

larger transmission and delay. However, we also must consider corrections to Ωc,

which are expected to be of the same order (see Eq. 3.23).

Obtaining an expression for Ωc to this order is more difficult. In general the

polarization governing the propagation of Ωp scales as c3c∗1, while Ωc sees a term c3c

∗2

(see (3.6)). Since |c2| ≈ |c1(Ωp/Ωc)|, the polarization governing the coupling beam

propagation vanishes where the probe is not present, and, in the weak probe limit,

is everywhere much smaller than the polarization governing the probe propagation.

This is the reason the coupling beam is approximately constant in the weak probe

limit.

We now calculate the polarization it does see. For simplicity, we will consider


the case where the two beams are co-propagating, although the basic ideas we will

derive here can be applied to other geometries and will in fact be discussed for the

orthogonal case in Chapter 4. Also, we will set ∆2 = 0 to simplify the algebra. The

propagation equation (3.6), plugging in our leading order solutions for the atomic

amplitudes (3.35,3.37) is:

∂

∂zΩc = −f23σ0n

|Ωp|2Ωc0

2

(t−W−1c D(z))

Wc(τ 20 +W−2

c D(z)), (3.39)

where we have set the speed of light in vacuum c→ ∞. We see that this polarization

on the RHS indeed scales with |Ωp|2.

The expression (3.39) must be integrated from where the coupling beam enters

(z = zin) to the spatial point of interest, z0. If we use our weak probe solution

(3.31), a simplifying approximation can be made when the density is slowly varying

compared to the pulse spatial length Lp (3.33). We are performing the integration at

a particular time t and at that t the probe will be centered about some position zt.

Then in a slowly varying region about zt, D(z) can be written as a linear function of

z: D(z) = D(zt) + (z − zt)σ0n. We then can write:

Ωp(z, t) ≈ τ0√τ 20 +W−2

c D(zt)Ωp0 exp

(−D(zt)

γ

Wc

)

×exp

−1

2

(t−W−1

c D(zt) − z−zt

Vg(zt)

)2

τ 20 +W−2

c D(zt)

. (3.40)

When this expression breaks down, far away from zt, Ωp vanishes due to the expo-

nential term so this introduces little error. We are essentially assuming the variation

of Ωp with z is primarily due to the group velocity, rather than non-adiabatic loss, or


the atomic density variation. With our approximate expression (3.40) and a similar

replacement for the other terms in the integrand (3.39), the integrand is seen to be

symmetric in z about zt. Then integrating from zin to z0 gives zero, if and only if

the point z0 is in a region where Ωp is nearly vanishing, i.e. clearly before or after

the region about zt. If, on the other hand, we stop the integration in a region of

non-vanishing Ωp, there is no symmetry and we get a contribution. Combining these

considerations we get an approximate solution:

Ωc(z0, t) = Ωc0 + Ωc01

2

f23f13

Ω2p0

Ω2c0

[exp

(− t

2

τ 20

)

− τ 20

τ 20 + 4DWc

−2 exp

(−2D

γ

Wc

)exp

(−(t−DWc

−1)2

τ 20 + 4DWc

−2

)], (3.41)

where D is evaluated at z0. From the first term in the brackets we see that the

coupling field increases from its initial value when the probe enters (near t = 0). The

photons which have temporarily disappeared from the probe field are in the coupling

beam, and lead to a small “hump”. Once there, these photons propagate nearly

freely through the cloud as the majority of the medium is simply |1〉 atoms. When

the probe pulse is exiting the region about z0 (which occurs at about t = D(z0)W−1c ),

the process is reversed and the coupling beam gives photons back, leading to a “dip”.

The extent to which the propagation is lossless is the extent to which the process

is completely reversible (and the dip and the hump are of equal magnitude). An

example of an adiabaton as a function of time at z0 = zout is shown in Fig. 3.8(a),

which compares (3.41) with numerical integration. The slight disagreement in the two

results for the dip is due to the somewhat strong probe causing errors in our weak

probe limit result (3.31). One sees in the figure that the adiabaton dip follows the


t s)

p,c|2

c0

2

z m)

a b

- 40 - 20 0 20 40

0.2

0.4

0.6

0.8

1

1.2

Figure 3.8: Adiabatons. Numerical result for the same parameters as Fig. 3.5except a larger probe Rabi frequency Ωp0 = (2π)4 MHz = Ωc0/2 so adiabatons arevisible. (a) The normalized coupling intensity |Ωc/Ωc0|2 (thick solid curve) at zoutfrom numerical propagation and the analytic result (3.41) (thin solid curve). Theprobe input (also normalized to Ω2

c0) (dotted curve) and output (dashed curve) areshown as well. (b) The spatial profile of the coupling and probe fields at two differenttimes. When the probe is entering (t = 0.77 µs, thin dashed curve), the coupling field(thin solid curve) is everywhere slightly higher than its input intensity. Once theprobe is inside the cloud (t = 6.69 µs, thick curves), the adiabaton spatial profile isthe negative of the probe profile.

numerically calculated probe profile quite well. In Fig. 3.8(b) we plot the adiabatons

as a function of space at two times. As the probe beam enters, one sees a small rise

everywhere in the coupling field intensity (due to the first term in the brackets in

(3.41)). At later times, once the probe is completely compressed within the medium,

Ωc = Ωc0 everywhere except for a dip which is the opposite of the probe intensity

profile (from the second term in the brackets).

The fact that the oscillator strength ratio enters in determining the magnitude

of the adiabatons leads to some interesting consequences. It enters in such a way

that the number of photons is conserved, rather than the quantity Ω2 = |Ωp|2 + |Ωc|2,

(which is proportional to the number of photons times the oscillator strengths). When

the probe is completely contained (so we can ignore the contribution of the hump):


|Ω(z, t)|2 = Ω2c0 + |Ωp(z, t)|2

(1 − f23f13

), (3.42)

so Ω2 is constant if and only if f13 = f23. Recall from Chapter 2 that it is Ω2 rather

than Ω2c0 which is determines the EIT preparation rate (and therefore the width of

the transmission profile, the group velocity, pulse length, etc.).

We now consider how this change in Ω2 acts back on the probe propagation.

In Fig.3.9(a) we plot the results of numerical simulations for probe output at the

PMT for a weak probe, as well as a strong probes with three different values of the

coupling oscillator strength (f23 = (2/3)f13, f23 = f13, and f23 = (4/3)f13). In the

case of equal oscillator strengths, the strong probe has the effect of spreading the pulse

(since the nonlinearity introduces dispersion in the group velocity) and to shorten the

delay (since Ω does not change while c1 decreases, leading to a net decrease in the

susceptibility). In contrast, when f23 > f13, Ω decreases according to (3.42). As a

result, the peak of |Ωp|2, where Ω2 is the smallest, lags behind the edges. This leads to

the distortion observed in the figure, where the peak of the pulse is more delayed than

the front and back edge. When f23 < f13 the converse happens and the distortion

bends the other way.

In Fig.3.9 (b) we plot the transmission as a function of (Ωp0/Ωc0)2 for the same

three oscillator strength cases. Overall, we see slightly larger transmission than the

weak probe limit (the horizonal line). As stated above, this is due to the smaller c1

giving a smaller susceptibility. Interestingly, the equal oscillator strength case sees the

largest increase in transmission. This is because in the other cases, the asymmetric

distortion introduces high frequency components which are more readily absorbed.


Figure 3.9: Strong probe propagation. (a) Numerically calculated Ωp outputfor a weak probe (thin solid curve) and stronger probes Ωp0 = Ωc0 = (2π)8 MHzfor three different cases. Other parameters are as in previous figures, with f13 =1/2 only now f23 takes on the values (1/2) (thick solid curve), (2/3) (short dashedcurve) and (1/3) (long dashed curve). The thin dotted curve is the reference pulse.(b) The integrated energy transmission T =

∫dt I(t)/

∫dt I0(t), where I(t) =

|Ωp(zout, t)|2, I0(t) = |Ω(in)p (t)|2 for f23 = 2/3 (solid squares), 1/2 (open squares),

and 1/3 (circles). In all plots, the horizontal line indicates the analytic result in

the weak probe limit. For the transmission, the analytic result is√

1 + τ 2sp/τ

20 . The

transmission is seen to increase with higher probe to coupling intensity. (c) Thedelays, defined as τd =

∫dt t I(t)/

∫dt I(t), are seen to decrease. (d) The spread,

wsp ≡√

(2∫dt (t− τd)2I(t))/(τ 2

0

∫dt I(t)) sees a significant increase. The analytic

result is√τ 20 + τ 2

sp/τ0 (e) The cube of the skew,∫dt(t− τd)3I(t)/(wspτ0)

3∫dtI(t),

a measure of the asymmetry, is zero for f23 = f13 and its sign in other cases dependson the relative size of the oscillator strengths.


In Fig. 3.9(c) we plot the delays, defined to be the mean of the intensity distribution

(see caption for the formula). In all cases, the delays are smaller than the weak probe

value, but the amount of decrease depends on f23/f13 in qualitative agreement with

(3.42). Fig. 3.9(d) shows the half widths of the output pulses, defined as the square

root second moment of the intensity, normalized to the input τ0. The horizontal line

shows the weak probe limit result wsp =√

1 + τ 2sp/τ

20 . In all cases we see additional

spreading at higher (Ωp0/Ωc0)2 due to nonlinear dispersion. Finally a dimensionless

measure of the asymmetric distortion, the skew, is shown in Fig. 3.9(e). It is defined

in the caption. The equal oscillator strength case is seen to have no skewness, while

the other two cases have opposite values of the skew due to asymmetric distortion in

opposite directions.

We conclude from these calculations that the qualitative features of USL and EIT

still hold up to values of (Ωp0/Ωc0)2 on the order of unity so it is possible to coherently

transfer a large fraction of the atoms. On the other hand, our analytic solutions for

the linear probe propagation begin to break down and it is no longer possible to

preserve the pulse shapes as they propagate.

After a sufficiently long propagation distance, the asymmetric distortion can cause

a severe sharpening of the front or back edge. This will continue until the edge has

a profile with a spatial width on the order of the wavelength of light, at which point

the slowly varying approximation breaks down. This could be an interesting way to

create and study sharp profiles in a light field or an atomic field.

The magnitude of all of all the effects are functions of optical density, as propaga-

tion through a large cloud will clearly cause more distortion for a given (Ωp0/Ωc0)2.


In fact, even a moderate nonlinearity can accumulate significant distortion after a

large optical density [67].

3.3.2 Considerations in four-level systems

To follow up on the effects of a fourth level discussed in Subsection 2.6.2 we

investigate here the consequences for the probe transmission. Consider again the

equations (2.26) (we will assume here, for simplicity that βp,c are real). Since there

are non-zero dipole matrix elements between the ground states and |4〉, this level will

contribute to the polarization of the medium. Carrying out the analogous calculation

to Subsection 3.1.2 we find that:

Ωp χ(4−lev) = −ρΓ (c3 + βpc4) . (3.43)

This is again a linear susceptibility, so we have worked in the weak probe limit and

set c1 ≈ 1. As in Section 3.2.4, we find it as a function of probe frequency ∆p + δ.

In Fig. 3.10(a) we plot χI(4−lev) very near the resonance, calculated according to this

method. We assume the presence of a fourth level with ∆4 = (2π)192 MHz, in

accordance with the F = 1 manifold on the D1 line of sodium. We show the lone

coupling (βp = 0), lone probe (βc = 0), symmetric (βp = βc = 1), and anti-symmetric

(βp = −βc = 1) cases. To demonstrate the AC Stark shift, the coupling beam has been

left on the bare resonance (∆c = 0) in each case. As expected, the anti-symmetric

case doubles the lone coupling case AC Stark shift, and the symmetric case brings it

back to zero. The normal three-level case (βp = βc = 0) is indistinguishable on the

plots from the symmetric case.


Figure 3.10: Susceptibility in the presence of a fourth level. Susceptibility infour-level atoms, for the same parameters as previous plots but with a level |4〉 with∆4 = (2π)192 MHz. The oscillator strengths are f13 = 1/12 and f23 = 1/4. Theseparameters are consistent with the D1 line of sodium. The calculations are based onthe FT, weak probe (c1 = 1) solution of Eq. (2.26), and (3.43). (a) χI(4−lev) in thevicinity of the EIT resonance. The solid curve shows the symmetric case βp = βc = 1,which is indistinguishable from the normal three-level case (βp = βc = 0). The loneprobe case, βp = 1, βc = 0 (thick dotted curve) sees a small non-zero χI(4−lev) on

resonance, in agreement with (3.44). The lone coupling case, βp = 0, βc = 1, (thindotted curve) has the same increase, but also an AC Stark shift, in agreement with(2.28). The anti-symmetric case βp = 1, βc = −1,(thick dashed curve) sees a minimumin χI(4−lev) four times as large as the lone cases, and an AC Stark shift twice as large

as the lone coupling case, in agreement with (3.44). (b) χR(4−lev) in the same vicinity.The zero-crossings are Stark shifted, but the slope, and therefore Vg, are not affectedto the order we are considering.

The smallest absorption coefficient occurs when χI is minimized. When ∆c = 0,

this minimum occurs at the frequency ∆p+δ corresponding the AC Stark shift. Using

our results of Subsection 2.6.2 for the most general case (both ground states coupled

to |4〉), we find, using (2.33) and (3.43):

(χI(4−lev))(min) = ρ

Γ2

4∆24

(β2c + β2

p − 2βpβc), (3.44)

which indeed vanishes in the symmetric case (βp = βc). The imaginary susceptibility

with a lone probe or a lone coupling transition to |4〉 are seen to be of the same

magnitude. In the anti-symmetric case (βp = −βc) one sees that the imaginary


susceptibility is four times as high as the two lone cases.

In all cases where a fourth level is the dominant contribution to loss of trans-

mission, T = exp(−D0(χI(4−lev))

(min)/ρ) gives the transmission at the Stark shifted

resonance and therefore:

Dmax =

((χI(4−lev))

(min)

ρ

)−1

(3.45)

provides a convenient estimate of the optical density D0 one can propagate through

before (1/e) attenuation of the probe occurs. This maximum is independent of either

light field intensity.

In Fig. 3.10(b) we plot χR(4−lev) for the same cases. We see that while the fourth

level induces small shifts, it does not affect the shape or slope near the resonance so,

to leading order, the group velocity is not changed.

We have also confirmed numerically that varying ∆c on scales much less than the

natural linewidth Γ simply shifts all the profiles χR,I(4−lev) such that it is the two-photon

detuning which is important.

3.4 Ultra-slow light experiments

3.4.1 Observation of ultra-slow light

We first performed USL experiments with an orthogonal geometry on the D2 line

[1]. The experiments were carried out on cold atom clouds in a 4-Dee magnetic trap

[39, 68], both above and below the transition temperature for BEC. The probe was

negatively circularly polarized (εp = (x − iy)/√2) and propagated along z and the


coupling beam linearly polarized (εc = z) and propagated along w = x. Fig. 3.11

shows an experimental schematic.

Details of the atomic trapping and cooling scheme can be found in [39, 68]. Briefly,

sodium atoms were evaporated in a candlestick source [69] and into a Zeeman slower

[21]. The Zeeman slower loaded a Magneto-Optic Trap (MOT) [22] which collected

∼ 1010 atoms and cooled them to 250 µK. Polarization gradient cooling [23, 24] further

cooled to 50 µK. The final step was to load them into the 4-Dee [39] trap and use

evaporative cooling [25] to bring the atoms to the desired temperature. The 4-Dee

magnets produce a harmonic potential with frequencies ωz = (2π)28 Hz, ωx = ωy =

15ωz, leading to a long cigar shaped atomic cloud. By varying the final cut of the

evaporative cooling, we created clouds both above and below the critical temperature

for condensation. Condensation was observed at 435 nK when the cloud contained

about 6×106 atoms. Further cooling to about half this temperature, resulted in very

pure (∼ 90% condensate fraction [70]) condensates of about 2 × 106 atoms. After

the cooling was completed, the magnetic fields were adiabatically ramped to a looser

trap with ωz = (2π)21 Hz, ωx = ωy = 3.3ωz, and the magnets produced an 11 G

bias field in the z direction, which defined the quantization axis for the probe and

coupling fields.

In Fig. 3.11(a), the small white dotted line on the CCD image indicates a 15 µm

region in the center of the cloud which was imaged onto the PMT. By choosing this

imaging region to be substantially smaller than the width of the cloud, we observed

only probe light which passed through near the center column (x = y = 0) of the

atomic cloud, minimizing the effects of the variation in the optical density of the


Figure 3.11: Diagram of first USL experiment. Figure taken from [1]. (a)A linearly polarized (εc = z) coupling beam propagates along x and a circularlypolarized probe pulse (εp = (x − iy)/√2) is input along z. The trapping magnetscreate an 11 G bias field along z. With a flipper mirror in front of the camera CCD 1,we direct this probe beam either to the camera or the photomultiplier tube (PMT).The pulse delays are measured with the PMT. For these we place a pinhole in anexternal image plan of the imaging optics and select a small area, 15 µm in diameter,of the probe beam centered on the atom clouds (as indicated by the dashed circle ininset (i)). The imaging beam propagating along the y axis is used to image the atomclouds onto CCD 2 to find the dimensions and densities of the atom clouds. Inset (ii)shows atoms cooled to 450 nK, which is 15 nK above Tc (Note that this imaging beamis never applied at the same time as the probe and coupling lasers). The position ofthe cloud and its diameter in the two transverse directions x and y are found withCCD 1. Inset (i) shows an image of a condensate. (b) Energy level schematic of therelevant levels.


Figure 3.12: Observation of Ultra slow light. Figure taken from [1]. (a) Pulsedelay measurement. The open circles are the reference pulse, with τ0 = 1.8 µs andΩp0 = (2π)2.0 MHz. The filled circles show the output recorded at the PMT. Thecurves show Gaussian fits, which are used to derive the delay τd = 7.05 ± 0.05µs. Anupper bound on the group velocity Vg = 32.5 ± 0, 5 m/s is inferred by the measuredlength of the cloud, 229 ± 3µm. The coupling input Rabi frequency here was Ωc0 =(2π)5.6 MHz. (b) Light speed versus atom cloud temperature. The speed decreaseswith temperature due to the atom density increase. The open circles are for a couplingintensity of 52 mW cm−2, corresponding to Ωc0 = (2π)12.0 MHz, and the closed circles12 mW cm−2, corresponding to Ωc0 = (2π)5.6 MHz. There is a sharp decrease in thegroup velocity observed at Tc, indicated by the vertical line, as the peak atom densitygreatly increases at this point. The atom density at 200 nK was 8 × 1013cm−3 andhere the condensate contained 60% of the atoms. The dephasing due to |4〉 preventedpropagation through the most dense clouds.

cloud at different x, y. A typical example of a delayed pulse in a thermal cloud is

shown in Fig. 3.12(a). The delay here is τd ≈ 7 µs, corresponding to a group velocity

of Vg = 32.5 m/s at the cloud center. Note also the complete separation between

reference and output pulse. During the period between the end of the reference pulse

and the beginning of the output pulse, the probe is completely contained inside the

medium. The reason for the finite attenuation of the pulse is coupling to |4〉, as

discussed below. In the case shown the optical density is D0 = 63, meaning that the


transmission without the coupling beam present would be e−63.

In Fig. 3.12(b) we show data taken at a series of temperatures and two different

coupling intensities. From the difference in the two data sets, one sees that Vg scales

approximately linearly with coupling intensity as expected. Lowering the temperature

of the cloud increases density at the center, leading to the observed reduction in Vg

at colder temperatures (see Eq. 3.30). The vertical line at T = 435 nK indicates

the observed condensate critical temperature. When the condensate forms, there is

a dramatic increase in the central density, leading to the large observed decrease in

the group velocities. At the lowest temperature data point, T = 80 nK, we observed

Vg = 17 m/s.

3.4.2 Measurement of dephasing

In Fig. 3.12(a), the pulse has been attenuated at the PMT, indicating a non-zero

dephasing (γ = 0). There is a non-zero coupling between |2〉 and |4〉 (see Fig. 3.11(b))

and, as discussed in Sections 2.6 and 3.3.2, this gives rise to a dephasing. Experimen-

tally, we can easily accurately estimate the dephasing using a method developed by

Kasapi, et al. [71]. If the beams satisfy two-photon resonance (∆2 = 0) and the pulse

is chosen to be sufficiently wide (τ0 τsp), then comparing the expected transmission

(3.32) with the expected delay gives:

−lnT

2τd= γexp . (3.46)

Thus measuring the delay and transmission of the pulse provides an experimental

measurement of the dephasing independent of the coupling intensity Wc or D0. In


the experiment [1] we measured a minimum dephasing rate on the order of γexp =

(2π)20 kHz, for the Ωc0 = (2π)5.6 MHz series. Assuming the coupling to |4〉 is the

dominant dephasing mechanism, Eq. (2.28) predicts a dephasing γ4 = (2π)43 kHz.

Thus the observed dephasing is of the right order of magnitude. The reason for the

disagreement (a factor of 2) is not clear. We did see the roughly correct scaling

γexp ∝ Ω2c0 as the measured dephasing for the Ωc0 = (2π)12.0 MHz series was γexp =

(2π)60 kHz.

In the more general case, we note from (3.32) that when the two-photon detuning

(from ∆2) or pulse spreading effects (from τsp) are comparable to the attenuation due

to γ, the experimentally observed dephasing is expected to be:

γexp = γ +2∆2

Wc

+1

2τdln

√1 +τ 2sp

τ 20

. (3.47)

So when γ is larger than the latter two terms, γexp is independent of Wc and when

the second and third terms contribute significantly, the observed dephasing increases

with lower Wc. In practice, inhomogeneous frequency shifts in the atomic cloud con-

tribute an effective average ∆2 seen by a probe. This issue is examined in Chapter 4.

The scaling of γexp with Wc can thus be a diagnostic as to possible origins of pulse

attenuation.

3.4.3 Experiments on the D1 line

From the arguments in Subsection 3.3.2 (see Eq. 3.45) we expected to be unable to

get through an optical density larger than Dmax = 288 with the set-up just discussed,

due to the coupling to |4〉. The large attenuation motivated us to set up a new


laser system on the D1 line (see Fig. 2.4), where the F = 3 manifold is not present.

In [4], experiments were carried in the orthogonal set-up, with the same levels as

just described, only on the D1 line. In that case, we expect no contribution from

off-resonant levels, and to be limited only by bad polarizations and inhomogeneous

frequency shifts.

In [2], we used a co-propagating geometry, with |2〉 = |3S1/2, F = 2,mF = +1〉

and |3〉 = |3P1/2, F = 2,mF = 0〉, again on the D1 line. Here we present some

more detailed, previously unpublished, experiments carried out in this co-propagating

geometry. They confirm some of the weak probe limit results (in particular (Eq. 3.31))

derived in this chapter.

An example of an USL experiment in this geometry is shown in Fig. 3.13. The

coupling beam Rabi frequency, Ωc0 = (2π)2.0 MHz, was lower than the initial exper-

iments. As a result, we get a longer delay, τd = 11.2 µs, and a group velocity on the

order of Vg ∼ 10 m/s. The smaller coupling intensity implies we must use a longer

pulse to avoid spreading and attenuation (i.e, keep τ0 > τsp in (3.31)).

The closest dephasing level in this geometry is due to the |4〉 = |3P1/2, F =

1,mF = 0〉, which is ∆4 = (2π)192 MHz away. Because of it, we have a double-Λ,

anti-symmetric system, so the increase in susceptibility, and therefore the observed

(χI4−lev)(min) is multiplied by 4 relative to the lone coupling case (see Subsection 3.3.2).

We would then expect γexp = (Ω2c0/∆

24)(Γ/2). The maximum theoretical optical

density which we can propagate due to this |4〉 level is, according to (3.45), Dmax =

4400.

We also measured the coupling field output on a separate PMT. The result, shown


0 20 40 60 80time( s)

0

0.2

0.4

0.6

0.8

1

1.2

1.4T

ransm

issio

n

Figure 3.13: Observation of USL and adiabatons on the D1 line. The exper-imental data shows the probe intensity at the PMT, normalized to the input peakvalue and the coupling intensity at the PMT, normalized to its input value. The Rabifrequencies were Ωc0 = (2π)10.6 MHz and Ωp0 = Ωc0/6. The cloud had an opticaldensity D0 = 300. A Gaussian fit to the input pulse is shown as a dotted curve andthe solid line is a Gaussian fit to the output data. The fits are used to derive a delayof τd = 11.2 µs. The measured transmission was T = 46%. These give a dephasingγexp = (2π)5.5 kHz. The coupling intensity shows the signature hump and dip ofadiabatons.

in Fig. 3.13, shows an experimental observation of adiabatons, in accordance with our

expectation in (3.41). A hump is observed when the probe pulse is input, and a dip

occurs as the probe pulse exits and registers on the PMT.

We carried out the experiment over a broad range of coupling intensities. A

summary of the results is presented in Fig. 3.14. In each plot, experimental results

are presented as filled circles. For comparison the open circles represent theoreti-

cal expectations based on (3.31) (with γexp chosen to be as just described). These

predictions use the experimentally measured D0 (from CCD pictures) and measured

coupling intensities Wc for each data point, and have no fitting parameters (the shot-

shot variation of D0 was responsible for the fluctuations in the theory points). When

these fluctuations are taken into account, the predictions for transmission, delays,

spreading, and dephasings are all in good agreement with the experimental values.


The figure shows that delays as high as 175 µs and dephasings significantly below

γexp = (2π)1 kHz were observed. The lowest observed group velocity was 50 cm/s.

The only significant disagreement between theory and experiment occurred when

Ωc0 ≤ (2π)1 MHz, (Wc ≤ (2π)0.1 MHz), where the transmission was less than ex-

pected. At this point, finite frequency effects in the atoms, which we discuss in the

next chapter, probably began to play a significant role, and the more complicated

expression (3.47) is appropriate.

Finally, we have explored the weak probe limit experimentally by varying the

probe to coupling intensity ratio. In contrast to the expected increase in transmission

with stronger probe strengths (see Fig. 3.9), we found that when Ω2p0/Ω

2c0 ≥ 0.1

we observed reduced transmission. We note that the number of photons and atoms

scattered into states outside the Λ system via spontaneous emission scales with this

parameter. These scattered photons and atoms in other levels could act back on the

fields and atoms still in the system [66]. We have done some numerical modelling

of this based on the full sodium structure (see Fig. 2.4) and found that these effects

can indeed reduce the transmission, but this discussion goes beyond the scope of this

thesis.


Figure 3.14: Comparison of experimental and theoretical pulse delay resultson the D1 line. We plot various experimentally observed values for a range ofcoupling intensities Wc = Ω2

c0/Γ. In each case, filled circles represent experimentalobservations, and the open circles theoretical predictions based on (3.31), calculatedwith measured values of D0 andWc. (a) Delays, obtained from Gaussian fits, are seento be in good agreement with theory and are observed as high as 175 µs, correspondingto Vg < 1 m/s. (b) The transmission is also in good agreement with theory based onspreading effects and γexp due to |4〉. Disagreement occurs at the lowest values of Wc

where we expect atomic dynamics may begin to play a role. (c) The fitted widths ofthe pulses, relative to the input pulse widths. (d) Dephasings γexp, calculated with(3.46) as low as (2π)400 Hz are observed, compared with the theoretical predictionfor γexp described in the text.

Chapter 4

Ultra-slow light in Bose-Einstein

condensates

We first discussed the evolution of internal atomic levels in the presence of light

fields, and then added the propagation of the light fields in the presence of the atoms.

We now add the third, and final, piece of our formalism: The external motion of

the atoms due to momentum kicks, trapping fields, and atom-atom interactions. We

specifically include these effects assuming the atoms are Bose condensed alkali atoms

[5, 26].

In experiments with atomic BECs, the Gross-Pitaevskii (GP) [18] equation has

been very successful in describing the dynamics [5, 27]. Our strategy will be to use

a second quantized Hamiltonian, described in Section 4.1, which includes external

atomic dynamics and the atom-light interactions discussed up to now, and use it

to derive the corresponding Gross-Pitaevskii (GP) equations. We will still treat the

light fields classically, and continue to describe their propagation according to the

102

Chapter 4: Ultra-slow light in Bose-Einstein condensates 103

SVE Maxwell equations in Chapter 3, but will have to update the notation to the

present formalism. The resulting set of self-consistent equations will form the basis for

a numerical code which we apply to problems throughout the remainder of this thesis.

In the limit that the light field coupling terms dominate the external dynamics we will

see a direct correspondence between the formalism developed here and the results of

Chapter 3. The external dynamics can then be viewed as a small perturbation on our

previous results. In the opposite limit, where the light fields are off or very weak, the

formalism reduces to two coupled GP equations (one for each of the ground states |1〉

and |2〉), a limit which will be relevant later in the thesis, when we discuss experiments

where the input coupling field is varied in time and space, even down to zero intensity.

In Section 4.2, we study these equations in the weak probe limit. Many of the

results of Chapter 3 are relevant here. In particular, the entire BEC evolves approx-

imately into the time and space dependent dark state. The results are compared

with USL experiments we have done in BECs. We then use the formalism to derive

how the external dynamics give the atoms inhomogeneous frequency shifts, leading

to non-zero two-photon detunings. Physically, these shifts arise from Zeeman shifts

from the trap, inhomogeneous broadenings due to mean field atom-atom repulsion,

and Doppler shifts due to non-zero momentum. Our model also includes losses due

to inelastic and high momentum elastic collisions. Due to these shifts and losses,

dropping Ωc0 below a critical value destroys the transmission, which in effect puts

a lower limit on achievable group velocities. We also look at how the light induced

momentum recoil results in a “moving dark state” which can act back to reduce the

probe propagation and increase the group velocities.


Further analytic analysis of our formalism is performed in Section 4.3 to show

how the dark/absorbing picture can be applied even in the strong probe limit. In

this way, it is seen how the light fields can be used to drive macroscopic excitations

in the condensate. We will develop an analysis based on energy flow and free energy

in the system, concepts which will be used in Chapter 8, when we discuss how this

compression of the probe pulse results in local topological defects.

4.1 Formalism

Here we outline our derivation of the coupled GP equations, starting from a second

quantized Hamiltonian (Subsections 4.1.1 and 4.1.2). These equations are then used

to calculate the ground state of the system, which we take to be our initial conditions

(Subsection 4.1.3). At this point the correspondence to our formalism in Chapter 3

will be clear and discussed in Subsection 4.1.4. Adiabatic elimination of the excited

state |3〉 is then used in Subsection 4.1.5 to reduce the system to two atomic field

and two light field equations, which form the formalism that the remainder of the

thesis is based on. These equations are the basis of a numerical code which we use to

analyze the problem. The numerical technique is outlined in Appendix E. To help

put the various contributions to atomic dynamics that we consider in perspective, we

outline the relevant parameters for the various experimental schemes and geometries

presented in this thesis (Subsection 4.1.6).


4.1.1 Second-quantized Hamiltonian

We formally describe our system with a second quantized, semi-classical Hamilto-

nian, in which the atoms are described by atomic field operators, Ψi(R, t) (Ψ†i (R, t)),

representing annihilation (creation) of an atom in internal state i at a position R.

The Hamiltonian is similar to the one used in [61] and consists of four contributions:

H = H(0) + H(TV ) + H(int) + H(dip), (4.1)

with the first term H(0) describing the internal energy due to the electron-nucleus

interaction:

H(0) =∑

i=1,2,3

∫dR ωi Ψ†

i (R, t)Ψi(R, t), (4.2)

The second term H(TV ) describes the external dynamics due to kinetic energy and

the magnetic trap:

H(TV ) =∑

i=1,2,3

∫dR Ψ†

i (R, t)

(−

2∇2

2m+ Vi(R)

)Ψi(R, t), (4.3)

where m is the atomic mass. We trap the condensates in |1〉 in a cylindrically

symmetric harmonic trap V1(R) = m(ω2r(x2 + y2) + ω2

zz2)/2. We will eliminate

V3(R) from the problem later, but V2(R) will play an important role. Generally

V2(R) = (1+α(Z))V1(R), with the constant (1+α(Z)) being the ratio of the magnetic

moments of states |2〉 and |1〉 (thus α(Z) parameterizes the fractional difference in

magnetic moments). The third term H(int) describes atom-atom interactions. Densi-

ties are sufficiently weak that atom-atom interactions can be described by two-body


collisions, and the extremely cold temperatures in BECs allow us to describe the

collisions in the s-wave scattering limit [72], replacing the two-body potential with a

point like interaction:

H(int) =∑

i,j=1,2,3

1

2Uij

∫dR Ψ†

i (R, t)Ψ†j(R, t)Ψj(R, t)Ψi(R, t). (4.4)

The interaction strengths are Uij = 4π2Naij/m, where aij are the scattering lengths,

and N is the total initial number of particles. For our trapped state in sodium

a11 = 2.75 nm [73]. Other ground state-ground state scattering lengths are of the

same order of magnitude and have been calculated [74]. We allow for complex aij, with

the imaginary parts representing inelastic collisions which cause atoms to populate

internal levels outside our three-level description. However Ima11 = 0 (on the time

scales of ∼ 100 ms, which we are considering) can be assumed since otherwise it would

be impossible to trap and create the BEC in |1〉. The scattering lengths involving

the excited state are generally larger and less precisely known but later we eliminate

them from the problem. The atom-light field interaction term H(dip) is done in the

same semi-classical dipole approximation as in Chapter 2, and again uses the RWA

to eliminate far off-resonant terms:

H(dip) = 12∫dR

[(Ωp(R, t)e

i(kpz−ωpt)Ψ†3(R, t)Ψ1(R, t) + h.c.

)+(

Ωc(R, t)ei(kcw−ωct)Ψ†

3(R, t)Ψ2(R, t) + h.c.)

− iΓΨ†3(R, t)Ψ3(R, t)

]. (4.5)

We are keeping the same convention of introducing a phenomenological decay Γ of


the excited state which causes loss from the system. It, along with imaginary parts

of aij, make the Hamiltonian H non-Hermitian.

We work in a Heisenberg picture, where the time evolution occurs in the atomic

fields according to [75]:

i∂Ψi(R, t)

∂t= [Ψi(R, t),H] (4.6)

and the fields satisfy the bosonic commutation relations [75]:

[Ψi(R, t), Ψ†i (R

′, t)] = δ(R−R′), [Ψi(R, t), Ψi(R′, t)] = 0. (4.7)

4.1.2 Gross-Pitaevskii equations

The second quantized Hamiltonian can be used to derive the coupled Gross-

Pitaevskii (GP) equations, which govern the evolution of the mean fields. We proceed

by writing each field operator as the sum of its expectation value (a c-number) plus

fluctuations:

Ψi(R, t) = 〈Ψi(R, t)〉 + δΨi(R, t), (4.8)

where 〈· · · 〉 indicates the quantum expectation value in the exact many particle state

[75]. The c-number expectation value component is known as the mean field and

has no operator character. The GP equations are an approximate method which

calculates only the evolution of the mean fields (rather than the full operators) and

only in terms of the mean fields themselves. Their validity then depends on the mean

fields dominating the fluctuations. We need two assumptions for this to be the case.


First the system must be weakly interacting, specifically (na3ij)1/2 1 where n is

the atomic density. Generally (na3ij)1/2 ∼ 10−3 for our experiments. This parameter

determines the fraction of atoms which occupy excited states at zero temperature

due to interactions [5, 27]. Second, to neglect thermal excitations, the temperature

must be significantly below transition temperature for Bose condensation (T Tc),

which is typically several hundred nano-Kelvin. For harmonic traps, the condensate

fraction is 1 − (T/Tc)3 [70] so we get a condensate fraction of ∼ 90% at half the

transition temperature, the regime in which we typically operate. When both of these

assumptions hold then initially almost all the atoms condense into the single particle

ground state. Then we describe the atomic operator with the mean field, which

essentially becomes a c-number wave function, neglect the fluctuations completely,

and calculate the ensuing dynamics.

As an aside we mention that, strictly speaking, there is an uncertainly relation

between number of particles and phase, making it impossible for there to truly be a

non-zero expectation value 〈Ψi〉 for a system containing an well defined total number

of particles N . However, in the limit of large N it can be shown that giving the

expectation value (and therefore the mean field) a well defined phase and number

introduces errors only of order 1/√N [19, 76].

We define our mean fields with the quickly varying phases (due to the probe and

coupling photon energies and momentum kicks) transformed away:


ψ(R, t) ≡

〈Ψ1(R, t)〉eiω1t

〈Ψ2(R, t)〉ei(−kpz+kcw+ωpt−ωct+ω1t)

〈Ψ3(R, t)〉ei(−kpz+ωpt+ω1t)

. (4.9)

(Recall w refers to the coupling field propagation direction, whether it be x,+z, or

−z.) Thus the total condensate wave function is a three component spinor. We

will refer to the three components as ψ1, ψ2, ψ3, respectively. Writing the Heisenberg

equations (4.6) with the Hamiltonian (4.1), and using the commutation relations

(4.7) yields the equation of motion for the atomic field operators. We then take the

expectation value of the equation and, using the definition in (4.9), find the equation

governing the evolution of the wave functions:

i∂

∂tψ(R, t) = HGPψ(R, t);

HGP =

H1 0 12Ωp

∗

0 H2 − ∆212Ωc

∗

12Ωp

12Ωc −i1

2Γ − ∆p + H3

. (4.10)

In this form, we see the equations are similar to the atomic amplitude evolution

equations (2.4), with the difference being the appearance of additional terms Hi on

the diagonals. They are defined as:


H1 ≡ −2∇2

2m+ V1 +

∑i=1,2,3

U1i |ψi|2 ,

H2 ≡ −2(∇2 + 2ik2 · ∇)

2m+ V2 +

∑i=1,2,3

U2i |ψi|2 ,

H3 ≡ −2(∇2 + 2ikp

∂∂z

)

2m+ V3 +

∑i=1,2,3

U3i |ψi|2 , (4.11)

where k2 ≡ kpz − kcw is the difference in the wavevectors of the beams. It appears

because in (4.9) we transformed away the quickly varying phases induced by photon

absorption and emission. When we propagate the equations (4.11), the term with k2

will cause ψ2 to have an overall velocity equal to the two-photon kick Vrecoil ≡ k2/m.

There is an analogous effect for ψ3. Due to this transformation, we also introduce the

(constant) recoil energy shifts in our detunings:

∆p ≡ ωp − (ω3 − ω1) −k2

p

2m, ∆2 ≡ (ωp − ωc) − (ω2 − ω1) − k2

2

2m. (4.12)

We see in (4.11) that in the mean field description, the atomic interactions give rise

to (possibly inhomogeneous) energy shifts proportional to the density in each state,

N |ψi|2. Different trapping potentials (V2 = V1) can also contribute inhomogeneous

energy shifts.

It will be useful later, when considering excitations created by the light fields, to

calculate various contributions to the energy 〈H〉 in the system. Calculating this in

our mean field description means simply replacing the field operators in Eqs. (4.2-4.5)

by their mean fields (Ψi → ψi). When we consider the energy we will generally be

focusing on the motion due to external dynamics. We will denote the kinetic and


trap contributions to the energy by E(T )i and E

(V )i , respectively, and the interaction

energy from |1〉 ↔ |1〉, |2〉 ↔ |2〉, and |1〉 ↔ |2〉 interactions by E(int)11 , E

(int)22 , and

E(int)12 , respectively. The total energy from these external contributions is then:

E ≡ E(T )1 + E

(T )2 + E

(V )1 + E

(V )2 + E

(int)11 + E

(int)12 + E

(int)22 . (4.13)

Before proceeding, we note that, very often in the analysis that follows, we directly

compare energy contributions with Rabi frequencies, detunings, and other frequencies.

Therefore to simplify notation through this chapter (and later), we will always use

the convention that if a quantity, f , has units of energy, then

f ≡ f. (4.14)

4.1.3 The initial ground state and the Thomas-Fermi approx-

imation

Our initial conditions in previous chapters consisted of putting all the atoms in

|1〉. As we now have external degrees of freedom, we take it to be all atoms occupying

|1〉 in the single particle ground state, since the condensates are always created by

cooling atoms into the ground state [5]. The wavefunction associated with this state

is found by varying ψ1 in order to minimize the energy E (4.13) [77], subject to the

constraint that:

∫dR |ψ1(R, 0)|2 = 1,

∫dR |ψ2(R, 0)|2 = 0. (4.15)


The constraints lead to the introduction of a Lagrange multiplier, µ, which is the

chemical potential of the system. The problem becomes one of solving the eigenvalue

problem [27]:

(−

2∇2

2m+ V1 + U11 |ψ1|2

)ψ1 = µψ1. (4.16)

This is also known as the time-independent Gross-Pitaevskii equation. It can be

obtained from (4.10) with the ansantz ψ1(R, t) = ψ(0)1 (R)exp(−iµt), so the dynamics

in the ground state simply consist of a homogenous phase rotation of an otherwise

steady state wave function. In numerical simulations, we always find the ground state

by propagating the time-dependent GP equation in imaginary time (t → −it) and

constantly renormalizing to satisfy (4.15) [78]. The numerical technique for solving

the GP equation is outlined in Appendix E.

To do analytic calculations, the Thomas-Fermi (TF) approximation, which com-

pletely neglects the kinetic energy term in (4.16), can be applied for BECs with a

sufficiently large N [79]. In a harmonic trap with weak interactions, or very small

N , the kinetic energy term would favor a larger wave function size (smaller spatial

gradients) while the trap term V1 favors a smaller size. The ground state is then

determined by a balance of the two (Virial’s theorem). When interactions U11|ψ1|2

become stronger, the repulsive interactions also favor a larger size. As the interac-

tions become stronger (through larger N or a11), this term dominates the kinetic

energy and the ground state is simply a balance between the interactions and the

trap. Neglecting the kinetic energy in (4.16) immediately yields:


ψ(0)1 (R) = Max

√µ− V1(R)

U11

, 0

(4.17)

For a harmonic trap this gives an inverted parabola for the density. The peak density

is determined by the normalization constraint (4.15), which fixes µ. This approxi-

mation can be applied in any number of dimensions. A three dimensional example

is shown in Fig. 4.1, along with the exact numerical solution obtained from imagi-

nary time propagation. The density profiles of the numerical and TF results agree

well in this case. The only visible deviation in the density comes near the border

of the condensate, where kinetic energy as determined from the TF approximation

diverges. The chemical potential µ, as determined by the 3D TF approximation, is

µTF = (2π)1.26 kHz and disagrees with the numerical result by only 0.04%.

The parameters in Fig. 4.1 are typical of BECs in experiments throughout this

thesis. It is for a N = 1.2 × 106 atom condensate in a 3D harmonic trap, with

ωz = (2π)21 Hz and ωx = ωy = 3.8 ωz. The density profile has half widths (half

max) of σz = 35.3 µm and σx,y = 9.3 µm, and a peak density N |ψ1(R = 0)|2 =

8.3 × 1013 cm−3. In the TF limit, the aspect ratio of the condensate σx,y/σz is equal

to the ratio of the trapping frequencies ωz/ωx,y.

In the TF limit the ground state energy is Eg = N(5/7)µTF , with the contri-

butions distributed according to E(V )1 = N(3/7)µTF and E

(int)11 = N(2/7)µTF . Nu-

merical simulations reveal these results hold quite accurately, with the kinetic energy

contributing only E(T )1 = 0.004Eg. A simple criteria to judge the validity of the

TF model is to compare the harmonic oscillator lengths a(HO)j =

√/2mωj (the

size of the ground state in the absence of interactions) with the length scale set by


Figure 4.1: Ground state condensate and the Thomas-Fermi approxima-tion.(a) Density profile in the y = 0 plane of the numerically calculated ground stateof a condensate with N = 1.2×106 sodium atoms, and in a trap with ωz = (2π)21 Hzand ωx = ωy = 3.8 ωz. The scattering length is a11 = 2.75 nm. The dotted linesindicate the cuts shown in (b) and (c). (b) A density profile of a cut along thex = y = 0 line. For comparison, the TF approximation result (4.17) is shown asa dotted curve. (c) A density profile along y = z = 0. The trapping potential istighter in this dimension and so there is a small but perceptible difference betweenthe numerical and TF results near the condensate border, due to the kinetic energyof the TF approximation diverging in this region.

the interactions Na11. The ratio (Na11/(a(HO)x a

(HO)y a

(HO)z )1/3) is the approximate ra-

tio of interaction to kinetic energy and whenever it is much larger than unity, the

TF approximation is good [27]. The kinetic energy eventually becomes an impor-

tant contribution near the boundary of the condensate. This region extends over

a width ∼ dj ≡ (σj/√

2)(a(HO)j /σj)

4/3, j = x, y, z [79]. This ratio also determines

the relative size of the TF wave function relative to the harmonic oscillator length:

σj/a(HO)j = (15Na11/a

(HO)j )1/5.

As mentioned above, in the ground state the external dynamics simply cause a

homogenous phase evolution of ψ1, corresponding to an energy shift of level |1〉 of µ.

This shift can be compensated by simply changing one of the light field so ∆2 = µ.

It is only when the atoms are significantly excited with the light fields that the Hi

contribute in an inhomogeneous and/or time-dependent fashion.

Often in numerical simulations, we capture the essential dynamics by calculating


only in the probe propagation dimension (z). In such a case, we must model the

condensate as occupying a cylinder of cross sectional area A with the wave functions

assumed to be constant in the radial directions, x and y. The numerically calcu-

lated density profile along z for a 1D condensate calculated with A = π(8.3 µm)2 is

indistinguishable from the 3D one plotted in Fig. 4.1. This A is chosen such that

µ (or equivalently the density at R = 0) mimics the 3D case. In this 1D case, we

get σz = 35.4 µm. The same technique can be applied to a 2D calculation. The

2D simulations will be particularly useful in modelling the effects associated with an

orthogonally propagating coupling beam. In this case we treat z and x dynamically,

and assume all quantities are homogenous in the y direction.

4.1.4 Correspondence with the previous formalism and light

field propagation

When the light fields couplings (∼ Ωp,c) are strong compared to (H1 − µ),H2,3,

we can neglect the contributions from the external dynamics terms we have just

introduced. In this case, the similarity of Eqs. (4.10) and (2.4) indicate that the

results in the previous chapters are still applicable. Physically, this means that the

time scales for the internal dynamics associated with the light field couplings and

propagation of the probe pulse are fast compared to any atomic motion which might

be induced.

When we completely neglect these terms, there is a formal equivalence between

the system of equations when one makes the correspondence:


√Nψi(R, t) ←→

√n(R)ci(R, t). (4.18)

We will discuss how the Hi terms can be treated as a perturbation to our previous

results later.

First we note that the light fields continue to be the described as dynamical

classical fields, though we make a slight change of notation to be consistent with our

GP formalism. Before, the atomic density n(R) entered the macroscopic polarization

when we summed over the atoms in the cloud (see Eq. 3.4 and paragraph before).

Now the atomic mean fields ψi(R, t) represent the many body wave function and

therefore incorporate the density naturally. Eqs. (3.6) can be rewritten:

∂

∂zΩp = −if13σ0

Γ

2N ψ3ψ

∗1,

∂

∂wΩc = −if23σ0

Γ

2N ψ3ψ

∗2. (4.19)

Here we have neglected the (1/c)(∂/∂t) terms from the equations since we saw in

Chapter 3 they never contribute significantly. This will generally be true whenever

the smallest time scale in the problem is much smaller than the time it takes light

in vacuum to traverse the condensate length σz/c, which is 0.1 ps for parameters of

interest.

Since in our Gross-Pitaevskii formalism, the atoms are treated as mean fields, the

problem has been reduced to the interaction of coupled classical (c-number) atomic

and light fields. Our equations allow loss of both atoms and photons. In both cases,

the physical origin of this loss is a scattering out of the macroscopically occupied mean

fields. For example, an absorption and subsequent spontaneous emission event will


take a photon from the laser and scatter it in a random direction with a different wave

vector and polarization so it is lost from the laser field. This will also cause a random

momentum recoil in an atom, so it occupies a different state than the macroscopic

wave functions and is lost from the condensate.

4.1.5 Adiabatic elimination of level |3〉 and final equations

We saw in Chapters 2 and 3 that adiabatic elimination of |3〉 has a wide range

of validity. Let us assume that all time variation of the light field inputs is large

compared to the inverse natural linewidth Γ−1 = 16 ns, so we can apply that here

as well. The external dynamics terms, as we saw in 4.1.3, are generally of order

µ ∼ (2π)1 kHz. Even when we consider solitons and vortices in BECs, which have

a much higher kinetic energy, the kinetic energy never exceeds the energy scale set

by µ. Therefore, it is well justified to neglect H3 as compared to Γ and adiabatically

eliminate ψ3 in (4.10):

ψ3 = −i(

Ωp

Γψ1 +

Ωc

Γψ2

)(1 + O

(µ

Γ,

1

Γτ0,∆p

Γ

)). (4.20)

Neglecting the small correction in (4.20) we then reduce the equations of motion for

the ground state atomic fields (4.10-4.11) to:

i∂

∂t

ψ1

ψ2

=

−i

2Γ

|Ωp|2 Ωp

∗Ωc

ΩpΩc∗ |Ωc|2

+

H1 0

0 H2 − ∆2

ψ1

ψ2

;

H1 = −2∇2

2m+ V1(r) + U11|ψ1|2 + U12|ψ2|2,

H2 = −2(∇2 + 2ik2 · ∇)

2m+ V2(r) + U22|ψ2|2 + U12|ψ1|2. (4.21)


In these equations we have neglected all contributions from collisions with |3〉 atoms,

since |ψ3|2 is always much smaller. We note that inelastic collisions cause losses via

ImU12,22. When using these equations in practice, we usually add one additional

loss term to each atomic field equation to account for losses due to elastic collisions

between high momentum |2〉 atoms and the nearly stationary |1〉 atoms [80]. This, in

effect, takes the next order term in the expansion in the small parameter (katoma12)

in cold collisions, which becomes comparable to other losses in the presence of a large

two-photon momentum kick. This is accomplished by defining ak212 = −ik2Rea122,

and letting a12 → a12 + ak212, thus adding a small additional imaginary scattering

length.

Under adiabatic elimination of ψ3, the light field propagation equations (4.19)

become:

∂

∂zΩp = −1

2f13σ0N(Ωp|ψ1|2 + Ωcψ1

∗ψ2),

∂

∂wΩc = −1

2f23σ0N(Ωc|ψ2 |2 +Ωpψ1ψ2

∗). (4.22)

These, along with Eqs. (4.21), form a full set of equations that are of quite general

validity as they can be applied for any range of light field strengths. In the limit

the light fields dominate the external dynamics, they reduce to the stationary atom

equations which were studied in Chapter 2 and 3. In the limit that the light fields

are off, they are simply GP equations describing two interacting condensates. When

the lights are off, there is no mechanism by which the ground states exchange atoms.

Because of their general validity Eqs. (4.21) and (4.22) are the basis of a Crank-

Nicolson [81, 82] based numerical code which we have developed to analyze and study


the system. It has been implemented in one and two dimensions generally and also

three dimensions when cylindrical symmetry can be assumed. We describe the algo-

rithm in detail in Appendix E. In all cases, the dynamical quantities were represented

on a Cartesian mesh with a spatial extent larger than the condensate. As mentioned

before, the initial conditions were determined by imaginary time propagation of ψ1

(beginning with a TF wavefunction), in the absence of light fields and with ψ2 = 0.

The boundary conditions for each light field are the given time envelopes along a

plane at one edge of the grid (zin, win). For generality, we allow the input probe and

coupling fields to have spatial profiles transverse to their direction of propagation.

4.1.6 Experimental geometries and parameters

The experiments discussed in this thesis [1, 2, 4] use two distinct schemes with

regards to the energy levels involved and the light field propagation directions. In

addition, there is a third which can be easily implemented. Each one offers different

advantages and experimental possibilities as well as drawbacks. In Section 4.2, we

will estimate the effect of various contributions to the external dynamics terms Hi

such as the Zeeman shifts from the magnetic trapping potentials, differences in the

ground state collisional scattering lengths, and the two-photon momentum kicks.

Therefore, it is useful here to summarize what these contributions are for the relevant

experimental schemes. In all the schemes used, we start with a trapped sample in

|1〉 = |3S1/2, F = 1,mF = −1〉 which has a scattering length a11 = 2.75 nm and

has no imaginary part (no collisional loss into other levels). The parameters for each

scheme are summarized in the following table (all |3〉, |4〉 refer to levels on the D1


line):Scheme Orthogonal Co-prop. Counter-prop.

|2〉 |F = 2,mF = −2〉 |F = 2,mF = +1〉 |F = 2,mF = +1〉|3〉 |F = 2,mF = −2〉 |F = 2,mF = 0〉 |F = 2,mF = 0〉εp (x− iy)/√2 (x+ iy)/

√2 (x+ iy)/

√2

εc z (x− iy)/√2 (x+ iy)/√

2Vrecoil = k2/m (2.9 cm/s)(z − x) ≈ 0 (5.8 cm/s)z

α(Z) -3 ≈ 0 ≈ 0a12/a11 1.2 0.963 − 0.052i 0.963 − 0.052ia22/a11 1.2 1.036 − 0.351i 1.036 − 0.351i

ak212/a11 −0.050i ≈ 0 −0.057i

nearest |4〉 N/A |F = 1,mF = 0〉 |F = 1,mF = 0〉config. of |4〉 N/A anti-symm. 2-Λ anti-symm. 2-Λ

∆4/(2π) ∼ 1500 MHz 192 MHz 192 MHz

To obtain the scattering lengths for the various ground states we use a Mathematica

notebook developed by Greene, et. al [74], which uses a complication of recent ex-

perimental measurements. Generally these quantities are slowly varying functions of

the background magnetic field (on the scale of Gauss), so we always quote the values

at B = 0.

The first observation of ultra-slow light [1], discussed in Chapter 3 used the or-

thogonal scheme, but on the D2 which had ∆4 = −(2π)60 MHz. Later experiments

[4] were on the D1 line in this scheme, and have the advantage of having no off-

resonant transitions closer than the ground state hyperfine splitting. But this scheme

has the disadvantage of repelling atoms in |2〉 (α(Z) = −3, i.e. V2 = −2V1). It also

has a large momentum kick, which can be advantageous or not depending on the

particular goal of an experiment. Because there are no available angular momentum

conserving state-changing collisions between |1〉 and |2〉 there are no inelastic colli-

sional losses. However, the momentum kick does introduce a loss rate due to high

momentum elastic collisions.


The co-propagating scheme, which was used in the stopped light experiments [2],

circumvents some of the problems with the orthogonal scheme. First, it has nearly

equally trapped ground states (V1 = V2). As we will see, this minimizes inhomoge-

neous broadenings. Also, the lack of a momentum kick is preferable for experiments

in which we wish the ψ1 and ψ2 atoms to overlap for a long time. However, the

ground states in this scheme have relatively large imaginary scattering lengths (listed

in the table) which lead to an effective loss rate γ via inelastic collisions. Also, the

scheme suffers from coupling to |4〉, which is ∆4 = (2π)192 MHz away and in an anti-

symmetric double-Λ configuration, as discussed in Chapters 2 and 3. It should be

noted that in Rubidium-87, these imaginary scattering lengths are smaller by about

2 orders of magnitude. Furthermore, the nearest ∆4 in this scheme in Rb-87 is more

than 800 MHz away. Therefore, this geometry in Rubidium could allow for a very

clean EIT system.

At present, we have not implemented the counter-propagating scheme, but it

would be useful for schemes where we want equal traps but still have a large two-

photon momentum kick.

4.2 USL propagation in BECs with a weak probe

With our formalism now in place, we first consider the weak probe limit and do

the analogous calculation for the ultra-slow light dynamics as we did in Section 3.2.

We will recover the result that the ground state superposition of the BEC evolves in

such a way as to keep it everywhere in the dark state. We illustrate the formalism

and numerical code with both 1D and 2D calculations. The 2D simulations demon-


strate some of the effects of transverse density gradients of the BEC. We also present

experimental results which corroborate our theoretical predictions.

We then consider corrections introduced by the external dynamics, which play an

increasingly important role as the light field intensities are reduced. Primarily we will

see they cause inhomogeneous broadening, which lead to two-photon detunings, and

loss rates, which lead to dephasings. Both effects introduce additional absorption

events as Wc is reduced, putting a lower limit on achievable group velocities Vg.

We will also see an interesting effect of how a two-photon momentum kick on the

atoms decreases the light-atom interaction and therefore increase Vg when Vg becomes

comparable to the recoil velocity Vrecoil. At the end of this section, we briefly consider

how our findings here can be applied qualitatively to non-condensed atomic samples.

4.2.1 Equations in the weak probe limit

Since obtaining an analytic solution in Section 3.2 required the weak probe as-

sumption, we start with that assumption here as well. Our calculation here will differ

from that one in two respects. First, we have already adiabatically eliminated ψ3 from

our equations, and already assumed we are near one-photon resonance. As a result

the equations have a slightly different form but lead to equivalent results. Second, in

previous chapters, we transformed away all phase evolution of c1 due to the detuning

∆p so we could treat it as a constant. Here H1 gives ψ1 an additional phase evolution

at a rate µ, amounting to a new term in the detuning. As we have seen, the quality

of the EIT is extremely sensitive to this detuning so we must be sure to include it in

our calculation. A simplification is possible when one considers (4.21) and (4.22) to


first order in Ωp. Then the solution for ψ1 is ψ1(R, t) = ψ(0)1 (R)e−iµt, where ψ

(0)1 (R)

is the original (real) ground state wave function. Then, introducing a transformation

ψ′2 ≡ ψ2e

iµt allows us to eliminate ψ1 as a dynamical variable. As before, Ωc is the

constant Ωc0 to this order. We are left with two equations:

i∂

∂tψ′

2 = (H2 − µ− ∆2)ψ′2 −

Ω2c0

2Γψ′

2 −ΩpΩc0

2Γψ

(0)1 ,

∂

∂zΩp = −1

2f13σ0N(Ωp|ψ(0)

1 |2 + Ωc0ψ(0)1 ψ

′2). (4.23)

The space dependent operator H2 generally prevents a closed form solution for Ωp(R, t),

which was possible in Sections 3.2.4 and 3.2.5. However, Fourier transforming (4.23)

still yields a good qualitative picture. Note that since the nonlinearity in the GP

equations is in their dependence on the density in each state |ψi|2 and ψ2 = O(|Ωp|)

then H2 is independent of ψ′2 and time to linear order in Ωp. The equations (4.23)

then become linear, allowing application of a Fourier transform. Solving the first

equation:

ψ′2 = − ΩpΩc0

Ω2c0 + 2iΓ

(FT[H2ψ′

2]

ψ′2

− δ − ∆2 − µ)ψ(0)

1 , (4.24)

where FT[X] represents the Fourier transform ofX. So our conditions for maintaining

the majority of atoms in the dark state is the same as before (see Eq. 3.28) with an

added requirement due to the contribution from H2. In the limit that H1 = H2, then

H2ψ′2 = µψ′

2 (at all R and the last two terms cancel. Any non-zero difference in the

real parts of H2 and H1 will give an additional detuning, and any imaginary part

of H2 gives a dephasing. There are five distinct possible contributions to this term


and their magnitude depends on the particular ground states chosen and the beam

propagation directions. They are:

• Different trapping potentials (V2(R) − V1(R) = 0) occur when |1〉 and |2〉 have

different magnetic moments, causing an inhomogeneous detuning.

• Different elastic scattering lengths. In the weak probe limit the difference is

ReU12 − U11|ψ(0)1 |2/, also leading to an inhomogeneous detuning.

• Inelastic and high momentum elastic collisions. These lead to a loss rate term

ImU12|ψ(0)1 |2 and therefore a dephasing.

• Atomic recoil. Except in the co-propagating case, the difference in the kinetic

energy terms i

mk2 · ∇, gives the |2〉 atoms a two-photon momentum kick.

• Whenever the pulse has spatial structure there is a small difference in the kinetic

energy (∼ ∇2). This is less important for pulses with a length scale comparable

to the condensate length since in the TF limit, the kinetic energy scale is quite

small.

If we assume that the all these differences are sufficiently small, we can again perform

a binomial expansion of the denominator of (4.24) to get:

ψ′2 = − Ωp

Ωc0

ψ(0)1

[1 − 2i

W 2c

(FT[H2ψ

′2]

ψ′2

− δ − ∆2 − µ)]. (4.25)

Upon inverting the FT, we would obtain an expression analogous to (3.37). As in

that equation, the small corrections to unity in the brackets are the component in the

absorbing state, and therefore determine the occupation of the excited level. When


the term in brackets is dominated by unity, the entire BEC approximately follows the

time and space dependent dark state:

ψ2(R, t) = −Ωp(R, t)

Ωc(R, t)ψ1(R, t). (4.26)

For later reference we will find it useful to then define the wave functions in the

dark/absorbing basis:

ψDψA

=MBEC

ψ1

ψ2

,

MBEC ≡ 1

Ω

Ωc −Ωp

Ω∗p Ω∗

c

(4.27)

(Recall Ω ≡√|Ωp|2 + |Ωc|2).

Turning now to the probe propagation, it is clear, that in the limit that the new

external dynamical terms are much smaller than τ−10 (which determines the scale of

δ in (4.25)), non-adiabaticity is the leading order contribution to ψA and we recover

exactly our results from Chapter 3 for the probe propagation. As an example, we show

in Fig. 4.2 a numerical calculation of USL propagation. We propagated Eqs. (4.21)

and (4.22) in a 1D Thomas-Fermi cloud of N = 1.2 × 106 condensate atoms in a

harmonic trap ωz = (2π)21 Hz and A = π(8.8 µm)2. The optical density of the ground

state wavefunction is then D0 = 409. The beams are chosen to be co-propagating

along z, the trapping potentials for the two ground states are equal, and there is

a small difference in the scattering lengths a12 = 1.20 a11. The Rabi frequencies

(Ωp0 = (2π)2.5 MHz, Ωc0 = (2π)8 MHz) and pulse width (τ0 = 1.5 µs) are such


z m)

a

0 5 10 150.00

0.01

1.18

c0.041.20

0.031.19

0.02

1.17

1.16

t s)

N,N

(10

)1

T

6 N(1

0)

2

6

p|2

p

0

2

t s)- 60 - 40 - 20 0 20 40 600

2

4

6 b0.8

1.0

0.6

0.4

0.2

151050

De

nsi

ty(1

0c

m)

13

-3

0.006

0.004

0.002

0.000

Figure 4.2: One dimensional USL numerical results. Numerical calculation ofUSL in a 1D condensate, with N = 1.2 × 106, ωz = (2π)21 Hz and a cross-sectionalarea A = π(8.8 µm)2. The coupling beam input Rabi frequency is Ωc0 = (2π)8 MHz,and a probe with width τ0 = 1.5 µs and peak Rabi frequency Ωp0 = 2.5 MHz is input.We choose a22 = a12 = 1.20a11. (a) The probe output intensity (circles) agree wellwith the analytic expression (3.31) using the calculated optical density D0 = 409 ofthe ground state TF wavefunction. (b) A snapshot of the atomic densities when thepulse is compressed inside the cloud, at t = 3.4 µs. The density N |ψ2|2 (dotted curve)follows the pulse Eq. (4.26)), leading to a small depletion in N |ψ1|2 (dashed curve).The density in the dark state N |ψD|2 (thin solid curve) is indistinguishable from theoriginal TF density. There is, however, a small density in the absorbing state N |ψA|2(thick solid curve, with an exaggerated scale set by numbers on the right hand sideof the plot). (c) Total number of atoms in the two ground states N1 = N

∫dR |ψ1|2

(dashed curve) and N2 = N∫dR |ψ2|2 (dotted curve) as a function of time. The total

N1 +N2 is plotted as a solid curve.

that the external dynamics terms have a negligible effect. Fig. 4.2(a) compares the

numerically calculated probe output with the analytic result (3.31) and we see very

good agreement. Fig. 4.2(b) shows the densities in the two ground states N |ψ1,2|2

as well as the dark and absorbing state densities N |ψD,A|2 as a function of z at a

time when the pulse is completely contained in the BEC. We see that, as before,

ψ2 follows the probe pulse shape (according to Eq. (4.26)). The density in the dark

state, |ψD|2, is at all times very close to the original Thomas-Fermi shape. The small

deviation from it, given by |ψA|2, is more than three orders of magnitude smaller.

The sharp kink is where ψA goes through zero and changes sign (just as c3 did in the

last chapter).


In Fig. 4.2(c) we track the number of atoms in the ground states N1, N2 as a

function of time. We expect from the calculation of the flow of photons in Subsec-

tion 3.2.1, that the peak number of atoms in |2〉, when the pulse is fully compressed,

will roughly agree with the number of photons input, which was 30, 700 in the case

plotted. This is indeed seen to be the case in the figure. There is a small amount of

loss due to the non-adiabaticity, and so the final number of atoms in |1〉 is slightly

smaller than N . This loss, 5380 atoms, is roughly equal to the total difference in

number of photons between the input (reference) and output probe pulse, 5120. This

is always the case in the weak probe limit when the susceptibility seen by the cou-

pling field is much smaller. By the arguments in Subsection 3.2.1, the total atomic

loss must always equal the total number of absorbed photons. From the curves in

Fig. 4.2(c) we also see that while the atomic losses are occurring, the reduction is

in N2 rather than N1 atoms. This is because as the probe is absorbed, the atomic

populations readjust themselves into the corresponding new dark state with a smaller

ψ2.

4.2.2 Transverse dimension effects

With our numerical code, we also performed 2D calculations and to learn how the

transverse gradients in the condensate density come into play. In Fig. 4.3 we present

a calculation with the same parameters as Fig. 4.2. Now the x direction is calculated,

using a trap with ωx = 3.8 ωz, and all quantities are still assumed to be homogenous

along y over the length of the system Ly = 2 × 8.8 µm. This leads to a cloud with

a central optical density D0(x = 0) = 386. Due to the density gradients in the x


direction, the light is faster and less compressed away from the center, leading to the

striking “crescent” shape of the density of |2〉 atoms shown in Fig. 4.3(a). This has a

measurable effect on the probe output as shown in Fig. 4.3(b). The thin solid curve

shows the probe intensity for light which has traversed the central column (x = 0) of

the BEC, and is in close agreement with the theoretical expectation (dashed curve).

In experiments, a pinhole is used to image the intensity integrated over the central

15 µm diameter circle. If one takes our numerical 2D numerical results and integrates

over the same region, assuming cylindrical symmetry, one gets the result plotted as

thick dashed curve. Because the condensate size is comparable to the pinhole size,

the optical density D0(x) varies over the area captured by the pinhole. The average

D0 seen by the probe pulse over this area is somewhat smaller than the D0(x = 0)

and this acts to shorten the observed delay compared to the expectation based on

the central density. The inhomogeneity of D0 over this region spreads the observed

pulse.

In Fig. 4.3(c-d) we plot the coupling and probe intensity along the cuts indicated

in Fig. 4.3(a). One sees the adiabatons (see Subsection 3.3.1) as before, with the

coupling field intensity dipping in regions occupied by the probe.

We also performed numerical calculations with the coupling beam propagating

orthogonally to the probe, along x. The corresponding plots for this case are shown

in Fig. 4.4. As seen in Fig. 4.4(a), there is little difference in the |2〉 density. Because

of the different coupling direction, the adiabatons behave somewhat differently. To

obtain them analytically, one would have to integrate the coupling field from the

bottom edge (at x = xin). Cuts of the coupling intensity in Fig. 4.4(c-d), show


Figure 4.3: Two dimensional effects in co-propagating USL. Numerical calcu-lation of USL in a 2D condensate, with the same parameters as Fig. 4.2, only withωx = 3.8 ωz and the length in the y direction chosen to be Ly = 2× 8.8 µm. (a) Thedensity N |ψ2|2 at t = 3.2 µs has a remarkable “crescent” shape. This results fromthe lower density nearer to the condensate edge causing the probe to travel faster andbe compressed less there. The dotted ellipse indicates the TF border of the originalground state density. The dotted lines indicate positions of cuts plotted (c) and (d).(b) The probe output intensity (normalized to the peak input intensity). The thinsolid curve shows the numerical result for the intensity at x = 0, which is the lightwhich traversed the central column of the cloud, and the dashed curve shows theanalytic results for the central optical density D0(x = 0) = 386. In experiments,we image a central, 15 µm diameter circle of probe light. The results of integratingover this area, assuming cylindrical symmetry, is shown as the thick solid curve. Thetransverse variation of D0 is seen to cause less delay and more spread. (c) A cut atx = 0 (indicated in (a)) of the coupling |Ωc|2 (dashed curve) and probe |Ωp|2 (solidcurve) intensities, both normalized to Ω2

c0. One sees an adiabaton dip in the couplingintensity. (d) The same plots along two cuts in the x direction (also indicated in (a)).


Figure 4.4: Two dimensional effects on orthogonal USL. The same calculationas Fig. 4.3, only now in an orthogonal geometry with the coupling field entering fromthe bottom (negative x). As a result, the adiabatons in (c) and (d) behave differently,leading to a slightly faster propagation speed than the co-propagating case, as seenin (b).

the behavior is qualitatively different. The most important difference we note is that

because the coupling field is propagating over a much shorter distance, the adiabatons

are substantially smaller, so the coupling intensity stays much closer to its input value

Ω2c0. Also the adiabatons are seen at points to be positive (humps as opposed to dips)

even after the probe is compressed. Due to both of these effects, the probe field

propagates faster than in the co-propagating case. As seen in Fig. 4.4(b), the probe

pulse, even at x = 0, is less delayed than the analytic estimate predicts. In the

weak probe limit, the adiabaton effects just discussed do not play a large role in the

pulse propagation, and so the results for the probe output are almost identical in the

orthogonal- co-propagating cases. However, these differences can play a significant

role when Ωp ∼ Ωc. In particular, because the coupling intensity can vary with x (see

the x cut of the coupling intensity in Fig. 4.4(d)), the pulse can acquire a transverse


asymmetry, whereby the x > 0 portion arrives faster than the x < 0 portion. This

issue is particularly relevant in the light roadblock experiments discussed in Chapter 7.

4.2.3 Experimental results

Experiments of USL in BECs have borne out this behavior. Using our “stopped

light” technique, discussed in Chapter 5, it is possible to freeze the light pulses and

take images of the atoms at various times during the propagation. An absorption

image of the |2〉 atoms when the pulse is compressed is shown in Fig. 4.5(a) and

one sees the crescent shape. Fig. 4.5(b) shows the pulse output on the PMT, when

a 15 µm pinhole was used so only the central portion of the pulse was imaged. As

discussed above, this leads to a larger spreading and a shorter delay than what one

would expect theoretically for propagation through the cloud center.

Even with the transverse variation in optical density, we can use the same mea-

surement of the dephasing discussed in Subsection 3.4.2, from the transmission and

delay, as the optical density is normalized out. In the case shown in Fig. 4.5(b), the

measured dephasing is γexp = (2π)28 kHz. Fig. 4.5(c) plots measured dephasings for

a variety of coupling intensities in both condensates and thermal clouds. The mecha-

nism of this dephasing is not completely understood, though we note that γexp scales

approximately linearly with coupling field intensity Wc, leading us to believe that

the origin of the effect is due to the other optical transitions, perhaps from impure

polarizations. They are also somewhat higher for condensates than thermal clouds.

Because multiple scattering events would be more important in a denser medium, this

seems to indicate that multiple scattering events could be playing a role. Supporting


Figure 4.5: Observation of USL in a BEC. (a) Absorption image of |2〉 atomswhen a probe pulse is compressed in a BEC. The BEC consists of N = 1.4 × 106

atoms, in a trap ωz = (2π)21 Hz, ωx = ωy = 3.8ωz. The Rabi frequencies wereΩc0 = 11.3 MHz and Ωp0 = 2.1 MHz and the probe pulse width was τ0 = 0.26 µs. Thecoupling field was switched off at t = 0.42 µs, freezing the atoms in the superpositionat that time. The trap was then switched off and the image was taken 0.5 ms later.The atoms were imaged with a 10 µs long imaging pulse propagating along y, resonantwith the F = 2 → F = 3 transition on the D2 line. The OD is defined to be the−ln(Tbeam) where Tbeam is the transmission coefficient of the imaging beam. (b) Apulse delay measurement (dots) in a condensate with 1.0 × 106 atoms and the sametrap, with an optical density D0(x = 0) = 278. In this case, Ωc0 = 8.2 MHz andΩp0 = 2.0 MHz and τ0 = 0.54 µs. Gaussian fits to the input probe and the datapoints are indicated with the dashed and solid curves, respectively. In this case thetransmission was T = 0.47 and the fitted delay was τd = 2.1 µs, leading to a measureddephasing of γexp = (2π)28 kHz. The width of the output pulse was 2.9 times thewidth of the input pulse and the predicted τsp = 0.4 µs so this can explain some ofthe spread. The transverse inhomogeneity is playing a role as well, as discussed inthe Subsection 4.2.2. The theoretical delay through the cloud center is 6.5 µs and weindeed see a somewhat smaller one, primarily due to the transverse inhomogeneouseffects. (c) Experimentally measured dephasing γexp for several different couplingintensities (Wc) for condensates (filled circles) and thermal clouds (open circles) inthe same experimental geometry.

this conclusion is that we also observed that the γexp scales approximately linearly

with the probe energy (regardless of whether the energy was varied via the width τ0

or intensity Ω2p0).


4.2.4 Limitations at low coupling intensities

We now consider what happens whenWc is made smaller so the external dynamics

have a measurable effect. To track the effect of different terms we will consider

different physical effects, one at a time in the following. In practice, as we outlined in

Subsection 4.1.6, usually several come into play in any given geometry. We will see

here how each contribution contributes either an effective inhomogeneous two-photon

detuning or dephasing. These in turn reduce the transmission and therefore put a

lower limit on the Wc which can be used (and therefore group velocity Vg).

In the following we will obtain analytic estimates losses in one dimensional systems

by assuming the Thomas-Fermi approximation accurately describes the density and

then compare the results to numerical simulations in 1D. In 1D, the Thomas-Fermi

density profile is:

|ψ(0)1 |2 =

µ

U11

(1 − z2

2σ2z

);

σz =1

ωz

√2µ

m, µ =

(3U11m

1/2ωz

4√

2A

)2/3

. (4.28)

For the following examples we use ωz = (2π)21 Hz and A = π(9 µm)2. Then µ =

54 ωz = (2π)1.13 kHz.

Zeeman shifts

One of the largest effects which occurs in practice is a large inhomogeneous Zeeman

shift due to V2(z) = V1(z) giving a space dependent two-photon detuning. Writing

V2 = (1 +α(Z))V1 and assume the (bare) the two-photon detuning at R = 0 is ∆2. In


[1, 4], α(Z) = −3. We will assume the transmission is of the form (3.32) but with ∆2

replaced by some effective ∆(Z)2−eff , which is determined by calculating the average of(

∆2 + V1 − V2

)2over the propagation, weighted by the density N |ψ(0)

1 |2. In the TF

approximation:

(∆(Z)2−eff )2 ≡

∫ +√

2σz

−√2σzdz |ψ(0)

1 |2(∆2 + V1 − V2)2∫ +

√2σz

−√2σzdz |ψ(0)

1 |2

= ∆22 +

2

5α(Z)µ∆2 +

3

35(α(Z)µ)2. (4.29)

This expression is minimized by choosing ∆2 = −0.2α(Z)µ, but not completely elim-

inated. At this detuning ∆(Z)2−eff = 0.21α(Z)µ. Thus the Zeeman shift causes a slight

shift in the position of the peak transmission but more importantly, even at this peak,

the inhomogeneous broadening destroys the perfect two-photon resonance, leading to

a reduction of the transmission. In Fig. 4.6, we show numerical results for the example

α(Z) = −3. To see how it becomes important as we turn down the coupling intensity

Wc, Fig. 4.6(a) shows the transmission as a function of Wc, keeping ∆2 = 0. At

high Wc the EIT profile is wide and the transmission is determined by non-adiabatic

losses, but near Wc = (2π)0.1 MHz (Ωc = (2π)1 MHz), the Zeeman broadening be-

comes comparable to the bandwidth Wc/√D0 and there is a dramatic decrease in the

transmission. The solid curve shows the theoretical expectation based on Eq. (4.29)

and (3.32). In Fig. 4.6(b) we plot the transmission as a function of ∆2 for the case

Wc = (2π)0.05 MHz, and compare to the case with no Zeeman shifts α(Z) = 0. The

analytic estimate is seen to closely predict both the resonance shift and the reduction

in the peak transmission.

This loss mechanism can be an important consideration in the ultimate lower limit


Figure 4.6: Transmission reduction and resonance shift due to inhomoge-neous Zeeman shifts. (a) The integrated pulse transmission based on numericalsimulation (circles) as a function of Wc. The cloud parameters are the same as inFig. 4.2 except a12 = a22 = a11. The Zeeman parameter α(Z) = −3. AsWc is reduced,the probe intensity is scaled down so Ωp0 = Ωc0/4 at each point and the pulse width τ0is increased so τ0 = 30.1W−1

c = 1.41τsp. In this way the expected transmission due tonon-adiabaticity alone is always T = 1/

√1 + (τsp/τ0)2 = 0.83 (horizontal line). The

analytic expectation based on (4.29) and (3.31) is shown as the solid curve. (b) ForWc = 0.05 we vary ∆2 for both α(Z) = −3 (filled circles) and α(Z) = 0 (open circles).Analytic expectations are plotted as the solid and dashed curves, respectively.

of observable group velocities. Lower group velocities can be achieved by turning down

the coupling intensity, and larger τ0 can be used to counteract the pulse spreading

and attenuation due to non-adiabaticity. However, in the case of an inhomogeneous

broadening, one must keep Wc > 2∆2−eff

√D0 to keep the transmission reasonable.

For the example we have been considering, this would correspond to a group velocity

at the center of the cloud of Vg = 0.58 cm/s.

Mean field shifts

In the case that U11 and U12 differ, the mean field shifts experienced by |1〉 and |2〉

atoms differ. The difference scales with the inhomogeneous density of atoms. This

is mathematically equivalent to the Zeeman case, but we replace α(Z) in Eq. (4.29)

with α(MF ), the fractional difference between the scattering lengths:


α(MF ) =ReU12 − U11

U11

. (4.30)

In practice, α(MF ) in sodium are substantially less than unity (see the table in Sub-

section 4.1.6). Because the α(MF ) are small, a difference in magnetic moment α(Z),

when present, will be a much larger contribution. However, when V2 = V1 (as in the

geometry in the experiment in [2]), it is possible for the mean field shifts to be a

leading order effect.

Inelastic and elastic collisional losses

In contrast to the above case, collisions between ground state particles which

populate other internal levels lead to losses from the condensate, which leads not

to frequency width ∆2−eff , but to a loss rate γeff . The imaginary part of U12, for

example, can be on the order of 0.1 U11 (see table in Subsection 4.1.6). Similarly,

the two-photon momentum kick can cause high momentum elastic collisions. Letting

α(coll−loss) ≡ −ImU12/U11, one can calculate a loss rate at the center of the cloud:

γ(coll−loss)0 = α(coll−loss)µ and a similar calculation of weighting this over the entire

cloud reveals γ(coll−loss)eff = 0.8γ

(coll−loss)0 .

As we saw in Chapter 3 and Eq. (3.31), the transmission loss due to a dephas-

ing rate γ scales faster with D0 than a given detuning ∆2. One needs to maintain

Wc > 2γD0 to get good transmission. Taking α(coll−loss) = 0.05 (see table in Subsec-

tion 4.1.6) yields a minimum possible group velocity of Vg = 1.0 cm/s.

A numerical calculation with the same parameters as Fig. 4.6 with Ωc0 = (2π)1 MHz

and V1 = V2 shows the transmission drops from T = 0.84 for α(coll−loss) = 0 to


T = 0.43 when α(coll−loss) = 0.1. The prediction according to the method outline

above is T = 0.41, so we see it provides an accurate estimate of the loss of transmis-

sion due to inelastic collisional losses.

Recoiling dark states and Doppler shifts

When a two-photon momentum kick is present (k2 = 0), the difference in the

kinetic energy contribution can be important. Then ψ′2 will recoil to satisfy the dark

state superposition. We will now show that when the recoil Vrecoil is comparable to

the group velocity Vg = k2/m the light-atoms interaction is reduced, thus increasing

the group velocity and transmission.

Consider the case where the only contribution to H2 − µ is the momentum recoil

term. For simplicity, consider a 1D geometry with counter-propagating beams. Then

the first equation of (4.23) becomes:

(∂

∂t+ Vrecoil

∂

∂z+

Ω2c0

2Γ

)ψ′

2 = −Ωc

2ΓΩpψ

(0)1 . (4.31)

Now, instead of Fourier transforming only in time, we will transform in time and space.

Calling our space frequency variable k, and assuming that ψ(0)1 is slowly varying in

space compared with Ωp and ψ′2, we have:

˜ψ′2(k, δ) = − Ωc0

˜Ωp

Ω2c0 − 2iΓ (δ + Vrecoilk)

ψ(0)1 , (4.32)

where ˜f denotes the double FT (time and space) of f . We next assume, as before, that

we are in EIT conditions so the denominator of (4.32) can be binomially expanded.


Plugging the result into the Fourier transformed equation for the probe propagation,

we get a dispersion relation between k and δ which the frequency components of ˜Ωp

must satisfy :

k

(1 +f13σ0N |ψ(0)

1 |2Wc

Vrecoil

)=δf13σ0N |ψ(0)

1 |2Wc

. (4.33)

To reconstruct the probe in real space and time we use our boundary conditions (the

input pulse’s temporal profile at z = zin) and this dispersion relation. It is only

possible to obtain an analytic expression when the BEC has a spatial profile slowly

varying compared to the pulse length. While this may not always be the case, the

solution we will obtain will be seen to be a good estimate of the recoil effect even then.

In analogy with our weighted average approach in (4.29), let us define the average

group velocity with no momentum kick Vg−eff ≡ Wc/(f13σ0N〈|ψ(0)1 |2〉) (taking the

weighted average of |ψ(0)1 |2 over the condensate) and the delay with no momentum

kick τd = D0W−1c . With these definitions our delay, calculated with (4.33) and the

boundary conditions, becomes:

τ(k2)d = τd

1

1 + Vrecoil

Vg−eff

. (4.34)

We indeed see that the correction simply scales with the ratio of the recoil velocity

to the group velocity.

As the coupling field intensity Wc is reduced, the recoil decreases the interaction

so the group velocity and pulse length increase. This reduction in interaction leads

to an increase in the transmission, in contrast to other effects we have considered.


The question then arises as to whether this behavior continues as Wc is turned down.

Does the medium eventually appear completely transparent or is the transmission

eventually reduced? It turns out that the rise in transmission occurs only until pulse

length reaches the length of the BEC and then another effect becomes important.

Since a BEC is a macroscopic number of particles in a single quantum state, the

momentum is found by multiplying by the derivatives of the macroscopic wave

functions ψ(0)1 and ψ′

2. The derivatives of ψ′2 give a factor on the order of σ−1

z or L−1p ,

whichever is larger. This gives an estimate for the momentum spread of the BEC in

accordance with Heisenberg’s uncertainty principle ∆P ∼ Maxh/σz, h/Lp. When

we ignore the ∇2 term but consider the (much larger) term due to the momentum

kick:

|(H2 − µ)ψ′2| = |k2

m· ∇ψ′

2| (4.35)

This is simply k2 times the atomic velocity operator (/m)∇, supporting an inter-

pretation of this term as a Doppler broadening. While the pulse is compressed inside

the medium, the recoil causes a lengthening of the pulse Lp, reducing the momentum

spread of ψ′2 in such a way that prevents the effect of this term from getting larger.

However, once the pulse is larger than the cloud size σz then this finite size puts a lower

bound on the velocity spread in ψ′2. This causes a finite inhomogeneous broadening

according to (4.35). Like the above effects, it leads to an additional detuning ∆(D)eff .

For a condensate with a spatial width ∼ 30 µm, and counter-propagating beams,

∆(D)eff = 280 Hz, which is comparable with the chemical potential µ and therefore

with the Zeeman broadening discussed above.


In Fig. 4.7 we demonstrate both the increase in the group velocity and transmission

due to the decreased light-atom interaction, as well as the transmission reduction due

to Doppler broadening. Figs. 4.7(a-b) contrast simulations of the co-propagating

and the counter-propagating geometries with the same parameters. In both cases

Vg−eff = 4 cm/s. The pulse (solid curves) is seen to be less compressed and faster in

the counter-propagating case (where Vrecoil = 6 cm/s). The absorbing state density

|ψA|2 is also seen to be smaller, indicating less atom-light interaction. However, there

is a sharp increase in this |ψA|2 near the cloud edge, which shows how the finite

cloud size limits the degree to which the interaction is reduced when the derivatives

of ψ′2 play a role. Fig. 4.7(c) shows the numerically calculated delays (filled circles),

normalized to the expected theoretical delay in the co-propagating case (horizontal

line) asWc is reduced (note the actual delays are increasing asWc is decreased). There

is very good agreement with the analytic prediction (4.34) (open circles). The Wc at

which Vg−eff = Vrecoil is shown as a vertical line. Fig. 4.7(d) shows the calculated

transmission (filled circles). As we discussed above, the transmission is increased

until the effect of the finite size of the BEC begins to play a role and the transmission

begins to drop significantly.

For comparison, we also show the same calculation with losses due to the high

momentum elastic collisions (which gives a finite γ as discussed above) included.

When these losses are included for our parameters they induce a transmission loss

which is larger than the rise we would otherwise see from the smaller interaction.

However, these simulations still show it should be possible to see a significantly smaller

delay due to the momentum kick before transmission losses are significant.


Figure 4.7: A recoiling dark state. Numerical simulations are for the same pa-rameters as Fig. 4.6 with Wc = (2π)0.05 MHz and equal traps, corresponding toVg−eff = 4 cm/s. (a) A co-propagating case. The probe field intensity (solid curve,normalized to its peak input value) is shown at t = 0.216 µs. The total atom den-sity profile (in arbitrary units) is shown as a dotted curve, and the absorbing statedensity N |ψA|2 is plotted as a dashed curve (according to the scale on the rightside of the plot). (b) The same plots in the counter-propagating case with the highmomentum collisional scattering term ak2

12 artificially set to zero. The recoil is thenVrecoil = 6 cm/s. (c) The numerically calculated delays (calculated from the firstmoment of the intensity) (filled circles) and the theoretical expectation based on(4.34) (open circles) as a function of Wc both normalized to the theoretical delaysin the absence of the momentum kick (horizontal line). The probe intensity Ω2

p0 andwidths τ0 are adjusted as in Fig. 4.6 so the weak probe and adiabatic requirementsare satisfied equally for all points. The vertical line here (and in (d)) shows the Wc

at which Vg−eff = Vrecoil. The arrows here indicate the case plotted in (b). (d) Thetransmission as a function of Wc (filled circles) For these points we do not includeatomic losses from high momentum elastic collisions (ak2

12 = 0). The open diamondsare the same calculation with the true value ak2

12 = πa212.


We also confirmed numerically that the peak transmission occurs for ∆2 = 0 so

there is no net energy shift due to the Doppler term (other than the recoil energy

k22/(2m) which we already included in our definition of ∆2 in (4.12)).

What is the ultimate limit on slow light?

The analysis in this subsection has provided us with lower limits on the group

velocities, delays, and pulse lengths based on contributions from Zeeman shifts, col-

lisions, and velocity recoils. In general, these terms are always finite (though can be

quite small in certain geometries). It should be noted however, that we will show how

two techniques we have developed, stopped light (Chapter 5) and the light roadblock

(Chapter 7), can be used to bypass many of the limitations that we found here by

lowering the effective optical density D0 through which the light must pass in the low

Wc regime.

We eventually would like to address the situation of group velocities on the order

of the BEC sound speed cs ∼ 0.5 cm/s and pulse compression on the order of the BEC

healing length ξ = 0.4 µm, thus getting into a regime where local BEC excitations

rather than macroscopic effects of the cloud are the dominant effect. The results

of this subsection are a good guide as to conditions needed for this to be possible.

Clearly, even if every contribution to H2 − µ that we have considered vanishes, there

will always be some small contribution introduced by the kinetic energy (∝ ∇2)

due to the pulse shape itself. The scales cs and ξ are the point at which this kinetic

energy becomes comparable to atom-atom interactions (this point is expounded upon

in Chapter 8) and it would be interesting to investigate if a qualitative change in the


light pulse propagation once these scales are reached.

4.2.5 Analogous limits for thermal clouds

The formalism we have developed is specifically applicable to BECs. However,

the physical origin of our loss mechanisms are quite clear and the physical arguments

can be applied to thermal clouds as well. Generally the densities in thermal clouds

are substantially smaller than in BECs so the importance of mean field shifts and

inelastic losses becomes much smaller. By contrast, the kinetic energy is generally

orders of magnitude higher, leading to larger Doppler and Zeeman shifts. For a given

temperature the thermal velocity is

vTh =

√3kBT

m, (4.36)

which is 3 cm/s for a 1 µK cloud. This leads to a Doppler width of (2π)100 kHz

in an orthogonally propagating geometry. While it is still possible to observe ultra-

slow light in cold atoms, even in this geometry [1] the Doppler shifts are easily the

dominant loss mechanism in many cases. Therefore in thermal clouds, one gains

substantially by going to the co-propagating geometry where k2 is negligible.

Similarly, the thermal occupation o f high energy levels causes a much larger

spatial extent of the cloud, leading to a larger spread in Zeeman shifts. The (1/e)

spatial size of a thermal cloud is:

σz−Th =1

ωz

√kBT

m. (4.37)


To estimate the resulting spread in Zeeman shifts for a given α(Z), simply replace µ

with kBT/ in (4.29).

4.3 USL propagation in BECs with a strong probe

While the above calculations gave good estimates of how the light induced atomic

motion acts back on the probe propagation, many interesting experiments involve

creating and watching the evolution of large, local excitations in the Bose-Einstein

condensate. These effects scale with the probe strength, and so it is worthwhile to

understand the behavior in the strong probe regime. In the following calculation, we

adiabatically eliminate the absorbing wave function ψA from the motion and derive

an effective Gross-Pitaevskii equation for the dark wave function ψD. Just as it was

not possible to solve the fully self-consistent problem in the strong probe regime in

Chapter 3, we do not attempt that here either. However, when Ωp,Ωc are given (and

they can often be estimated), it is possible to derive a GP equation for ψD which

gives a good intuitive guide to the behavior. We demonstrate the basic features with

several numerical calculations.

4.3.1 Dark state following and the dark state GP equation

We saw in the Section 4.2 that if the light field couplings dominate the external

dynamics, we recover the result that the two component condensate amplitudes will

follow the time- and space-dependent dark state superposition (4.26).

To see this in the strong probe case, we transform our coupled GP equations

(4.21) into the dark/absorbing basis with (4.27). The matrix MBEC is both space-


and time-dependent, which must be accounted for in the transformation. The result

is:

i∂

∂t

ψDψA

=

−iΩ2

2Γ

0 0

0 1

+ i

0 −ΩNA

2

Ω∗NA

20

+ MH − i ·M∇

ψDψA

, (4.38)

where

MH ≡ 1

Ω2

|Ωc|2H1 + |Ωp|2(H2 − ∆2) ΩcΩp(H1 −H2 + ∆2)

Ω∗cΩ

∗p(H1 −H2 + ∆2) |Ωp|2H1 + |Ωc|2(H2 − ∆2)

,

M∇ =

iΩk2

(p,∇p∗) + Ω(p,∇2p∗) + Ω(c,∇2c∗) −iΩk2

(p,∇c) + Ω(c,∇2p) − Ω(p,∇2c)

−iΩk2

(c∗,∇p∗) + Ω(p∗,∇2c∗) − Ω(c∗,∇2p∗) iΩk2

(c∗,∇c) + Ω(c∗,∇2c) + Ω(p∗,∇2p)

;

and Ωk2m,∇n ≡ k2

m· Ωm

Ω

(∇Ωn

Ω

), Ω(m,∇2n) ≡

2m

Ωm

Ω

(∇2 Ωn

Ω

),

ΩNA = 2ΩpΩc − ΩpΩc

Ω2. (4.39)

The first matrix represents the preparation of the dark state, as it damps the absorbing

component ψA and leaves the dark component ψD unaffected. In the second matrix,

the non-adiabatic transfer rate ΩNA couples ψD and ψA (just as in (2.11)). When the

light fields dominate, these first two terms are the largest and we recover our results

of Chapter 3, where the atomic dynamics were ignored.

The third matrix contains the external dynamics in the new basis. From the

diagonal terms, one sees that it evolves ψD and ψA according to weighted averages


of H1 and H2, depending on the fraction of their atomic densities in |1〉 and |2〉.

From the off diagonal terms, one gets the generalized two-photon detuning which

we discussed in detail in Section 4.2.4. Just as in Subsection 2.5, the detunings and

dephasing which are introduced cause a couple between ψD and ψA.

The fourth matrix arises because the transformation does not commute with the

kinetic energy operator. We see that, in analogy to the non-adiabatic coupling ΩNA

for fast temporal pulses, the spatial derivatives of the wave functions also cause non-

zero coupling between ψD and ψA.

Returning to our assumption that the light fields dominate the external dynamics

and the external dynamics are a perturbation, we can adiabatically eliminate the

strongly damped absorbing amplitude ψA. If we assume |ψD| |ψA| then setting

ψA = 0 in (4.38) yields:

ψA =2Γ

Ω2

(−iΩ∗

∆ +1

2Ω∗

NA + iΩk2c∗,∇p∗ − Ω(p∗,∇2c∗) + Ω(c∗,∇2p∗)

)ψD, (4.40)

where we have defined the “detuning operator”

Ω∆ ≡ ΩcΩp

Ω2

(H1 − H2 + ∆2

). (4.41)

Plugging (4.40) back into (4.38) gives us a single equation for the propagation of the

dark state amplitude:


i∂

∂tψD = (HD − iγD)ψD;

where HD ≡ |Ωc|2H1 + |Ωp|2(H2 − ∆2) − (Ωk2p,∇p∗ + iΩ(p,∇2p∗) + iΩ(c,∇2c∗)),

γD ≡(iΩ∆ +

1

2ΩNA − iΩk2

c,∇p + Ω(p,∇2c) − Ω(c,∇2p)

)

×2Γ

Ω2

(−iΩ∗

∆ +1

2Ω∗

NA + iΩk2c∗,∇p∗ − Ω(p∗,∇2c∗) + Ω(c∗,∇2p∗)

). (4.42)

The first term HD is the dominant one, and is the weighted average Hamiltonian that

the dark state feels as the probe pulse passes through. This GP equation governs ψD

when the light fields are given functions of space and time. Under conditions of good

EIT, the majority of the condensate is in the dark state and thus it evolves according

to this equation. Note that the momentum kick effect is accounted for with the terms

involving k2.

The second term, γD is generally much smaller but is important to consider as

it includes all the losses that ψD suffers due to non-zero coupling with ψA. In the

weak probe limit, it reduces to results we found previously. For example, when the

light fields completely dominate the external dynamics, ΩNA is the largest term in

the summations in γD and it leads to a term describing atomic losses due to non-

adiabaticity (analogous to (2.15)). When other terms become important it gives the

losses due to Doppler shifts, Zeeman shifts, and collisions.

4.3.2 Numerical results

As a brief demonstration of these concepts, we present three numerical simulations

in the strong probe regime, with Ωp0 = Ωc0 = (2π)0.35 MHz. In the first, the traps


are equal and the fields are co-propagating. In Fig. 4.8(a) a snapshot of the densities

N |ψ1|2, N |ψ2|2, and N |ψD|2 are shown. As one would expect, the dark state wave

function, ψD is nearly equal to the original ground state. Fig. 4.8(b) tracks the energy.

The total energy E (defined in Eq. 4.13) is nearly constant though it is converted

from primarily E(V )1 and E

(int)11 , into a combination of these and E

(V )2 , E

(int)22 and E

(int)12 .

The energy E does experience a slow reduction due to atomic loss. If one defines Eg

to be the ground state energy with all atoms in |1〉 for the given number of atoms

remaining as a function of time, then in the case plotted, E ≈ Eg at all times. In

other words, no excitations are induced.

In contrast, Fig. 4.8 (c-e) are the corresponding plots for a case with V2 = −2V1.

Now many more losses are induced. As expected from (4.42), the loss of density

is most severe near the condensate edge, where the Zeeman shift causes Ω∆, and

therefore γD, to be largest. The losses are significant enough that the wave function

ψD begins evolving (according to (4.42)) as the probe passes through. After the

probe pulse has passed through the wave function continues to slosh around. From

an energetic point of view, this is because E > Eg and the available energy goes into

breathing kind of motion at this point.

It is interesting to note that free energy G ≡ E − Eg is present (and therefore

excitations) even though the energy of the cloud is smaller than the initial energy.

The excitation induced in the case presented here is simply a breathing. More inter-

esting excitations occur from the removal of small wavelength density perturbations,

a subject we discuss in Chapter 8. The free energy analysis presented here is relevant

there as well.


Figure 4.8: USL induced atomic dynamics with a strong probe. (a) A strongprobe simulation, with the same cloud parameters as previous figures and equal trapsand scattering lengths. Here Ωp0 = Ωc0(2π)0.35 MHz. The pulse width is τ0 =0.768 ms. The plots shows the densities at t = 0.864 ms of N |ψ1|2 (thick solid curve),N |ψ2|2 (dashed curve), N |ψD|2 (dotted curve), and the original ground state density(thin solid curve). The dark state never deviates far from the original density inthis case. (b) The total energy E (thick solid curve) as function of time, along with

E(int)11 +E

(V )1 (thick dashed curve), E

(int)22 +E

(V )2 (thin dashed curve) and E

(int)12 (dotted

curve). The ground state energy for the given number of atoms remaining (Eg) isindistinguishable from E (note that initially Eg = 0.6µ in a 1D TF condensate). (c)The same plot as (a), in the case V2 = −2V1. Here absorptions have caused the darkstate to deviate from the original density. (d) The same plot as (c) at t = 7.584 ms,which is after the probe has passed and all remaining atoms are in |1〉. The wavefunction here, once the probe has passed, is not in the ground state and thereforecontinues to evolve. (e) The same plot as (b) in this latter case. Here we also plotEg (thin solid curve) and see that there is excitation energy available (E > Eg) afterthe probe has left at t = 6.5 ms.


Note also that while the pulse is still compressed in the BEC, E(V )2 + E

(int)2 is

actually negative due to the large repulsive potential, and therefore E < Eg for some

of the time. This is because light fields can pump energy in and out of the system in

way so E is not a conserved quantity.

Finally, we look at a case where the two-photon momentum kick plays a significant

role in the dark state dynamics. Results of a numerical calculation in the counter-

propagating geometry, with a two-photon recoil Vrecoil = 6 cm/s, are presented in

Fig. 4.9. In the Fig. 4.9(a), the pulse is compressed. Atoms in the dark state travel

with a fraction of the recoil velocity, depending on the fraction of atoms in |2〉, in

accordance with (4.42). As a result, when the atoms are returned purely to |1〉 they

have travelled some distance, leading to a net displacement of the cloud, which we

see in Fig. 4.9(a). This process continues until the pulse exits outside the other side.

At this point, the BECs center of mass has clearly moved in the positive z direction,

as shown in Fig. 4.9(b). The trap then causes a dipole oscillation about z = 0 and

so several milliseconds later its center of mass has moved back in the z < 0 direction.

In the absence of dissipation, this oscillation will continue at the same amplitude.

The position of the cloud’s center of mass is plotted in Fig. 4.9(c). The magnitude

and speed of the fast initial movement to the right is governed by the recoil velocity

and relative probe to coupling field strength, while the ensuing oscillation period is

governed by the trap frequency. From the plot we note that the direction of the atoms

reverses almost immediately upon the exit of the pulse. In Fig. 4.9(d) we plot the

energies. A striking feature here is that the energy of the system actually increases

after the probe pulse has passed. As discussed before, the light fields can cause energy


z m)

c

De

nsi

ty(1

0c

m)

13

-3

d

a

b

tms)

d

Mean

positio

n

m)

E/

c

Figure 4.9: Moving dark state with a strong probe. Numerical simulations arefor the same parameters as Fig. 4.8 but in a counter-propagating geometry (k2 =(4π)/λ), a12−k(eff) = 0 and τ0 = 0.38 ms. (a) At t = 0.43 ms, the density N |ψ2|2(dashed line) is recoiling, and center of the mass of the density N |ψ1|2 (thick solidcurve) has moved to the right because some of the atomic probability has recoiledwhile in |2〉. Most of the atoms are in the dark state N |ψD|2 (dotted line). Forreference, the original density is shown as a thin solid curve. (b) The atomic densityN |ψ1|2 at t = 11.5 ms (thick solid curve), after the probe field has passed. It thenoscillates back and is shown at t = 18.0 ms (dashed curve). (c) The mean position(first moment in z of the total density) as function of time (solid curve). The meanposition of the atoms in |1〉 is shown as a dashed curve. The point at which the twocurves join indicates when the probe pulse has exited and ψ2 = 0 everywhere. Theinitial increase in the mean position is due to the light field induced momentum kickand the subsequent oscillation back is due to the trap potential. (d) The energy(with the same convention as Fig. 4.8). Here we see the momentum kick causes anincrease the in total energy of the system.

to flow in and out of the system, and in this case the recoil adds a substantial amount

of kinetic energy. In the case presented, less than 4% of the atoms are removed from

the condensate via spontaneous emission, and a significant free energy G = E − Eg

has resulted.

As we discussed in Subsection 4.2.4 the atomic recoil reduces the overall light-atom

interaction as Vg approaches Vrecoil, which acts back to increase the group velocity.


Therefore, the group velocity can never be reduced below this atomic velocity, and

the light will always eventually “outrun” the atoms.

Chapter 5

Stopping, storing and reviving light

While we developed a formalism in Chapter 4 that was quite general, all our

calculations to this point have studied our initial ultra-slow light experiments of a

c.w. coupling beam and a pulsed probe. We have used our formalism to show how

the atoms respond to the pulse propagation and eventually to derive the limitations

for slow group velocities and creation of local defects in BECs. We have taken the

viewpoint that the coupling beam, along with the atoms, determine the linear probe

susceptibility and we’ve seen that the coupling field intensity, via Wc ≡ Ω2c0/Γ, is an

important parameter in this susceptibility. Because we have the ability to precisely

controlWc it is natural to ask about the possibility of drastically changing the suscep-

tibility in either time or space. The technique of spatial engineering of the coupling

beam is explored in Chapter 7.

Immediately upon publishing our initial USL results we noted that, because our

pulses are completely contained inside the atomic medium, it would interesting to ex-

plore the possibility of switching the coupling beam off while the probe was inside and

153

Chapter 5: Stopping, storing and reviving light 154

predicted that this would halt the probe pulse and freeze the BEC in a superposition

of |1〉 and |2〉 [83]. This motivated an exploration (both experimental and theoreti-

cal) of the consequences of a time-dependent input coupling intensity, the results of

which we present in this chapter. While it is not surprising that varying the coupling

intensity as the pulse propagates varies the group velocity (according to Vg ∝ Wc),

there are issues which are not immediately obvious. First, what if the coupling field

is switched completely off, making the EIT width Wc → 0? Second, how fast can the

coupling intensity be changed (i.e. what is the adiabatic requirement for the coupling

field changes?

In regard to the first question, we (and others independently [84]) found the sur-

prising result that, upon variation of Wc, the probe adjusts its amplitude to keep

the ratio of the two intensities constant in time. In the extreme limit that the cou-

pling field is switched off (Wc → 0) the probe intensity ramps to zero, and all the

information (both amplitude and phase) which was in the original pulse is written

onto and stored in the atomic wave functions ψ1, ψ2. This coherent information can

be retrieved by switching the coupling beam on after an arbitrary storage time, at

which point the probe pulse is revived to its original form and then propagates out

the other side. Importantly, we realized both the switch-off and switch-on are com-

pletely coherent and reversible as the probe is constantly adjusted to keep ΩNA (see

Eq. (2.11)) small [2]. Just as in a regular USL experiment, the output probe pulse

maintains the input’s amplitude and phase to a good approximation. Thus one can

temporarily bring Vg → 0 without degrading coherence in the system.

These results were first seen experimentally in our original “stopped light” exper-


iments [2], where pulses were compressed in an atomic medium and stored for several

milliseconds, before being revived and propagating out the other side. Pulses which

were stored for times on the order of ∼ 100 µs were indistinguishable from regular

USL pulses, indicating that the original probe information was coherently transferred

into the atoms, then perfectly maintained in the atomic sample during the storage,

and subsequently transferred back onto the probe field. A later experiment at ITAMP

[13] demonstrated similar effects, though in that case, the probe was not completely

contained.

In regards to the second question, in [2] we were the first to note the surprising

and important result that the coupling field can be switched off and on with time

scales much faster than the adiabatic time requirement for the input probeWc τ−10 ,

discussed in Chapter 3. This point is expounded upon in detail in this chapter.

USL with completely contained pulses, combined with switching the coupling

field intensity off and on, provides a powerful and robust method to transfer coherent

information between lights and atoms (and vice versa), and store coherent optical

information in atomic media. In computing langauge, the switch-off is a “writing

process” (from the light fields to the atoms) and the switch-on is a “read process”. The

fact that the coupling field can be switched quickly is important from the standpoint

of the speed and robustness of these processes. The important difference between this

system and conventional computing is that it is completely reversible and coherent (in

the sense that the amplitude and phase information are preserved). We emphasize

mean fields in this analysis, but the transfer of quantum information (i.e. beyond

the mean field) between atoms and light may work equally well [14]. In the present


chapter, we concentrate on storage times with time scales sufficiently short that the

atomic dynamics play no role. In Chapter 6 we will consider some of the unique

opportunities provided by using a coherent medium (a BEC) in which to store the

light, and storing it there for times on the scale of atomic dynamics. As we will

see, this leads to novel two component BEC dynamics and coherent processing of the

information originally input on the pulse.

In Section 5.1 we study analytically (with supporting numerical simulations) what

happens when one quickly changes the input coupling field in time while the probe

pulse is contained in the BEC. The results are applicable in both the weak and

strong probe regimes. We will see the probe responds in a way that maintains the

dark state Ωp = −(ψ2/ψ1)Ωc. This is contrast to USL propagation with constant

Wc, studied in Chapter 3, where it was the wavefunctions which adjusted to light

fields. Our analysis will clearly demonstrate the somewhat surprisingly result that,

as long as the probe pulse is completely input into the medium, the coupling beam

input Wc can be switched arbitrarily fast to any value (even Wc = 0) without causing

absorption or affecting the atomic wave functions ψ1 and ψ2.

In Section 5.2, we present our initial “stopped light” experiment, which demon-

strates the stopped and revived pulses, with storage times of up to several milliseconds

in a cold cloud just above the critical temperature for condensation. The experiment

also demonstrated switching the coupling field on to different intensities, as well as

“double” and “triple” read out where the information was written back onto the probe

field in several distinct pulses.

Finally, in Section 5.3 we consider how using more complicated temporal dynamics


of the coupling beam can be used to achieve group velocities significantly below the

lower bounds on group velocities found in Chapter 4. Dynamically changing Wc can

temporarily reduce the bandwidth, allowing the pulse to propagate at the lower group

velocity, but only through a small optical density. This bypasses the role that the

large total optical density D0 usually plays in the pulse transmission width.

5.1 Fast switching of the coupling field

In this section we show analytically how the group velocity Vg, being proportional

to Wc, dynamically changes when Wc is varied. At the same time, the probe pulses’

temporal width τ0 adjusts in a way that prevents the adiabatic condition from ever

being violated, even as Wc → 0.

We will see the probe’s amplitude changes in such a way as to maintain the dark

state Ωp = −(ψ2/ψ1)Ωc. Remarkably, this dark state following of the probe will

occur even when coupling field is changed much faster than the adiabatic time scale

for the probe pulse we previously obtained for EIT (τ0 W−1c ). It turns out that

this adiabatic requirement is only relevant while the probe is still being input into the

medium. In terms of the dark/absorbing pictures developed in Section 2.3, we will see

Ωp quickly and smoothly adjusts in way that always keeps the non-adiabatic coupling

rate OmegaNA ≈ 0 (defined in Eq. 2.11). When the probe is already compressed

in the atomic cloud, its amplitude immediately adjusts with a fast switching of the

coupling field input as long as the switching is slow compared to the (extremely short)

excited lifetime Γ−1(= 16 ns in sodium). In this case, the switching is a completely

coherent (stimulated) process. Importantly, the wave functions ψ1, ψ2 are completely


unaffected by the switching.

When the switching is faster than Γ−1, the light fields no longer maintain the

dark state. Then the probe adjustment is dissipative rather than coherent and can

not happen on a time scale faster than Γ−1. However, it still ultimately adjusts

to the dark state, and the number of absorption events during the adjustment are

still negligible so ψ1, ψ2 are still completely unaffected by the process. The physical

origin of the robustness of ψ1, ψ2 derives from a fundamental difference between this

situation and the ultra-slow light experiments we have considered up to this point.

In those experiments, the light fields acted to drive the atomic samples into the dark

state as the probe enters the medium. However, once the probe is compressed, we

saw in Chapter 3 that all but a tiny fraction of the probe energy is in the coupling

field and the atoms. At this point the atoms act as a reservoir which drives the probe.

Therefore, when the coupling field is changed, the probe is driven into the new dark

state.

To see this behavior we will, instead of making the a dark/absorbing transfor-

mation of the atomic fields, make a dark/absorbing basis transformation of the two

light fields (Subsection 5.1.1) and concentrate on the co-propagating case. In Sub-

section 5.1.2, we will see that absorption becomes significant only when the length

scale of variations in the ψ1 and ψ2 is so short that the atoms can not drive propa-

gating light into the dark state superposition. If the input pulse satisfies the original

adiabatic requirement τ0 W−1c and is completely contained in the medium, then

the requirement on the length scale of variations in ψ1, ψ2 is automatically satisfied.

The analysis will also recover our previous result that when the pulse is still being


input into the medium (i.e. before it is contained), it must satisfy the EIT adiabatic

condition to avoid a large number of absorption events.

We consider switching fast compared to the radiative lifetime Γ−1 in Subsec-

tion 5.1.5. In Subsection 5.1.6, we turn to the more complicated question of dynam-

ically changing Wc in counter- and orthogonally-propagating geometries. Numerical

simulations of these geometries will show the coherent switching is still very successful.

5.1.1 Dark/Absorbing field basis

We consider ramping the input coupling Rabi frequency at a chosen switch time

ts from its initial value Ω(1)c0 to some new value Ω

(2)c0 over the course of a short time

interval [ts, ts + τs]. The probe propagation dynamics occur with a time scale τ0 and

so we will assume τs τ0 to prevent having to consider complications associated with

propagation during the switching. Note that this means we are specifically considering

fast switching (relative to the EIT bandwidth) and in fact at this point, we make no

assumption on how small τs can be. We will also assume τs µ−1 so we can safely

ignore atomic dynamics during the switching process. We will show that, to a good

approximation, the switch-off of the coupling field at the input position propagates

throughout the cloud with very little attenuation or delay. When this happens, the

probe switches in a way which maintains the dark state (4.26) at all R.

We proceed by considering Eqs. (4.10) and (4.19). For this calculation, we assume

co-propagating (k2 = 0), resonant (∆p = ∆c = 0) beams, and only consider the

dynamics in the z dimension. The generalization to the orthogonal and counter-

propagating cases is done in Subsection 5.1.6. We assume that the ground state


amplitudes ψ1 and ψ2 are constant in time during the switching interval and are

simply given by their (space dependent) values at ts:

ψ1,2(R, t) ≈ ψ01,2(R) ≡ ψ1,2(R, ts) for t ∈ [ts, ts + τs]. (5.1)

By the assumption τs µ−1, the Hi in (4.10) do not change the wave functions,

however, it is not immediately clear that the light field coupling terms do not. We

make the assumption (5.1) now, and show apostori that it is well satisfied. It is easiest

to consider the remaining three equations of (4.10) and (4.19) in frequency space (FT

t → δ via (3.7)) . Fourier transforming and solving for the equation for ψ3 in (4.10)

and plugging the result into the Fourier transforms of (4.19) yields:

d

dzΩp = −1

2f13σ0Nψ

01

1

1 − iδ (Ωpψ01 + Ωcψ

02),

d

dzΩc = −1

2f23σ0Nψ

02

1

1 − iδ (Ωpψ01 + Ωcψ

02),

where δ ≡(δ

Γ/2

). (5.2)

So we can consider fast switching times τs ≤ Γ−1 (which would introduce δ ≥ 1), we

have not adiabatically eliminated ψ3. Adiabatically eliminating ψ3 would amount to

setting δ = 0. We now perform a transformation into the “dark” and “absorbing”

light fields :

ΩD

ΩA

=

1

ψ0

−ψ0

2 ψ01

ψ01 ψ0

2

Ωp

Ωc

,

where ψ0 ≡√|ψ0

1|2 + |ψ02|2 (5.3)


so named because the absorbing field ΩA is quickly damped due to absorption by the

atoms, while the dark field ΩD, in the absence of spatial dependence of the atomic

fields ψ01, ψ

02, propagates freely. In Chapter 2 we saw that time-dependent light fields

introduce a non-zero coupling between the dark and absorbing amplitudes cD, cA.

Here, conversely, spatially dependent atomic fields ψ01, ψ

02 cause non-zero coupling

between the dark and absorbing light fields ΩD,ΩA. We note that the matrix in (5.3)

can also be used to define ΩD,A in terms Ωp,c in the time domain (we will use ΩD,A

later). Writing (5.2) in our new basis gives:

d

dzΩD =

(−αNA +

α12

1 − iδ), ΩA(

d

dz+αA

1 − iδ)

ΩA = αNAΩD,

where αA ≡ Nσ0

(f13(ψ

01)2 + f23(ψ

02)2),

α12 ≡ Nσ0(f13 − f23)ψ01ψ

02,

αNA ≡ ψ01

ψ0

d

dz

(ψ0

2

ψ0

)− ψ

02

ψ0

d

dz

(ψ0

1

ψ0

)

−ψ01

ψ0

d

dz

(ψ0

1

ψ0

)− ψ

02

ψ0

d

dz

(ψ0

2

ψ0

). (5.4)

The length (αNA)−1 determines of rate at which the ΩA gains amplitude because the

superposition that determines it is changing in space. One can show when the phases

of ψ1 and ψ2 are π apart, as is the case for many of the USL experiments we have

analyzed to this point (namely whenever ∆2 = 0 and H1,2 are negligible), the last two

terms in the definition of αNA cancel, and the first two terms sum to an expression

which is determined by the pulse length Lp (which is typically Lp ∼ 10 µm):


αNA ≈ 1

(ψ01/ψ

0)

d

dz

(ψ0

2

ψ0

)

∼(

Ωp

Ωc

)2

Lp

(5.5)

We will see below that this length is to be compared with (αA)−1, which we call the

preparation length, and is analogue in space to the EIT frequency width W . In the

weak probe limit (|ψ2| |ψ1|), it is the normal light field absorption length in a

two-level medium and even in the strong probe case it is of the same order. At the

center of our BECs (αA)−1 is less than 1 µm for typical densities. However, it grows

and actually diverges at low densities near the condensate edge.

5.1.2 Switching slowly compared to the natural linewidth

The simplest case, and most common one in practice, is switching slowly compared

to the radiative lifetime, τs Γ−1. Then we consider (5.4) only to first order in δ.

When (αA)−1 (αNA)−1, we can ignore the spatial derivative in the equation for

ΩA, in a method analogous to adiabatic elimination and solve for ΩA:

ΩA =αNA

αA(1 − iδ)ΩD, (5.6)

thus giving us (αA)−1 (αNA)−1 as a criterion for keeping the ΩA ΩD. Using

our weak probe result for Lp (3.33) in the new mean field notation (see Eq. (4.18)),

we see that maintaining a small ΩA is equivalent to maintaining a small absorbing

amplitude ψA:


(αA)−1 (αNA)−1 ⇔ W (1)c τ0

Ωp

Ωc0

, (5.7)

The latter inequality is actually the requirement we found in Chapter 4 to keep ψD

ψA . However, recall that we have already had to make the much stronger assumption

W(1)c τ0 √

D(z) in order to propagate the pulse to the point z (see Eq. (3.31)). In

other words, if the input probe pulse satisfies the bandwidth requirement set by the

original coupling intensity W(1)c , then the coupling field can always be switched (even

with τs much faster than τ0) and introduce no additional absorption.

We can then plug (5.6) back into (5.4) to see that the dark field will then propagate

according to:

dΩD

dz=

[−α

2NA

αA(1 − iδ) +

αNAα12

αA

]ΩD. (5.8)

Whenever we assume the weak probe limit, we are actually completely neglecting

terms which are of the order of the right hand side of (5.8). In that limit ΩD is a nearly

freely propagating field: changes introduced at the input (zin), will propagate nearly

instantaneously across the entire medium so ΩD is always approximately constant in

space. Even in the strong probe regime, the polarization seen by ΩD, according to

(5.8) is very small. The result (5.8) gives us finite corrections of O(Ω2p) which give

a finite absorption coefficient and group velocity for the dark state field ΩD. This is

accomplished by picking out the terms of zeroth and first order in δ (as we discussed

in 3.1.3 and did in the regular USL case with (3.29)). We find (in the case of equal

oscillator strengths a12 = 0) that they are:


αD =Reα2

NAαA

, VD =Γ

2α−1D , (5.9)

respectively. When α12 = 0 this result must be modified but is qualitatively the

same. The imaginary part of αNA contributes phase shifts. It interesting to note that

if ψ1, ψ2 are not changing during the switching, then αD, VD, and therefore the ratio

ΩA/ΩD (see Eq. 5.6)), remain constant in time at each position.

To translate this result back into the Ωp,c basis, we note when the pulse is contained

in the condensate, then (ψ02/ψ

0) = 0 at the condensate edge zin. So at that point,

(5.3) gives ΩD = Ωc. In the region occupied by the probe pulse, Ωp, ψ2 = 0, so

ΩD,A each contain admixtures of both Ωp,c. The probe must adjust with the coupling

field in way that to keeps ΩD constant in space and ΩA ≈ 0. Note that in the time

domain, this is equivalent keeps the non-adiabatically ΩNA ≈ 0, thus indicating that

the system continues to primarily undergo coherent exchange rather than absorption

events. We will return to this point in Subsection 5.1.3. In the case that the coupling

field is ramped to zero, this adjustment continues until both fields have completely

vanished. At this point we are left with the atoms in superpositions of |1〉 and |2〉

with no light fields present.

An example of a switch-off in a case where (5.7) is well satisfied is shown in

Fig. 5.1. The coupling field is switched off with τs τ0 when the probe is completely

contained. The absorbing field ΩA is originally non-zero in the region occupied by the

probe. In Fig. 5.1(a) we see that the ratio |ΩA|2/|ΩD|2 remains small and constant

at each z as the switch-off occurs. The only exception is near the entering edge

of the condensate, where the atoms have not yet driven the incoming light into a


Figure 5.1: Coherent fast switching of the light fields. (a) Spatial profiles of the

dark field intensity |ΩD|2 (dashed, normalized to (Ω(1)c0 )2) and absorbing field |ΩA|2

(solid, same normalization and scale set on the right hand side) at different timesduring a switch-off. The coupling input (at zin = −50 µm) is switched off at ts =3.5 µs (once the probe is contained within the cloud), with an error function profilewith a width τs = 0.1 µs. Successively thinner lines refer to t = 3.52, 3.65, 3.68, and3.75 µs. The calculation is done with the numerical code of Chapter 4 in 1D (with thesame parameters for the condensate). The dotted curve shows the original condensatedensity profile (in arbitrary units). In the simulation, τ0 = 1.5 µs, Ωp0 = (2π)3.5 MHzand Ωc0 = (2π)8.0 MHz. (b) Spatial profiles of the probe field intensity |Ωp|2 (solidcurves, normalized to Ω2

p0) and coupling field |Ωc|2 (dashed curves, normalized to

(Ω(1)c0 )2) at the same times as (a) for the same calculation.

purely dark field. While ΩA eventually diverges there, it is inconsequential because

it only occurs where there is negligible density. The attenuation and delay due to

(5.8,5.9) are not visible. There is a small inhomogeneity in |ΩD|2 due to the oscillator

strength difference f13 = f23, which makes α12 = 0 in (5.4). Translating back into the

probe/coupling picture in Fig. 5.1(b), we see the coupling and probe fields switch-off

everywhere together. The probe constantly adjusts itself both in time and space so

ΩA ≈ 0. We note that (5.4) does not depend on the weak probe limit. The probe in

Fig. 5.1(b) is in fact strong enough for the adiabatons to be visible in |Ωc|2 and they

are seen to play no role in the switching.

While the light fields are off, the probe’s information is stored completely in ψ1,2.

We label the time of the switch-off t(1)s . At an arbitrarily later time, which we label


t(2)s , the coupling field can be switched on again to a new input value, which we call

Ω(3)c0 . Then probe turns on, again in a way that keeps ΩA ≈ 0. If the atoms have

not significantly evolved in the interim and the coupling field is switched on to its

initial value (Ω(3)c0 = Ω

(1)c0 ), then the probe is restored to the amplitude and phase

profile it had before the switch-off. A numerical simulation of such an experiment,

is shown in Fig. 5.2. Fig. 5.2(a), which plots a regular USL simulation and a stored

light simulation on the same plot, shows that, remarkably, the output (revived) probe

pulse is indistinguishable from a case where the coupling is kept on, except for an

additional delay equal to the time the coupling field is kept off. The output from the

revived pulse has the same intensity and width as the normal USL pulse, and any

attenuation from the input occurs during the propagation rather than switching. In

Fig. 5.2(b), we plot ψ2 as a function of time at a particular position z′ within the

region occupied by the probe pulse at the switch-off time. The wave function ψ2 is

driven by the light fields during the propagation, but does not change amplitude or

phase during the switching itself. During the storage time, it undergoes a slow phase

rotation, due to external dynamics, at a rate µ, as discussed in Chapter 4.

If the atomic wave functions do evolve considerably during the storage, the probe

will be revived in a way that keeps ΩA ≈ 0 (i.e. Ωp = −(ψ2/ψ1)Ωc) with all quantities

evaluated at the switch-on time t(2)2 , assuming that (5.7) is still satisfied. This is

essence of the coherent processing discussed in Chapter 6.

Fig. 5.2(c) shows that the evolution of ψ′3 follows a path determined by the non-

adiabaticity of the probe during regular USL propagation, as discussed in Subsec-

tion 3.2.7. It then abruptly and smoothly goes to zero with Ωc0 during the switch-off


Figure 5.2: Stopping, storing and reviving a probe pulse. (a) We plot theinput and out output fields for the same numerical simulation as Fig. 5.1. The inputprobe pulse (thick dotted curve) is fully input before the coupling beam (thin dotted

curve) is abruptly switched off at t(1)s = 3.7 µs. It is then switched on again at

t(2)s = 15.3 µs. The output coupling field (thin solid curve) is seen to closely follow

the input. They are indistinguishable during the switching and only differ wherethe adiabatons occur (see Subsection 3.3.1). After the switch-on, we see the revivedprobe pulse (thick solid curve). For reference, we plot the output probe (thick dashed)and coupling (thin dashed) for a regular USL case (the coupling input kept constantfor the entire time). (b) At the point z′ = −12 µm, we plot the wave function ψ2

normalized to the original condensate wave function at that point ψ(0)1 (z′). The thick

solid curve is the real part, and the thin solid curve the imaginary part. The dashedcurves again show the case with no switching. The switching itself is seen to have noeffect on ψ2. During the storage time, there is a slow evolution (primarily a phaseevolution at µ as discussed in Chapter 4). (c) The wave function ψ3, calculated withthe adiabatic elimination expression (4.20) (which is valid whenever τs Γ−1), withthe same convention as (b). The excited level amplitude is seen to scale smoothlywith the coupling field as it switches off and on, showing that the switching causesno additional absorption. (d) The thick solid curve shows the total fractional loss ofatoms as a function of time, as compared to the case without switching (thick dashedcurve). The total loss in the end is seen to be the same, only part of it is delayed bythe storage time. The other two curves plot cases discussed later in the text. Thethin dashed curve shows a case where the coupling field is switched to twice its valueinstead of off (see Fig. 5.3 for parameters and discussion). The thin solid curve showsthe much greater losses which occur in the case when the probe is not contained (seeFig. 5.4).


and remains at zero during the storage. Upon the switch-on it then recovers its value

before the switch-off and continues to evolve as in normal USL. The fractional total

loss of atoms from the system in case shown is only ∼ 0.008 (Fig. 5.2(d)). This

loss are purely due to non-adiabaticity of the probe and is therefore the same as the

regular USL case.

The coupling field can be switched to arbitrary values and drive the probe field

to follow it. For example, in Fig. 5.3 we show a case where the coupling intensity is

switched on to twice its original value (W(2)c = 2W

(1)c ) instead of switched off. The

figure again compares to a case with a constant coupling field (regular USL). The

probe doubles its intensity and its group velocity doubles as a result. In Fig. 5.3(a)

we see the output peak intensity is larger than the constant coupling case, but its

temporal width τ is smaller by the same factor, so the total energy of the output

is the same. More generally, because Lp = 2τVg, and the Lp is constant during

the switching, the pulse’s temporal width τ adjusts with Vg: τ = τ0W(1)c /W

(2)c . This

dynamic change in τ is the basic reason that the switch-off works even whenW(2)c = 0

and the EIT width vanishes. During the switch-off, the EIT width Wc is narrowing,

but the pulse’s frequency ∼ τ−1 narrows to keep it within the shrinking EIT window.

The evolution of ψ2 is nearly the same in either case (Fig. 5.3(b)). Fig. 5.3(c) shows

the amplitude of ψ3 increases when the light fields are turned up, and therefore the

absorption processes are faster during this time, but the total time that ψ3 is non-zero

is smaller so the total integrated loss is again the same (see Fig.5.2(d)).

As we noted before in the case of a storage experiment, where Ω(2)c = 0 for some

time, the revived probe is determined by the relative wave functions and the coupling


Figure 5.3: Dynamically changing the group velocity. All conventions are thesame as Fig. 5.2 except the coupling field is switched to double its original intensityat a time t

(1)s (W

(2)c = 2W

(1)c ). (a) We see that, upon this doubling of the coupling

intensity, the probe intensity and group velocity double, while its temporal width τhalves. (b) The wave function ψ2 is almost indistinguishable in the two cases excepta slightly faster return of atoms to the |1〉 state due to the larger group velocity. (c)The excited state amplitude ψ3 increases at the switch but the faster group velocityat that point leads to a shorter time that ψ3 is excited, leading to the same overallloss. The integrated loss is plotted in the previous Fig. 5.2(d) for comparison withthat case.

field at the switch-on time Ωp(t(2)s ) = −(ψ2/ψ1)Ω

(3)c0 . The revived probe pulse will

then look like the original pulse only to the extent that ψ1, ψ2, and Ωc are the same

before and after the storage. The results of a non-trivial evolution of ψ1, ψ2 during

the storage is the subject of Chapter 6. The discussion of the previous paragraph

can be applied to the case of switching the coupling intensity on to arbitrary values.

Alternatively, we can turn the coupling field on at a slightly different frequency,

ω(3)c = ω(1)

c . Then the revived probe will be shifted by ω(3)c − ω(1)

c relative to its input

frequency ωp. In general, because the probe is completely off during the storage,

and it is the coupling field we turn on, there is no preference for the original probe

amplitude, frequency or polarization and new pulse is completely determined by the

system at the switch-on time. A thorough discussion of the possibilities for changing

the second coupling pulse’s frequency or polarization go beyond the scope of this

thesis.


5.1.3 Why are the ground state wave functions so robust?

We now investigate why ψ1,2 are unaffected by the switching, and justify our

assumption that they are constant. Physically, it can be understood by the following

picture of switching process. As Ωc0 is ramped down, the new dark state contains

a larger ψ2, so atoms are stimulated from |1〉 to |2〉. Associated with this process

is the absorption of probe photons and stimulation of photons into the coupling

field. However, since the ratio of the number of probe photons to |2〉 atoms is ∼

Vg/c, the probe field is completely depleted before any appreciable change in ψ1, ψ2

have occurred. The process just described, being completely coherent when (5.7) is

satisfied, is reversed upon turn on of the coupling field. The degree to which the

condition (5.7) is broken determines the number of absorptions into |3〉, leading to

losses of both atoms and photons through spontaneous emission.

To see this mathematically, consider the GP equations (4.21). We still neglect

external dynamics (set Hi = 0) and for simplicity take ψ1,2 and Ωp,c to be purely real,

which we found to be the case for USL propagation in Chapter 3 so long as ∆2 = 0.

Transforming into the dark/absorbing field basis (5.3) and assuming ∆2 = 0:

ψ1 = −Aψ1 + S ψ2,

ψ2 = −Aψ2 − S ψ1,

where A ≡ Ω2A

2Γ, S ≡ ΩAΩD

2Γ. (5.10)

The first term in each equation represents a loss rate A into state |3〉, while the

second terms represent the stimulated exchange rate S between the ground states.


It is evident from the definitions in (5.10) that the ratio of absorbing to stimulated

events A/S remains small so long as ΩA ΩD.

To see that the stimulated events (S) do not change ψ1,2 during the switching we

calculate:

∫ ts+τs

ts

dt S =

∫ ts+τs

ts

dtΩAΩD

2Γ

∼ τs2

αNA

αAWc

∼ τs2τ0

(Ωp0

Ω(1)c0

), (5.11)

From (5.10) and (ψ02/ψ

01) ∼ (Ωp0/Ω

(1)c0 ) we see the fractional changes in the two wave

functions are :

∆ψ1

ψ1

∼ τs2τ0

(Ωp0

Ω(1)c0

)2

∆ψ2

ψ2

∼ τs2τ0. (5.12)

Both of these are indeed small for fast switching times τs τ0 and can in fact be

made arbitrarily small (except that our result does depend on τs Γ−1, since ψ3 was

adiabatically eliminated in (4.21)). The fact that the total number of events (5.11)

grows as the switching time grows reflects the fact that transitions do occur due to

pulse propagation (which takes place on a time scale τ0). No additional transitions

are induced by the switching itself. The total number of lossy events from the system

during the switching is obtained in a similar manner:


∫ ts+τs

ts

dtA =

∫ ts+τs

ts

dtΩA

2

2Γ

(αNA

αA

)2

∼(

Ωp0

Ω(1)c0

)2τs

2Wcτ 20

, (5.13)

which is smaller than the number of stimulated events by 1/(W(1)c τ0)(Ωp0/Ω

(1)c0 ).

We can draw two important conclusions from these estimates. First, we’ve learned

that neither the simulated nor absorbing rates are fast enough to effect the ground

state wave function ψ1, ψ2 upon the switching. For the examples we showed in Fig. 5.2,

the number of integrated events in both (5.11) and (5.13) are smaller than the nu-

merical noise. Second, because ψ1,2 are approximately constant, so is αNA/αA (which

we have already seen numerically in Fig. 5.1) and we can conclude stimulated events

dominate the absorbing events so long as the original adiabatic time scale for the

input probe pulse is satisfied. This is true even as Wc → 0 and for any time scale τs

for the switching (down to Γ−1). Thus the switching and probe adjustment involve

coherent, reversible processes, just as in regular USL.

5.1.4 Storing an incompletely compressed pulse causes losses

When (5.7) breaks down, then A in (5.10) can become quite large. One case where

a large amount of loss occurs is when the pulse is not compressed inside the medium,

but still entering. At the condensate edge, the non-adiabatic length scale (αNA)−1

will still be large. However, because of the small density there, the preparation length

(αA)−1 is also large. In fact, our assumption that we can ignore the spatial derivative

d/dz in (5.4) in obtaining (5.6) breaks down. Physically, the fields see a non-zero ψ2


as soon they enter the medium and do not have the benefit of a large atomic density

to damp the absorbing field ΩA. Atoms are then lost during the switching process at

the edge where the light fields enter.

An example is shown in Fig. 5.4. Fig. 5.4(a) shows that, unlike the completely

contained probe case, the output from the revived probe is reduced from the regular

USL case. Fig. 5.4(b) shows the dark and absorbing fields during both the switch-on

and switch-off. The ratio |ΩA|2/|ΩD|2 is significant until the fields propagate some

distance into the cloud. In the figure, we see this ratio is especially large during the

switch-on when it is the larger coupling field (rather than the probe) which is being

input into the medium without the other light field present to establish the dark

state. In both cases, eventually the absorptions drive ΩA to zero and the dark field

is established, but it is through absorption rather than coherent, stimulated events.

Fig. 5.4(c) contrasts the probe and coupling intensities before the switch-off, when

the probe is only halfway into the cloud (thin curves), and after the switch-on (thick

curves). The probe revival is clearly attenuated and distorted from its input shape.

Fig. 5.4(d) shows the atomic densities of the two ground states. There is a per-

ceptible density of atoms lost from both components after the switch-on, due to

absorption near the entering edge of the BEC. (We also note that during the period

when the probe is still being input into the medium, but after the switch-off, there

is an increase of |2〉 atoms near the condensate border in order to attempt to restore

the dark state). Returning to Fig. 5.2(d), we see the fractional loss of atoms (through

spontaneous emission) as a function of time, in comparison to the contained case.

The losses are much greater in this case, and are most acute during the switch-on


Figure 5.4: Losses due to stopping a incompletely compressed pulse. (a)Conventions and parameters are the same as Fig. 5.2 except the coupling field isswitched off while the probe pulse is entering (t

(1)s = 0.3 µs). As a result, the coupling

field is seen to not completely and immediately switched off or on at the output.More significantly, there is a visible reduction in the output probe pulse comparedto the regular USL case. (b) The intensities |ΩD|2 (dashed) and |ΩA|2 (solid) dur-ing the switching. The thin curves show them during the middle of the switch-off(t = 0.48 µs). Note that here |ΩA|2 is plotted on the same scale as |ΩD|2 (both arenormalized to Ω2

c0). At this point it is seen to be relatively small. The thicker curvesshow the fields during the middle of the switch-on (t = 15.31 µs) and we see |ΩA|2is rather substantial for a few microns into the cloud. (c) The probe (solid) andcoupling (dashed) intensities before the switch-off (thin lines) and after the switch-on(thick lines), showing the rather severe attenuation and shape change in |Ωp|2. (d)The density Nc|ψ2|2 before the switch-off (t = 0.31 µs, dashed), after the switch-off(t = 0.71 µs, dotted), before the switch-on (t = 15.11 µs, thin solid), and after theswitch-off (t = 15.51 µs, thick solid). The inset shows Nc|ψ1|2 at the same times nearthe condensate edge.


and switch-off.

In a case such as this one, the fields are not adjusting to keep ΩNA small. Therefore

the normal adiabatic condition on the switch time scale τs W (W ≡ Ω2/Γ) would

have to be maintained to avoid losses. Note also that if instead the switching is done

as the pulse is exiting, this problem does not occur since the dark field has been

established by the time the fields reach the far end of the cloud.

5.1.5 Switching fast compared to the natural linewidth

We now consider the case of extremely fast switching times τs ≤ Γ−1. Recall

our equations (5.4) are still valid in this regime. We will see that in this case, the

condition governing the ability the Ωp to follow Ωc and maintain ΩA = 0 becomes

more restrictive. However, even when ΩA = 0 is not maintained, the atomic fields

ψ1, ψ2 and their coherence are still unaffected by the switching process.

We use the same arguments as before to arrive at an equation analogous to (5.6),

but note that the damping of ΩA in (5.4) αA/(1 − δ) is significantly reduced when

δ ≥ 1. Since the magnitude of the frequency components present are δ ∼ τ−1s , our

requirement (αNA αA) is then replaced by:

(αA)−1

√1 +

4

Γ−2τ 2s

(αNA)−1 (5.14)

Physically, the reduction in the damping of ΩA occurs because, when τs ∼ Γ−1, the

one-photon detuning of the frequency components due to the fast switching become

large compared to the natural linewidth. But this large detuning also reduces ab-

sorption events into |3〉. In fact, the decrease in the damping of ΩA, which tends to


increase |ψ3| and the decrease in the coupling to |3〉, which reduces |ψ3|, scale together

and cancel each other. The result is that |ψ3| is of the same magnitude as before and

the atomic fields ψ1, ψ2 are still almost completely unaffected by the switching. This

is contrast to the situation in the last subsection, where an incompletely compressed

pulse caused many absorption events to occur.

When (5.14) is satisfied, we still keep ΩA ΩD and we still get primarily coherent

exchanges. The inequality fails when τs ≤ (ΓW(1)c τ0)

−1. If the smallest possible input

pulse is used τ0 ∼ τsp then this corresponds to τs ≤ Γ√D0. For τs in this regime, we

will have ΩA ∼ ΩD, A ∼ S, and Ωp will not follow Ωc during the switching. But the

important point is that the number of absorption events ∝ ∫ ts+τs

tsdtA does not grow

due to the cancellation.

To check the ultra-fast switching regime numerically, we can not use the numerical

code which includes atomic dynamics because in it, we adiabatically eliminated |3〉

under the assumption that all time scales were large compared to Γ−1. However, the

numerical code used in Chapter 3 (and outlined in Appendix D) did not assume this

and so we use it here. An example with τs = 2 ns < Γ−1 = 16 ns is shown in Fig. 5.5.

Fig. 5.5(a) shows that ΩA grows due to the lack of damping. It then oscillates (or

rings) with a period on the order of the τs and is damped to its expected value

over a time scale Γ−1. However, as we argued above, the magnitude of the excited

state amplitude |ψ3| never exceeds its value before the switching (seen in Fig. 5.5(b)),

though it also oscillates and damps with the same time scales as ΩA.

Returning the probe and coupling basis, we note from our arguments show that

we expect neither field to strongly interact with the medium. We expect, and indeed


Figure 5.5: Switching faster than Γ−1. The plots are based on numerical simu-lations of a switching with τs = 2 ns < Γ−1 = 16 ns. (a) We plot the absorbingamplitude ΩA(z′, t) normalized to Ωp0 at the same particular position (z′ = 12 µm)as plots in Fig. 5.2 and zoom in on times around the switching. ΩA is seen to jumpabove its value during the propagation and oscillate at a frequency ∼ τ−1

s while beingdamped at a rate ∼ Γ. (b) The amplitude ψ3(z

′, t) (normalized to original value

ground state wave function at that point ψ(0)1 (z′)), undergoes similar oscillations,

but never jumps above its value before the switching. (c) The probe (solid curves,

normalized to Ωp0) and coupling (dashed curves, normalized to Ω(1)c0 ). The coupling

field switches off immediately, while the probe undergoes damped oscillations untilit reaches the its dark state value. (d) The wave functions ψ1(z

′, t) (dashed) and

ψ2(z′, t) (solid), both normalized to ψ

(0)1 (z′), are still unaffected by the switching.


see numerically in Fig. 5.5(c), that the coupling field turns off smoothly and quickly.

The probe field oscillates and only adjusts to its dark state value in a time scale Γ−1.

The total fractional loss of atoms in the end is 0.008, just as in the slow switching

cases above with the pulse contained.

We performed simulations for a series of τs < Γ−1 and confirmed that |ΩA| in-

creases with decreasing τs until it reached |ΩA| ∼ |Ωp| at which point it saturates.

We also confirmed that neither the total number of absorption events nor the peak

magnitude of |ψ3| varies with τs.

5.1.6 Orthogonal- and counter-propagating geometries

The results derived so far have been for the co-propagating geometry. However,

many interesting experiments use orthogonal- and counter-propagating geometries

and so we briefly consider the effects of switching in these cases. The transformation

(5.3) is more difficult to apply in this case. Here we will show numerically that the

basic results will hold.

In Fig. 5.6 we compare results of numerical simulation of stopped and revival

experiments in several cases, both in a counter-propagating and co-propagating ge-

ometry. In the contained case, shown in Fig. 5.6(a-b), the revivals are good in both

geometries. For pulses still entering the medium during the switching (Fig. 5.6(c-d)),

both see a similar amount of attenuation. When the pulse has begun exiting the far

edge during the switching (Fig. 5.6(e-f)) the revivals are both relatively good. As we

might expect in this case, there is slightly more attenuation in the counter-propagating

case because the coupling field is entering from the z > 0 side and therefore sees |2〉


Figure 5.6: Revivals in a counter-propagating geometry. In each plot we showthe probe (solid curves) and coupling (dashed curves) directly before switching off

(thin curves) and after switching on (thick curves). In all cases, Ω(1)c0 = (2π)8 MHz,

and Ωp0 = (2π)3.5 MHz, τ0 = 1.5 µs, and we switch the coupling field off and then

back on to its original value (Ω(2)c0 = 0,Ω

(3)c0 = Ω

(1)c0 ). In the top row (a,c,e) the beams

are co-propagating, and the in the bottom row (b,d,f) we show the case of counter-propagating beams. We see the quality of the switching is mostly the same in eithergeometry. In (a,b) the switch-off is at t

(1)s = 3.7 µs (when the pulse is completely

contained), τs = 0.1 µs, and the switch-on is at t(2)s = t

(1)s + 10 µs. In (c,d) the

pulse is still entering (t(1)s = 1.3 µs, t

(2)s = t

(1)s + 10 µs). In (e,f) the pulse is exiting

(t(1)s = 8 µs, t

(1)s = 7 µs, respectively, and t

(2)s = t

(1)s + 10 µs).

atoms before ΩA is damped.

In Fig. 5.7 we plot 2D numerical simulations in an orthogonal geometry. We see

that the revival quality is still quite good. Because of the two-photon momentum

kick, there is a small but perceptible kick during the off time. Examining the cuts

along z we note the revival is almost perfect (Fig. 5.7(c,d)). However, because the

probe occupies the entire transverse width of the BEC, the coupling field sees a non-

zero ψ2 immediately upon entering the x < 0 side. This causes some small loss and

asymmetry. This is most evident in Fig. 5.7(f) where we note that the absorbing

amplitude during the revivals is slightly higher on the side where the coupling field


Figure 5.7: Revivals in a orthogonal-propagating geometry. Numerical resultsof a 2D calculation for a stopped, stored and revived pulse. In the calculation,the condensate and trap parameters are the same as Fig. 4.3. The coupling beampropagates along x, Ωc0 = (2π)8.0 MHz, Ωp0 = (2π)3.5 MHz and τ0 = 1.5 µs.

(a,b) The density plots show the density Nc|ψ2|2 at the switch-off t(1)s = 4.1 µs and

the switch-on t(1)s = 14.1 µs, respectively. Both have a width both with a width

τs = 0.1 µs. The dashed curve shows the TF boundary of the original condensate in|1〉 and the dotted lines show the cuts along z and x presented in (b-e). (c) A cutthrough the center along z (indicated in (a,b)) of the normalized probe intensity |Ωp|2right before the switch-off (dashed curve) and right after the switch-on (solid curve).

(d) The absorbing intensity |ΩA|2 (normalized to (Ω(1)c0 )2) along the same cut. (e,f)

Cuts along x (indicated in (a,b)) of the same quantities as (c,d), respectively, withthe same conventions.

enters. As a result, in Fig. 5.7(e) there is slightly more asymmetry in the revived

probe than the probe before the switch-off.

5.2 Stopped and revived light experiments

We experimentally observed storage and revivals of completely contained light

pulses for up to several milliseconds last year [2] in a cold thermal cloud (just above

the critical temperature for condensation) in a co-propagating geometry. Although we

have analyzed the problem above in a BEC formalism, the analysis of the switching

in thermal samples is exactly the same since the time scales are generally much


Figure 5.8: Schematic of light storage experiment. Figure taken from [2]. (a)Relevant energy levels. Note level |4〉 causes the system to be in an anti-symmetricdouble-Λ system, which introduces dephasing and reduces transmission. (b). Exper-imental set-up. We apply a 2.2 mm diameter, negatively circular polarized couplinglaser (εc = (x − iy)/√2) and a co-propagating, 1.2 mm diameter positively circularpolarized probe pulse (εp = (x + iy)/

√2). The two laser beams start out with or-

thogonal linear polarizations (two-headed arrows and filled circles show the directionsof linear polarization of the probe and coupling lasers, respectively). They are com-bined with a beam splitter, circularly polarized with a quarter-wave plate (Λ/4), andthen injected into the atom cloud. After leaving the cloud, the laser beams pass asecond quarter-wave plate and regain their original linear polarizations before beingseparated with a polarizing beam-splitting cube. The atom cloud is imaged first ontoan external image plane and then onto a CCD camera. A pinhole is again used soonly those portions of the probe and coupling laser beams that have passed throughthe central region of the cloud are selected and monitored simultaneously by twophotomultiplier tubes (PMTs). States |1〉 and |2〉 have identical first-order Zeemanshifts so the two-photon resonance is maintained across the trapped atom clouds.

faster than atomic dynamics. Therefore the results of the previous section hold (with

the correspondence (4.18)). The differences between condensed and non-condensed

samples become apparent when one considers sufficiently long storage times that the

atomic dynamics become important, an issue which is analyzed in Chapter 6. Later,

another group saw storage and revivals of parts of pulses in a hot Rubidium-87 vapor

[13, 85].

We now present our storage experiments, which were published in [2]. The energy

level structure and an experimental schematic is shown in Fig. 5.8. In the experiment,

pulses with τ0 = 2.85 µs were input and stored in thermal clouds of N = 11 × 106


atoms at T = 0.9 µK with peak density 1.1 × 1013cm−3. Fig. 5.9(a) shows the result

of a regular USL experiment, with a input constant coupling Rabi frequency of Ω(1)c0 =

(2π)2.57 MHz. Delays of 11.8 µs were observed, in agreement with the theoretical

expectation of τd = 12.2 µs. This corresponds to a group velocity of 28 m/s. This Ω(1)c0

corresponds to W(1)c = (2π)0.26 MHz and an intensity of 3.2 mW/cm−2. Some of the

pulse attenuation is due to the dephasing from level |4〉 (we have an anti-symmetric

double-Λ system, diagrammed in Fig. 5.8(a)) and some is the result of using pulse

widths τ0 on the order of τsp = 3.4 µs. This was necessary because thermal clouds

have substantially smaller densities than condensates and f13 was only 1/12 in this

case, making D0 = 51 through the center of the cloud. With this fairly small D0 one

needs τ0 ∼ τsp in order to achieve complete compression of the pulse (τ0 < τd/2). One

sees experimentally that compression was accomplished since the input and output

pulses in Fig. 5.9(a) are clearly separated. The arrow in the figure indicates the time

that this occurs. The input pulse contained 27,000 photons, and the energy of the

pulse when fully compressed in the cloud was equal to the energy of 1/340 of a photon.

We then performed the experiment under the same conditions but switched the

coupling beam off (W(2)c = 0) with a time scale τs ∼ 1 µs, while the pulse was

completely contained (at t(1)s = 6.3 µs). Thus Γ−1 τs τ0, putting us in the

regime analyzed in Subsection 5.1.2. A case where the coupling field was switched

off and then switched on after a storage time of about t(2)s = 40 µs is shown in

Fig. 5.9(b). Remarkably, there is no detectable difference in the pulse peak intensity

or width from the regular USL propagation in Fig. 5.9(a). The information in the

probe was frozen and stored in the atomic medium for this time with little or no


-Time ( s) C

ouplin

gIn

tensity

(mW

/cm

)2

0

2

3

4

0 20 40 60 8020

0

0.2

0.4

0.6

0.8

1

1

a

Norm

aliz

ed

Pro

be

Inte

nsity

0 250 500 750 1000 1250 1500

0.05

0.1

0.2

d

Time ( s)

Tra

nsm

issio

n

20 0 20 40 60 80

0

0.2

0.4

0.6

0.8

1

0

1

2

3

4b

- 20 0 20 820 840 860

0

0.2

0.4

0.6

0.8

1

0

1

2

3

4c

Time ( s)

Norm

aliz

ed

Pro

be

Inte

nsity

Norm

aliz

ed

Pro

be

Inte

nsity

Couplin

gIn

tensity

(mW

/cm

)2

Couplin

gIn

tensity

(mW

/cm

)2

Figure 5.9: Measurements of delayed and revived probe pulses. Figure takenfrom [2]. Open circles (fitted to the dotted gaussian curves) show reference pulsesobtained as the average of 100 probe pulses recorded in the absence of atoms. Dashedcurves and filled circles (fitted to the solid gaussian curves) show simultaneouslymeasured intensities of coupling and probe pulses that have propagated under EITconditions through a 339 µm long atom cloud cooled to 0.9 µK. The measured probeintensities are normalized to the peak intensity of the reference pulses. Typically,Ωp0/Ωc0 = 0.3. (a) Probe pulse delayed by 11.8 µs. The arrow at 6.3 µs indicates thetime when the probe pulse is spatially compressed and contained completely withinthe atomic cloud. (b,c) Revival of a probe pulse after the coupling field is turnedoff at t = 6.3 µs and turned back on at t = 44.3 µs and t = 839.3 µs, respectively.The time constants for the probe and coupling PMT amplifiers are 0.3 µs and 3 µs,respectively. The actual turn on/off time for the coupling field is 1 µs, as measuredwith a fast photodiode. (d) Measured transmission of the probe pulse energy versusstorage time. The solid line is a fit to the data, which gives a (1/e) decay time of0.9 ms for the atomic coherence.


degradation. To investigate the time scale for which these revivals could occur, the

experiment was performed for a series of storage times and eventually a reduction in

the output amplitude was observed. An example of an 850 µs storage is shown in

Fig. 5.9(c).

The transmission of the probe energy as a function of storage time is shown in

Fig. 5.9(d). A fit to the data yields a (1/e) time constant of 0.9 ms. It should be noted

that this storage time is an order of magnitude larger than our largest observed delays

τd ∼ 0.15 ms in the regular USL experiments presented in Chapter 3. This is because

the leading order cause of dephasing during USL propagation is often related to light

induced transitions to bad levels, and these processes do not occur when the light fields

are switched off. The time scale of possible storage times is then set purely by atomic

dynamics, which can have a completely different time scale. Chapter 6 considers

how the evolution of ψ1, ψ2 in condensates can eventually cause inequality (5.7) to

break down. In a thermal cloud, a similar logic applies, but is complicated by the

fact that one must consider the evolution phase coherence between the ground states

in an ensemble of atoms occupying a range of energy levels. We do not attempt a

quantitative analysis here, though we note that even relatively small differences in the

Hamiltonia governing |1〉 and |2〉 atoms can cause a loss of coherence for sufficiently

long times and high temperatures. In this experiment, for example, the |1〉 and |2〉

states were equally trapped to first order but had slightly different quadratic Zeeman

shifts. This leads to a slow dephasing of the coherence between |1〉 and |2〉 and may

be responsible the observed decay of the revivals with storage time.

More complicated switching is demonstrated in Fig. 5.10. In Fig. 5.10(a-c) we


Co

up

ling

Inte

nsity

(mW

/cm

)2

No

rma

lize

dP

rob

eIn

ten

sity

50 1500 100

0

0.2

0.4

0.6

0.8

1.

0

0.2

0.4

0.6

0.8

1.

0

0.2

0.4

0.6

0.8

1.

01234

01234

01234

a

b

c

Time ( s)

- 10 0 10 20 30 40

0

0.5

1

1.5

2

2.5

0

10

20

d

Time ( s)

Figure 5.10: Varying the intensity of the second coupling pulse. Figure takenfrom [2]. Meanings of lines and symbols are the same as in Fig. 5.9. Measurements

of revived probe pulses varying the switch-on coupling intensity W(3)c . The initial

coupling intensity W(1)c is held constant. (a-c) The figures show data for W

(3)c /W

(1)c

ratios of 2, 1, and 0.5, respectively. (d) A case where the intensity of the revivedprobe pulse can exceeds that of the original input pulse, in this instance by 40%. (Theobserved peak-to-peak fluctuation of laser intensity is less than 10%.) The energy inthe revived probe pulses is the same in all panels (a-d), owing to the fact that thetotal stored amplitude of state |2〉 atoms (available to stimulate photons into theprobe field) is the same in all cases.

see the coupling intensity switched off to W(2)c = 0 and then on to values W

(3)c =

2 W(1)c , 1 W

(1)c , 1/2 W

(1)c , respectively. The probe’s output intensity scales with W

(3)c

as expected. The output probe pulse widths τ (3) scale inversely with the coupling

intensity W(3)c . Fig. 5.10(d) shows a case where the coupling field was turned on high

enough that the output probe intensity exceeded the input intensity. In all cases the

total integrated output energy is approximately the same as it is always determined

by the number of |2〉 atoms during the storage time, which is turn determined by

the number of photons input. We further note that the delays are in the figures


Couplin

gIn

tensity

(mW

/cm

)2

Norm

aliz

ed

Pro

be

Inte

nsity

Time ( s) Time ( s)

- 20 0 20 490 510 830 850

0

0.2

0.4

0.6

0.8

1

0

1

2

3

4a

- 20 0 20 40 60 80 100

0

0.2

0.4

0.6

0.8

1

0

1

2

3

4b

Figure 5.11: Measurements of double and triple read-out of the atomicmemory. Figure taken from [2]. Meanings of lines and symbols as in Fig. 5.9. Todeplete the atomic memory in these cases, we use two (a) and three (b) short couplingpulses. The total energy in the two or three revived probe pulses is measured to bethe same as the energy in the single revived probe pulse obtained with a single, longsecond coupling laser pulse (as used in Figs. 5.9 and 5.10).

demonstrate that higher W(3)c cause higher group velocities Vg of the revived probe

pulse as it exits.

One can also pulse the coupling field and read out the probe energy in pieces as

shown in Fig. 5.11. This demonstrates that the coupling field really can be used as

a control field which processes the probe information. In the case shown, the energy

is released in incremental pieces, in a manner programmed by the time dependence

of Wc. One could imagine making Wc an arbitrary function of time (with an almost

arbitrarily fast time scale!).

5.3 Bypassing band-width requirements with par-

tial switching

The storage and coherent reading of light is extremely useful from the standpoint

of harnessing and using the coherent nature of a BEC as we will discuss in Chapter 6.


However, we also want to mention possible applications of dynamically altering the

group velocity to studies of superfluidity in BECs. In Section 4.2.4 we discussed

various cases where external dynamics became comparable to the light field driving

terms and saw cases (such as Vg ∼ Vrecoil) where the dynamics severely effected the

propagation. However theoretically we also saw how many effects, such as the Zeeman

shifts, can make it difficult to reach a group velocity where these effects would become

observable. Furthermore, observed dephasings in BECs (Fig. 4.5(c)) were even higher

than the theory predicts, making the lower limits even more restrictive.

Of particular interest in BECs is the study of superfluidity, a subject which we

investigate in Chapter 8. The relevant velocity scale in a superfluid is the sound speed

cs which is typically about 0.4 mm/s at the center of BECs and one might expect

a qualitatively different response of the BEC when Vg reaches this velocity scale. In

some geometries this goes beyond the realm of experimental accessible Vg, according

to our analysis in Section 4.2.4.

We now outline a method of using dynamical change of Wc to possibly get into

these lower Vg regimes. As we saw in Section 4.2.4, the external dynamics put a lower

bound on the Wc (and therefore Vg) by introducing an effective frequency width

∆2−eff and dephasing γeff which reduced the transmission T . The transmission of

the probe in the presence of them is (according to (3.32)):

Teff = exp

(−2D0

γeffWc

)exp

(−D0

4∆22−eff

W 2c

)(5.15)

which, as we have emphasized, is dependent on the cumulative effect of many atoms

via D0 ∼ 300.


However, it is easy to see that with dynamical control of the group velocity, one

can bypass this rather stringent requirement. This is accomplished by turning the

coupling field to a low, though non-zero intensity W(2)c once the probe is in the center

of the medium. The dark state following (4.26) will still occur so long as:

W (2)c γeff ,∆2−eff . (5.16)

To demonstrate the method, we show results of 1D numerical a simulation with

α(Z) = −2 in Fig. 5.12. There the coupling field is kept at value W(2)c = W

(1)c /512 for

about 1 ms, temporarily reducing the group velocity to 1.25 cm/s at the center of the

cloud. Fig. 5.12(a) demonstrates significant transmission. Fig. 5.12(b) shows that

the pulse propagates during the period with a lower intensity Wc(2). By contrast, a

pulse input with the coupling field originally at this low value is almost completely

absorbed by the time it reaches the center of the cloud (Fig. 5.12(c)). We also note

that the two-photon detuning from the Zeeman shift is actually zero in the center of

the cloud, so restricting the low coupling field to a region near the center reduces the

magnitude of the inhomogeneous Zeeman shift in addition to bypassing the effect a

large D0.

Note that because the coupling beam dynamically changes the pulse width, the

ratio of probe field frequency width to the EIT width (τWc)−1 will remain constant

so we cannot use this method to change our minimum allowed pulse width τ0 ∼ τsp.

Because of this, there is no way, for example, to compress beneath our lowest allowed

Lp (3.34) using this method. Similarly, the adjustment of the probe intensity with

the coupling intensity prevents us from using it to temporarily enter the strong probe


Figure 5.12: Partial switching to circumvent bandwidth requirements. (a)Numerical results in the case of a large Zeeman shift, V2 = −2V1. We turn the couplingbeam down once the pulse is compressed to W

(2)c = W

(1)c /512 at t

(1)s = 3.7 µs, with a

time scale τs = 1 µs. This reduces the group velocity at the center of the cloud from6.4 m/s to 1.25 cm/s. It is then switched back on at t

(2)s = 992.5 µs and propagates

out and we see significant transmission. (b) The intensity profile of the probe (solidcurves, normalized to Ω2

p0) and coupling (dashed curves, normalized to the original

intensity (Ω(1)c0 )2) just after the switch-down (thick curves) and just before the switch-

on (thin curves). We see that there is little attenuation during the propagation. (c)

A numerical calculation with the coupling beam kept on at the lower value W(2)c

the entire time. The pulse input width and intensity adjusted appropriately (so theweak probe and bandwidth requirements are satisfied). The plots show the fields att = 225 µs and t = 1200 µs. There is significant attenuation due to the large Zeemanshift and large optical density. There is no visible transmission at the output. In thecase shown, the probe detuning was ∆p = −(2π)226 Hz to maximize the transmissionin accordance with (4.29).


regime. The methods of spatial engineering of the coupling field, the subject of

Chapter 7, can actually accomplish both of these.

Chapter 6

Coherent two component dynamics

and pulse processing

We now turn our attention to studying and harnessing the coherent dynamics of

the BEC in stopped light experiments. As we saw in Chapters 3 and 4, ultra-slow

and compressed light allows us to create superpositions of |1〉 and |2〉 atoms with

spatial structure. Then, in Chapter 5, we saw that the stopped light technique can

freeze these superpositions at an arbitrary time and then write the atomic states back

onto light pulses at a later time. However, there we never considered cases where the

storage times were long enough that the ensuing atomic dynamics were significant.

We now investigate the various kinds of dynamics which can occur and also consider

how these dynamics affect subsequent revivals.

In Section 6.1 we see present experimental results where the beams are in an

orthogonal geometry, and the momentum kick causes the |2〉 atoms to be quickly

ejected from the region occupied by the |1〉 condensate. This produces an atom laser

191

Chapter 6: Coherent two component dynamics and pulse processing 192

of coherent |2〉 atoms. The experiment presented here has the highest phase space

density flux reported for an atom laser to date. Also, by tailoring our input probe

temporal length and transverse width, we see how the stopped light method can be

used to output couple rather arbitrary spatial regions of the BEC. We then present

a theoretical analysis and consider the ultimate limits on this technique of creating

atom lasers. We find that our experimental results are actually very close to the

limitations due to the finite size of the atomic cloud and the Heisenberg uncertainty

relation.

It is clear that the two ground states’ wave functions ψ1, ψ2 will overlap for much

longer periods, when the beams are co-propagating and no momentum kick occurs. In

Section 6.2 we first consider generally the ensuing dynamics and introduce an intuitive

picture which uses an effective potential V2−eff seen by the |2〉 atoms, due to both the

trap and interactions with |1〉 atoms. We then consider how these coherent dynamics

effect revived probe pulses. To do this, we will use the arguments we developed in

Chapter 5 which showed (1) that the revived probe will be determined by the wave

functions ψ1, ψ2 at the switch-on time and (2) that the length scale of variations ψ1, ψ2

determine the quality of the revivals.

We then apply these considerations to investigate a variety of specific different

trapping potentials and interaction strengths. Depending on these parameters, and

the probe to coupling intensity ratio, a variety of dynamics occur in the two compo-

nent BEC. For example, depending the relative interactions strengths, ψ2 can either

gently breathe or undergo a quantum reflection, where it is forced to the condensate

boundary and interferes with itself when it reflects there. In each case, the relative


evolution of ψ2 and ψ1 determine the output pattern of revived pulse’s amplitude and

phase, so the condensate acts as a “pulse processor”. Near the end of the chapter

(Subsection 6.2.6) we address a particular set of parameters which promises to offer

the maximum possible storage times. This situation offers the possibility of using the

coherent BEC dynamics to process the probe phase. We concentrate always on coher-

ent mean field dynamics (the GP equation), which our formalism assumes. However,

we conclude the chapter (Subsection 6.2.7) by anticipating ways in which the results

presented here can be applied to probing and creating quantum states (beyond the

mean field description) such as squeezed states and possibly using stopped and revived

light in a BEC as an element in quantum computing technology [37, 38].

6.1 Using stopped light to make an atom laser

Switching the coupling laser off leaves us with a two component condensate with

interesting spatial structure and it is worthwhile to explore how the two component

condensate evolves. When we use an orthogonal (or counter-propagating) geometry

there is a large two-photon velocity recoil Vrecoil = k2/m ≈ 4.1 cm/s (5.9 cm/s).

The |2〉 atoms are quickly ejected from the region occupied by the stationary |1〉

condensate (within about 1 ms). Because the switching completely preserves the

coherence of the |2〉 atoms, this effectively output couples a portion of the condensate

into a high momentum state, creating an atom laser [33, 34].



We have performed a preliminary experimental investigation of this by stopping

light pulses in the cloud center in an orthogonal geometry (w = x), then taking a

series of absorption images of the |2〉 atoms at different times. Some of the images

are shown in Fig. 6.1(a). When the light fields are turned off the |2〉 atoms have

the crescent shape due to the transverse gradients in the atomic density profile (as

discussed in Subsection 4.2.2 and seen in the t = 0.5 ms image in Fig. 6.1(a)). The

length (along z) of the output coupled |2〉 is proportional to the input pulse length,

and can thus be adjusted experimentally. Furthermore, we note that while in the

present case, we illuminate the entire transverse width of the BEC, generally, we can

spatially select this width by imaging the probe pulse in this direction with a pinhole

before it reaches the BEC.

Returning to the case in the figure, the trap is turned off at the time the cou-

pling field is switched off, allowing a free expansion of both components. The series

of absorption images shows that indeed the |2〉 atoms have a large 4.1 cm/s recoil

velocity with a velocity spread far below this average value. The recoil demonstrates

that atoms have indeed undergone one probe absorption and one stimulated coupling

emission. For the times shown the width in the x and z direction expand very slowly.

We plot the half-width, half max through along z (through the center of the cloud)

for a series of many images in Fig. 6.1(b). We also plot the half-width half-max along

x (at the z corresponding to the widest point). Fits to the data indicate expansion

rates of ∆Vz = 0.078 cm/s and ∆Vx = 0.064 cm/s. This small velocity spread shows

that there have been no additional photon absorptions. We also note an interesting


Figure 6.1: Experimental demonstration of an atom laser. In the experiment,we begin with N = 1.4 × 106 atoms in the trap ωz = (2π)21 Hz, ωx = ωy = 3.8ωz, acoupling Rabi frequency Ωc0 = (2π)11.3 MHz propagating along x and inject a probewith Ωp0 = (2π)2.1 MHz and τ0 = 0.26 µs along z. At t = 0.42 µs the coupling beamis switched off, and the trap is turned off, allowing free expansion of both components.The |2〉 atoms fly off at the recoil velocity of 4.1 cm/s in the diagonal direction. (a)Absorption images of |2〉 atoms as they are ejected from the larger |1〉 condensate.The |2〉 atoms are imaged resonantly on F = 2 → F = 3 transition on the D2 linewith a 10 µs resonant pulse propagating along y at the times indicated. The ODrefers to the optical density seen by the imaging laser. Because the images before6 ms were saturated, quantitative OD information was not possible for the earliesttimes. The 0.5 ms and 1.5 ms pictures have been scaled differently (than the grayscale bar) so the density features are visible. (b) The full-width half max in z (filledcircles) and x (open circles) of OD cuts at various times after expansion. The z cutsare along cloud’s central column, while the x cuts are taken at the z correspondingto the widest width. (We do not find the x widths at early times, where the densitydistribution has the “crescent” shape). The lines indicate linear fits to the data, whichgive expansion rates of ∆Vz = 0.078 cm/s, ∆Vx = 0.064 cm/s. (c) Measured F (6.1),using the average velocity, 〈V 〉 = Vrecoil = 4.1 cm/s, and using velocity widths fromthe linear fits in (b) (we assume cylindrical symmetry ∆Vy = ∆Vx). To determine thedensity, we assumed a Gaussian distribution, and counted the total number of atoms(about 30,000) from the absorption images and the known absorption cross section.The weighted average density 〈n2〉 was then taken to be the central density dividedby 23/2 in accordance with a Gaussian distribution.


evolution of the density profile. The two thin arms of the crescent are then seen to

drift towards each other while the main body expands slowly. This causes the density

to take on a triangular shape (at t ≥ 6 ms).

The figure of merit for an atom laser is the phase space density flux, i.e. the

velocity times the density divided by the velocity spread in each dimension:

F =〈V 〉〈n2〉

(∆Vz∆Vx∆Vy), (6.1)

where 〈V 〉 is the measured average velocity of the sample (which is, in fact, almost

exactly the two-photon momentum kick Vrecoil), and 〈n2〉 is the average density of |2〉

atoms N |ψ2|2. The spreads are ∆Vj =√〈V 2

j 〉 − 〈Vj〉2. The measured peak density

N |ψ2|2 at t = 6 ms was 7 × 1011 cm−3 (resolution and saturation effects prevented

accurate quantitative estimate of the density at times t < 6 ms) and slowly decreased

as the cloud expanded, staying larger than 1 × 1011 cm−3 for 20 ms. Combining this

with our fits to the velocity spreads yields a measurements of the F , which are plotted

in Fig. 6.1(c). At times t = 6 ms (where we had accurate density measurements) it

is seen to be F = 3.3 × 1015 s2cm−5 though one can obviously extrapolate larger

values at earlier times. It slowly decreases due to the decrease in the cloud density,

but maintains a value of about 5 × 1014 s2cm−5 for the duration of the measurement

(25 ms). The value of F at early times (t ∼ 6 ms) compares very favorably with other

published atom lasers. For example, in Bloch, et al. [33] a phase space density flux

of 2 × 1014 s2cm−5 is reported.


6.1.2 Theoretical description of output coupled |2〉 atoms

To understand the observed dynamics and learn the fundamental limits on the

phase space density flux, we have performed numerical simulations of an atom laser.

In order to model all three dimensions numerically, we must assume three dimensional

symmetry. Therefore, we consider here a counter-propagating geometry, where w =

−z and the two-photon velocity recoil is therefore Vrecoil = 5.9 cm/s along the +z

direction.

From the results of Chapter 5 we know that the switch-off preserves the coherence

in the ground state wave functions and the ensuing dynamics are then described by the

coupled Gross-Pitaevskii equations with the light fields off. Within one millisecond,

the |2〉 has completely left the region of the |1〉 condensate and is essentially a freely

expanding single component condensate with a large average velocity Vrecoil.

In Fig. 6.2(a) we show snapshots of the density in the y = 0 plane for the numerical

calculation. The |2〉 condensate starts out with the characteristic crescent shape. In

the ensuing evolution we see, as we saw in the experiment, the entire sample moves

at the same recoil velocity Vrecoil = 5.9 cm/s with the momentum spread far below

this level. We also notice, as was seen experimentally, that the two thin edges of

the crescent drift towards each other and the density eventually takes on a triangular

shape, indicating that the experimental density evolution is qualitatively described by

the GP equation. However, we also see a feature which was not seen experimentally:

upon reaching each other, the two arms interfere due the phase coherence of the

macroscopic wave function. The integration over the column density which occurs

in experimental absorption images may have washed out some of the interference if


Figure 6.2: Numerical simulation of an atom laser. Numerical simulations in3D (assuming cylindrical symmetry) with N = 1.2 × 106, ωz = (2π)21 Hz, ωx =ωy = 3.8ωz. The coupling beam has Ωc0 = (2π)8.0 MHz and propagates along −z(counter-propagating). The probe has Ωp0 = (2π)2.0 MHz and τ0 = 1.5 µs. Thecoupling field is switched off at 6.1 µs at which point the trap is switched off, bothcomponents expand, and the |2〉 atoms are ejected at Vrecoil = 5.9 cm/s. (a) Crosssections of the density through y = 0 at the times indicated (relative to the switch-off time). The dotted ellipse denotes the original condensate’s boundary. The two“arms” collide and interfere while the entire density distribution undergoes a slowexpansion in both dimensions. A total of N2 = N

∫dR |ψ2|2 = 15, 000 atoms (1.26

% of the total initial number N) are in the |2〉 state. (b) The kinetic energy per |2〉particle E

(T )2 /N2 subtracting off the large kinetic energy from the velocity recoil (solid

curve) and interaction energy E(int)22 /N2 (dotted curve) as a function of time (both

normalized to the chemical potential µTF ). (c) The velocity widths ∆Vx (dottedcurve) and ∆Vz (solid curve), calculated with (6.3). (d) The calculated phase spacedensity flux F (6.1), using (6.2) and (6.3). Initially some of the decrease is due to theincreasing momentum spreads seen in (c). After the first millisecond, the primarycause of reduction is the finite number of atoms spreading and reducing the density.


it occurred. We could imagine doing the experiment with slightly shorter pulses,

thus causing more pronounced arms and getting into a regime where interference

fringes are visible. Alternatively, we could use more sophisticated imaging (e.g. the

slicing technique in [4] and in Chapter 8) which images only particular planar cuts

of the density. The contrast of interference fringes could conceivably be used as an

experimental diagnostic of the degree of phase coherence in the atom laser. (The GP

equation assumes mean fields and will always produce the result corresponding to

perfectly coherent matter waves).

The numerically calculated kinetic energy per particle of the |2〉 condensate, is

plotted in Fig. 6.2(b) (solid curve). The plot is of the energy above the large recoil

energy 2k2

2/2m = 19.6 µTF . To understand the behavior, we note at the original time

the kinetic energy is simply determined by the shape of the macroscopic wave function

(i.e. it’s finite momentum spread due to the Heisenberg uncertainty relation). During

the first 1 ms there is an increase of about a factor 1.5 due to both interaction with

the repulsive |1〉 condensate atoms and its own self interaction. After the first 1 ms, it

has left the region of the |1〉 condensate and experiences only a gentle increase as the

remainder of its self interaction energy (plotted as a dashed curve) is converted into

kinetic energy. Its final value is only roughly a factor of 2 larger than the Heisenberg

determined value at the time of release.

Numerically, we can calculate the velocity spreads, and therefore the phase space

density flux F by averaging the appropriate operators over the wave function ψ2. The

velocity and density of |2〉 atoms are thus calculated:


〈V〉 =

m

(k2 − i

∫dRψ∗

2(∇ψ2)∫dR |ψ2|2

), 〈n2〉 =

N∫dR |ψ2|4∫dR |ψ2|2 ,

and the spread in velocity in a particular direction is :

∆Vj =

√√√√−∫dRψ∗

2( ∂2

∂j2ψ2)∫

dR |ψ2|2 +

(∫dRψ∗

2( ∂∂jψ2)∫

dR |ψ2|2)2

(6.3)

Fig. 6.2(c) plots these calculated values for the simulation. The increase in kinetic

energy is seen to correspond to increases in the ∆Vj. We note that ∆Vz in particular

remains very small at all times. Using Eqs. (6.2-6.3) we calculate F (6.1) as a function

of time and plot the result in Fig. 6.2(d). We see that we gain by coupling the

atom laser into another element at earlier times, F is maximized. We have F =

60 × 1015s2/cm5 at t = 1 ms, at which point the |2〉 condensate has left the region

occupied by the |1〉 condensate. At later times F decreases, primarily due to the

loss in density because of the finite size of the cloud, though there is also a small

contribution from the increase in the ∆Vj. At 6 ms F = 17 × 1015s2/cm5, about five

times higher than the experimentally measured value at this time.

The optimal parameters for an atom laser will clearly depend on the particular

application, but the analysis presented here demonstrates some of the considerations.

To determine F there is an interplay of four considerations: the initial Heisenberg

determined velocity spread, the peak density, the interactions with |1〉 atoms and

initial self interaction energy. The initial size, and therefore the original Heisenberg

uncertainty spread, can be adjusted by changing the pulse input. The ratio (Ωp0/Ωc0)2

will play a role in both the peak density and the amount of self-interaction energy

among the |2〉 atoms.


6.2 Coherent two component dynamics and pulse

processing

We saw in Chapter 5 that the wave functions of the two ground states ψ1, ψ2 at

the switch-on time determine the revived probe’s amplitude and phase. Also, we

saw the magnitude of the spatial derivatives of ψ1, ψ2 determine αNA (see Eq. (5.4))

and therefore the quality of the transfer of coherent information between the light

and atomic fields. However, there we never considered storage times sufficiently long

that ψ1, ψ2 evolved significantly during the storage. We now consider how these wave

functions evolve when the coupling field is left off for a sufficiently long time that the

external dynamics play an important role in determining both of these quantities. As

we will see, it is important to consider not only quality of the revival, but the ability

of the revived pulse to subsequently propagate out of the BEC.

In this section we will first make a few general observations about the evolution

of ψ1, ψ2 in the absence of the light fields and how this evolution effects probe pulse

revivals and their subsequent propagation. We will then specifically consider, one

by one, different terms which contribute to H1 and H2, including momentum kicks,

trapping potentials, and real and imaginary scattering lengths. In all cases we will

investigate the issues numerically in 1D and compare with simple analytic estimates.

This will clarify the ultimate limits on the possible storage times for probe pulses in

BECs and demonstrate some of the ways in which the BEC dynamics can actually

process the amplitude and phase of the pulse. These are essential issues in eventu-

ally applying this kind of system to creation of quantum states of light or quantum


processing of light pulses.

6.2.1 General considerations for dynamics and revivals

Let us first consider the evolution of ψ1,2 once the light fields are switched off. For

simplicity, consider the weak probe limit. In this limit the wave function ψ1 is equal

to its initial value at all times, modulo the phase evolution: ψ1 = ψ(0)1 exp(−iµt),

where ψ(0)1 is the original ground state. In this chapter, whenever we plot phases of

the wave functions, we will be plotting phases of ψ′1,2 ≡ ψ1,2exp(+iµt). During the

storage time ψ′2 will evolve according to:

∂

∂tψ′

2 =

(−

2(∇2 + ik2 · ∇)

2m+ V2−eff

)ψ′

2 ;

where V2−eff ≡ V2 + U12|ψ(0)1 |2 − µ, (6.4)

since we can ignore the density |ψ′2|2 in the weak probe limit. The initial condition

will be ψ′2 = −(Ωp/Ω

(1)c0 )ψ

(0)1 , where Ω

(1)c0 is initial input coupling Rabi frequency and

Ωp is evaluated just before the switch-off at t(1)s . This equation will provide us with a

remarkably simple intuitive picture of the evolution of ψ′2 in the following examples.

Note that since |ψ(0)1 |2 is constant, there is no nonlinearity in the evolution of the ψ′

2

and V2−eff is independent of time. In the limit H2 = H1 the interaction and trap

terms cancel in the region occupied by the |1〉 condensate and V2−eff is as plotted

in Fig. 6.3. This method has been used previously by Park, et al. [86] to study

two component dynamics in JILA experiments [29]. In the Thomas-Fermi limit there

is a large flat bottom and two steep walls on either side, forming almost a box-like

potential. The sharpness of the boundary and the steepness of the wall both increase


Figure 6.3: Effective potential V2−eff . The potential V2−eff of Eq. (6.4) whenV2 = V1 (α(Z) = 0), and U12 = U11 (α(MF ) = 0), normalized to the chemical potential.It looks almost like a box potential in the Thomas-Fermi limit.

as the TF approximation becomes better. The small decrease in the potential near

the condensate border occurs because there is more kinetic energy and therefore a

smaller density than the TF prediction in this region. The characteristic width dz

(see Subsection 4.1.3) can be taken as a measure of the sharpness of the boundary

region. Over this length scale the slope of the potential goes from dV2−eff/dz = 0 to

dV2−eff/dz = 2µ/σz.

If, after some arbitrary evolution time, at t(2)s , the coupling beam is then switched

on to Ω(3)c0 , the probe will revive according to Ωp(R, t

(2)s ) = −(ψ′

2(R, t(2)s )/ψ

(0)1 )Ω

(3)c0 . In

the weak probe limit, the success of the probe revival and its subsequent propagation

will be determined by the scale of the spatial derivatives of ψ′2. To examine this issue

of the subsequent propagation, let us first recall that if the dominant cause of losses

during pulse propagation is the presence of high frequency components in the input,

then the transmission of the pulse will be (see Eq. (3.32)):

T =1√

1 +τ2sp

τ20

=1√

1 + 4D0(τ0Wc)−2. (6.5)

In Chapter 5 we converted time and frequency criteria into their spatial counterparts


(as in Eq. (5.7)) by using the relation between the pulse length Lp, time width τ , and

the absorption length αA: Lp = 2Vgτ0 = 2Wcτ(αA)−1, where the preparation length

(αA)−1 is defined in (5.4) and is ≈ f13σ0N |ψ(0)1 |2 in the weak probe limit.

To derive the transmission of a revived pulse, we will use that same strategy here

by noting that large spatial derivatives in the ψ′2 get transferred onto the revived

probe pulse, which in turn imply high frequency components. The spatial derivatives

of ψ′2 have a length scale, which we call Lψ′

2∼ 2|(dψ′

2/dz)/ψ′2|−1. The atomic dy-

namics can cause Lψ′2

to be significantly smaller than the length of the pulse before

the storage, which we call L(1)p ≡ 2W

(1)c τ0(αA)−1. If the resulting high frequency

components are too high, the pulse will experience attenuation analogous to (6.5).

Converting equation (6.5) into spatial parameters and replacing τ−10 with the frequen-

cies corresponding to Lψ′2, the condition for transmission of the revived probe pulse

becomes:

T =1√

1 + 4D(Lψ′2αA)−2

, (6.6)

where D is the optical density through which the revived pulse has to propagate

(which isD ∼ D0/2 if the pulse is stored and revived in the center of the cloud). In the

examples we study below, Eq. (6.6) will serve as a practical estimate for transmission

of revived probe pulses. Note that just as the condition for pulse transmission is

more restrictive than the EIT adiabatic condition Wcτ 1, due to the factor of

optical density D0, we have here that the condition for successfully outputting the

revived pulse with a high transmission (according to (6.6)) is more restrictive than

the condition for achieving the initial revival at switch-on.


In the strong probe limit, the considerations will be qualitatively similar, except

the dynamics become significantly more complicated and nonlinear. In addition, the

spatial scale of both ψ1 and ψ2 will play a role in determining the nature and quality

of the revival. The full expression for (αNA)−1 in Eq. (5.4) has to be used in place of

Lψ′2

and Eq. (5.5).

6.2.2 Revivals with a momentum kick

The first case we examine is the one encountered in the atom laser : that of a

large two-photon momentum kick. What happens when one switches the coupling

beam back on after the |2〉 condensate has recoiled? As the |2〉 atoms move through

the larger |1〉 condensate, the coherence between the two ground state is moving,

but is still perfectly preserved. In the atom laser, we turned the trap off in order

to prevent additional inhomogeneous forces on the output coupled |2〉. Here we will

consider keeping the trap on after the lights are switched off, so the |1〉 atoms continue

to be trapped. The recoil kinetic energy possessed by the |2〉 atoms, 2k2

2/2m =

()24.8 kHz, completely dominates the energy scale of the V2−eff potential, µTF ∼

(2π)1.2 kHz, so the atoms do not experience a significant effect from the |1〉 atoms,

but merely shoot out of the region. If the |2〉 atoms are trapped by the potential,

they will turn back at their classical turning point, which is far outside the original

condensate.

We first consider times where the |2〉 atoms are still in the region of the original

condensate. For simplicity, consider the case V2 = V1, U12 = U11. An example of a 1D

numerical calculation of the evolution of the density and phase of ψ′2 with a weak probe


is shown in Fig. 6.4(a). The phase of ψ′2 is exactly π when the light fields are first shut

off (the dark state for real light fields) and very little phase gradient develops as the

|2〉 atoms move across the cloud, until it reaches the condensate border. Therefore, so

long as the ψ′2 wave function is still contained in the |1〉 condensate, we get a nearly

full revival. For example, a revival at t(2)s = 0.42 ms is shown in Fig. 6.4(b). The

dominant form of loss in this case is the loss due to elastic scattering of the quickly

moving |2〉 atoms. In Fig. 6.4(c) we plot the transmission of the output probe relative

to the input probe and see a gradual reduction for increasing storage times. We see

this agrees with the expectation based on losses of |2〉 atoms from collisional loss until

the ψ′2 wave function starts to leave the |1〉 condensate.

As ψ′2 reaches the edge of ψ

(0)1 , we see that the revived probe pulse significantly

changes shape (Fig. 6.4(b),0.89 ms). This is because the ratio ψ′2/ψ

(0)1 becomes large

as |2〉 atoms enter the region of low density near the condensate edge, and we indeed

see the revived probe pulse has a higher peak intensity than the input pulse. As

the |2〉 atoms actually leave the region occupied by ψ(0)1 , the transmission suddenly

decreases (Fig. 6.4(c)). To understand this, we note that in the extreme limit that the

two condensates have completely separated spatially, then there is no local coherence

anywhere between them and the coupling field, upon being switched on, is simply

absorbed when it encounters the resonant |2〉 atoms.

From this discussion, we conclude that, apart from the usually moderate elastic

collisional loss rate, the momentum kick is not an effect which will significantly effect

the revivals so long as the two condensates are overlapping. One could even imagine

letting the |2〉 atoms evolve for a full trap period. In this case they would shoot far


Figure 6.4: Revivals with a momentum kick. (a) Two component dynamics ina counter-propagating geometry. The solid curve shows the density N |ψ′

2|2 at thetimes shown, and the dashed line shows the phase of ψ′

2. The density N |ψ1|2 dividedby 10 is shown as a dotted curve. Results are based on a 1D numerical calculationwith a N = 1.2 × 106, ωz = 21 Hz, and A = π(8µm)2. We use Ω

(1)c0 = (2π)8 MHz,

Ωp0 = (2π)2 MHz, τ0 = 1.5 µs and switch the coupling off with at t(1)s = 3.7 µs with

τs = 0.1 µs. (b) Revived probe pulses at zout (solid curves) when the coupling fieldis switched on at the times plotted in (a). For reference, the dashed curves show theoutput in the limit that no atomic dynamics take place during the storage time. (c)The integrated transmission of the revived pulses as a function of the switch-on time(circles). For comparison, the solid curve shows the expectation based on the loss of|2〉 atoms due to high momentum elastic collisions. The agreement is seen to be gooduntil the |2〉 atoms begin to exit the original condensate region. The squares showthe widths of the output pulses, which get narrower as the |2〉 condensate approachesthe |1〉 condensate edge.


in the positive z > 0 direction until they reach their classical turning point at which

point they will shoot back, and go through the |1〉 atoms in the opposite direction after

half a trap period. They will arrive back in their original position and momentum

after a full trap period, at which point a revivals could still take place, assuming that

elastic collisions have not eliminated too many atoms. Numerical simulations have

confirmed this. One could also output couple a sample of |2〉 atoms from as an atom

laser from one condensate and inject them into nearby second condensate. Again a

revival should occur since both the output condensate and the new condensate are

phase coherent (even if there is a random phase relationship between them). In fact,

the phase of the revived pulse would be determined by the relative phases of the

first and second condensates (and the easily calculated phase evolution of the atom

laser during the transfer). This could thus be a tool to measure phase coherence and

relative phases between distant objects.

6.2.3 Differently trapped |2〉 atoms (V2 = V1)

All our discussion to this point in the chapter have involved recoiling |2〉 atoms,

which obviously puts rather severe limits on the time the wave functions ψ1 and ψ2

overlap. If we use co-propagating beams, by contrast, we can create superpositions

which will live together for much longer times, allowing possibilities for significant

processing to take place. We now consider a number of different situations with no

momentum kick (k2 = 0). We start with the case of a large Zeeman shift (V2 =

(1 + α(Z))V1).


Figure 6.5: Dynamics for anti-trapped |2〉 atoms. Dynamics when α(Z) = −3and there is no momentum kick (k2 = 0), based on 1D numerical simulations. The

pulse width is τ0 = 1.1 µs, the switch-off time is t(1)s = 4.6 µs and other parameters

are the same as in Fig. 6.4. (a) The effective potential V2−eff . The repulsive potentialis the dominant contribution. (b) The densities of both condensates N |ψ1|2, N |ψ2|2and phase of ψ′

2 (with the same conventions as Fig. 6.4(a)). The repulsion causes alarge phase gradient (and therefore velocity) to quickly develop and eventually the|2〉 atoms begin to exit. The phase at 5.6 ms is not shown since it consists of avery large number (> 12) revolutions of the phase across the cloud. (c) The energy

per |2〉 particle from interactions E(int)12 /N2 (dotted curve), the repulsive magnetic

trapping fields E(V )2 /N2 (dashed curve) and kinetic energy E

(T )2 /N2 (solid curves) as

the condensates evolve. As |2〉 atoms are pushed out, they experience an increasinglynegative Zeeman shift energy and a corresponding gain in kinetic energy.

Dynamics

As we will see, cases where the |2〉 atoms are trapped (α(Z) > −1) are qualitatively

distinct from the case where it is repelled (α(Z) < −1).

In the region occupied by the |1〉 atoms the effective potential in (6.4) can be

written:

V(Z)2−eff (z) = α(Z)µ

(z2

2σ2z

). (6.7)

This is plotted in Fig. 6.5(a) for the repelled case α(Z) = −3. Assuming pulses have


length scales Lp on the order of the condensate size σz then, in the spirit of the

Thomas-Fermi approximation, we temporarily ignore the kinetic energy term in (6.4)

and have the approximate solution:

ψ′2(z, t) ≈ ψ′

2(z, t(1)s )exp

[−iα(Z)µ

(z2

2σ2z

)(t− t(1)s )

]. (6.8)

Initially, this evolution equation causes an increasing steep parabolic phase gradient

to develop, corresponding to an increase in the velocity spread of the atoms due to

the force from the potential gradient. Eventually this causes the density to begin to

roll down the potential hill and the expression (6.8) begins to lose validity.

The dynamic evolution of the magnitude and phase of ψ′2 in the repelled case

is shown in Fig. 6.5(b). The phase gradient mentioned above is seen to be quite

significant within 1 ms, and actually goes through several cycles of 2π before there is

any measurable change in the density. Eventually the imparted velocity causes the

|2〉 atoms to be pushed down both sides of the potential “hill” (5.6 ms). From an

energetic point of view, the unbounded repulsive potential continues to feed energy

into the kinetic energy of the |2〉 atoms (plotted in Fig. 6.5(c)).

In the case where the |2〉 atoms are trapped but with a different potential than

V1, the wave function will undergo a breathing motion. Fig. 6.6 shows the evolution

in the case V2 = 2V1(α(Z) = 1). We see that for the first 13 ms the wave function

contracts. At this point it then begins to expand again, nearly exactly reproducing

its initial density at 26 ms. We do not plot the phase, but is it becomes quickly

varying at 13 ms, then relaxes to a nearly homogenous phase at 26 ms. Significantly,

the confining potential prevents the kinetic energy from ever reaching above a certain


Figure 6.6: Dynamics for a differently trapped |2〉 atoms. Dynamics whenα(Z) = 1 and otherwise the same as Fig. 6.5. The conventions are the same as there.In (c), we add an additional curve (thin dotted) showing the self-interaction energy

E(int)22 /N2.

value.

In practice, it should be noted that the effects of gravity are often significant on

the scale of harmonic traps used in current experiments. The effect of gravity is to

displace the center of V2 relative to V1 when their trap frequencies are significantly

different. This coupled in additional relevant dynamics in two component experiments

at JILA and has also been discussed theoretically in the literature [29, 86].

Revivals

We now turn to the question of revivals in these cases. From our approximate

expression for ψ′2 (6.8), we estimate length scale Lψ′

2by taking the spatial derivative

of ψ′2 and get:


Lψ′2∼ 2σ2

z

L(1)p

1

α(Z)µ (t− t(1)s )(6.9)

where we have replaced z → L(1)p /2 because the original pulse width will determine

the initial range of the z with a significant density of |2〉 atoms. We can plug (6.9) into

(6.6) to get an estimate for the transmission of the revived pulse. Fig. 6.7(a) shows

the numerically calculated transmission of the revived pulse in the anti-trapped case

(filled circles) as a function of switch-on time t(2)s , along with the analytic estimate

using (6.9) and (6.6). Good agreement between the two is seen.

The analytic results slightly overestimates the revival transmissions at longer

times. This is because (6.9) assumes, by neglecting the kinetic energy, assumes the

amplitude |ψ′2| stays constant and does not account for the atoms being pushed out

to a larger range of z at later times (which is clearly visible, for example, in the 5.6 ms

frame of Fig. 6.5(c)). Eq. (6.9) also assumes the pulse is stored in the cloud center.

For longer storage times or if the pulse is stored on one side, the estimate (6.9) must

be modified appropriately.

When the pulse occupies nearly the full length of the condensate (L(1)p ∼ 2σz), the

estimated characteristic time in the transmission drop off is simply (α(Z)µ)−1. One

sees from (6.9) and (6.6) the time scale allowing transmission is increased for a highly

compressed pulse (L(1)p σz). We saw in Subsection 3.2.6 that this is only possible

for large D0. In the case shown in Fig 6.7, Lp ≈ σz/3 and we observe a time scale of

3 ms, a marked improvement over (α(Z)µ)−1 = 0.044 ms.

In Fig. 6.7(b) we plot the dark and absorbing field intensities |ΩD,A|2 just before

the switch-off, and just after a switch-on at 7.1 ms (the case indicated by an arrow


Figure 6.7: Revivals in trapped and anti-trapped cases. (a) For the casesin Figs. 6.5 and 6.6 we plot the transmission of the output probe pulses for variousstorage times. Filled circles show the anti-trapped case, and the solid curve shows theanalytic prediction based on (6.6) and (6.9). The open circles show the transmissionof the revived pulses for the trapped case. The dashed curve shows an estimate basedon (6.6) with Lψ′

2calculated numerically from the kinetic energy as described in the

text. The widths of the revived pulses (normalized to the input width) in the trappedcase are plotted as open squares. (b) In the anti-trapped case, the absorbing fieldintensity |ΩA|2 just before the switch-off (thick dashed curve) and just after a switch-on at 7.1 ms (thick solid curve), which is the simulation indicated by an the arrow in(a). Both are normalized to Ω2

c0 with the scale set on the right-hand side of the plot.The higher absorbing field leads to a reduction in the revived probe. The dark field|ΩD|2 is plotted as thin lines (same times and normalization) with scale set on theleft. The dotted line shows the original condensate density (in arb. units). (c) Theprobe intensity (thick lines) normalized to Ω2

p0 and coupling (thin lines) normalizedto Ω2

c0 at the same times as (b) (again dashed curves corresponding to before theswitch-off and solid curves after the switch-on). There is a visible decrease in therevived probe, but not as severe as the reduction in the output transmission seen in(a).

in Fig. 6.7 (a)). One sees there is a dramatic increase in the absorbing field due to

the spatial derivatives of ψ′2 which develop during the storage time. As a result, the

revived probe is seen to be lower than the probe before the switch-off, as we see in

Fig. 6.7(c). However, it is notable that the revived pulse energy is nearly 50% of

the pulse before the switch-off, while the transmitted revived pulse (see the arrow

in Fig. 6.7(a)), is down by a factor of more than 7 from a pulse with no storage


time. The majority of the attenuation of the transmitted pulse then occurs during

the propagation rather than the switching, which is why the optical density D enters

our estimate (6.6).

In contrast, revivals in the trapped case (α(Z) = 1) can occur indefinitely. The

revival transmission for this case is also plotted in Fig. 6.7(a) (open circles). As could

be expected from the dynamics, we see a decrease in transmission as the wave function

is compressed and gains kinetic energy (and therefore higher spatial gradients in ψ′2).

We then see an increase when ψ′2 begins to expand back to its original shape. The

total free energy in the system puts a bound on the kinetic energy, which in this case

always stays low enough that there is some significant amount of revival. After each

period of oscillation the transmission is nearly equal to the transmission in the absence

of any storage. We also plot a prediction for the revivals based on the numerically

calculated kinetic energy and Eq. (6.6). To obtain it, we take the kinetic energy per

particle in |2〉 and convert it into a length scale: Lψ′2

= 2(2mE(T )2 /

2N2)−1/2. We see

this gives a very good estimate of the transmission.

We also observe in Fig. 6.7(a) (open squares) how the output pulse width changes

as ψ′2 compresses and relaxes. This is a simple example of how BEC dynamics during

the storage time are written onto the output pulses.

Experimental revivals

In connection with atom laser experiment discussed above, in the orthogonal ge-

ometry, we also attempted revivals after short storage times, keeping the trap on.

The energy levels involved corresponded to α(Z) = −3 and there was the additional


complication a two-photon momentum kick a 45 degree angle. Experimentally the

revived pulse transmissions were observed to decay with a time scale on the order of

50 µs.

Using our above criteria to estimate this time scale theoretically, we saw that the

momentum kick will have very little impact the first 50 µs (since ψ1, ψ2 still overlap).

By contrast the large Zeeman shift will quickly have an effect. To estimate this, we

must consider the fact that the stored pulse covered the entire transverse width of the

condensate, so it experiences the full inhomogeneous Zeeman shift of the condensate

and it is irrelevant that the length Lp along z was smaller than σz. In this case, the

time scale, as we mentioned before, will be simply given by (α(Z)µ)−1 ∼ 44 µs, in good

agreement with the observed time scale. We note that this effect of the transverse

width could be avoided by imaging a small radial profile probe pulse before it enters

the BEC, so it only interacts a small central column, where the Zeeman shifts are

smaller.

6.2.4 Equally trapped |2〉 case with U12 > U11

Dynamics: Quantum Reflections

More interesting dynamics occur in the case where the traps are nearly equal and

the dominant source of difference between H1 and H2 is a difference in the scattering

lengths. We again concentrate on the weak probe limit. When U12 > U11 the |2〉 atoms

will experience a slight additional repulsion due to the stronger interaction with |1〉

atoms and the potential V2−eff will look like Fig. 6.8(a), with minima located at each

end of the condensate (The plot shows two cases: U12 = 1.2U11 and U12 = 1.05U11).


In two and three dimensions, the minimum will be along a continuous trough along

the condensate border.

The initial dynamics, where the |2〉 atoms are still inside the original condensate

region, are mathematically equivalent to the anti-trapped case, only now on a much

lower energy scale (and therefore larger time scale) since α(MF ) = (U12 − U11)/U11 is

generally much smaller than unity (we considered the case α(Z) = 1 in the previous

subsection). The |2〉 condensate gains kinetic energy as it travels towards the bound-

ary of the |1〉 condensate. Upon reaching it, it sees a sharp corner and a potential

hill. As a result of the sharp boundary, there is a reflection of part of the coherent

matter wave. The part of the wave which has reversed direction interferes with the

part still travelling outwards. Fig. 6.8(b) shows snapshots for the case α(MF ) = 0.2.

The interference fringes are seen to have 100% contrast. The asymmetry between

positive and negative z is a result of stopping the pulse slightly to the z < 0 side. In

an experiment, the contrast of interference fringes can again be used as a diagnostic

of phase coherence.

The period of the fringes is twice the momentum of atoms at the reflection point

(the factor of two because it is the relative momentum of opposite travelling waves).

This momentum can easily be estimated by the kinetic energy gain. For for atoms

which start at z = 0, this is equal to the difference in the potential V2−eff at the

cloud center and the condensate border: ∆V2−eff = α(MF )µ. This estimate predicts

a fringe wavelength at the boundary of:

λint ≈(a

(HO)j )2

σz√

2α(MF ), (6.10)


Figure 6.8: Quantum reflections due to high repulsive interactions. (a)The potentials V2−eff for the cases U12 = 1.2U11(α

(MF ) = 0.2) (thick curve) andU12 = 1.05U11(α

(MF ) = 0.05) (thin curve), with equal traps, and other parametersas in Fig. 6.5. The potentials have minima at the condensate edges and a barrier inthe middle. (b) Dynamics of N |ψ2|2 (solid curves) and N |ψ1|2 divided by 20 (dottedcurves) after a switch-off with α(MF ) = 0.2. As a result of the shape of V2−eff , the|2〉 atoms are initially pushed out but then bounce back at the condensate boundary.Oppositely oriented momenta then interfere and give fringes with a length scale λint(6.10). At 50 ms, we see the atoms have gone part of the way up the repulsivewall to their classical turning point (6.11). After about 104 ms the atoms return tothe center but, due to small nonlinearities, ψ′

2 never completely recovers its originalshape. (c) The corresponding plots in the case α(MF ) = 0.05 show larger λint fringes,a slower time scale, and also shows |ψ′

2|2 nearly attains its original form at 239 ms.(d) Thick curves show various contributions to the energy in the α(MF ) = 0.2 case:

kinetic energy E(T )2 /N2 (solid curve), energy due to V2−eff , (E

(V )2 +E

(int)12 )/N2 (dashed

curve), and the total energy of |2〉 atoms (E(V )2 + E

(int)12 + E

(T )2 + E

(int)22 )/N2 (dotted

curve). Thin curves show the corresponding plots for α(MF ) = 0.05.


where a(HO)z is the harmonic oscillator length defined in Subsection 4.1.3. We also

note that, because the |2〉 atoms have some kinetic energy at the boundary and the

potential outside the condensate is not infinitely steep, they can reach regions outside

the original condensate. In the TF limit, the classical turning point for the atoms

will be

zTP = ±√

2σz

(1 +α(MF )

2

)(6.11)

in the limit α(MF ) 1. In the third frame of Fig. 6.8(b) we see there is a significant

density outside the original condensate region.

We observe numerically that the atoms slosh about the potential minimum on both

sides but never actually repeat or revive the original wave function ψ′2(z, t

(1)s ). The the

wave function envelope oscillates but always maintains some of its interference fringes

as seen in the last frame of Fig. 6.8(b). In terms of energy, we see in Fig. 6.8(d) that

energy is exchanged between the effective potential V2−eff (thick dashed curve) and

kinetic energy (thick solid curve), though the oscillations of this exchange are damped

and not completely periodic. Part of the reason is the small amount of nonlinearity

introduced by the small but finite number of |2〉 atoms, N2 = 0.011N1. As a result of

this effect, the total energy per |2〉 particle (thick dotted curve) is seen fluctuate at

about the 1% level.

Fig. 6.8(c) shows the corresponding plots for the evolution of |ψ′2|2 in a case with

α(MF ) = 0.05. Because the repulsive barrier at the center of the cloud is much smaller

in this case (see Fig. 6.8(a)), there is a much smaller amount of momentum gain, and

therefore much larger interference fringes (as predicted by (6.10)). The time scale


Figure 6.9: Transferring quantum reflections onto light pulses. (a) Transmis-

sion of revived probe pulses versus storage time t(2)s in the case α(MF ) = 0.2 (filled cir-

cles) and an estimate based on the numerically calculated kinetic energy (Fig. 6.8(d))(solid curve). We see a damped oscillation. Open circles and the dashed curve showthe case α(MF ) = 0.05. (b) For the case we show α(MF ) = 0.05, we show the outputprobe intensity (solid curve) versus the expectation with no atomic dynamics (dashed

curve) at for revivals at times t(2)s = 46 ms and t

(2)s = 141 ms. Arrows in (a) indicate

the two cases shown. The phase of the output pulse is shown with circles (The phasewith no atomic dynamics is zero everywhere). At 46 ms we see simply a spread ofthe pulse because the ψ′

2 has been pushed towards the borders. At 141 ns see theinterference fringes are transferred onto the probe and successfully propagate out,though somewhat washed out due to the high frequencies. Large phase jumps occurat the minima of the fringes.

for the dynamics is also proportionally larger. We also see in Fig. 6.8(c) that ψ′2

comes much closer to repeating its original form after returning to the vicinity of the

condensate center for the first time at 239 ms.

Revivals

In the initial stages, we can make a direct correspondence to the anti-trapped

case in terms of the transmission of the revivals, however, it is interesting to con-

sider revivals after fringes have appeared. Once we enter the regime with interference


fringes, Lψ′2

will never be below the minimum fringe size λint. Fig. 6.9(a) shows the

revivals as a function of time for the case shown in Fig. 6.8. The initial behavior

can be modelled with the arguments used in the anti-trapped case. However, compli-

cated damped oscillations of the transmission occur after the occurrence of quantum

reflections. We see that the transmission nearly goes to zero in the α(MF ) = 0.2

case due to the small λint. At later times, when most of the atoms return to the

center, the kinetic energy is smaller and we get partial revivals. In the α(MF ) = 0.05

case, λint is sufficiently large that we always get > 25% transmission in the revived

pulses. Fig. 6.9(b) shows two cases (indicated by arrows in (a)). At 46 ms there is

a spread in the width of revived pulse with respect to a pulse with no storage, as

well as a non-trivial, inhomogeneous phase pattern. At 141 ms, ψ′2 has interference

fringes which are transferred onto the output pulse. They are partially washed out

during their propagation out but maintain their basic shape. This demonstrates the

degree to which sensitive coherent information from the condensate dynamics can be

efficiently and robustly written back onto the probe light field.

Strong probe regime: Phase separation

It is interesting to consider effects which are added when Ωp0 ∼ Ωc0 and study

the nonlinear two component dynamics. Fig. 6.10(a) shows an example where Ωp0 =

Ωc0/√

2 and U12 = U22 = 1.2 U11. About 9% of the original atoms are in |2〉 when the

switch-off occurs. We see that most of the |2〉 atoms, instead of getting pushed out, get

pulled into the gap in the |1〉 condensate. This is an example of phase separation [87],

an effect which has been observed experimentally in multi-component condensates


Figure 6.10: Phase separation dynamics and revivals. (a) The density profilesof |ψ1|2 (dotted curves) and |ψ2|2 (solid curves) at various times for a case set upby a probe Rabi frequency Ωp0 = Ωc0/

√2 and relative interaction strengths U12 =

U22 = 1.2 U11. Other parameters are the same in previous figures. We see a phaseseparation leading to two domains of |2〉 atoms which then slowly drift within the

larger |1〉 condensate. (b) The effective potentials V(SP )1,2−eff (Eq. 6.12) leading to

dynamics seen in (a). (c) The kinetic energy E(T )2 /N2 (solid curve) and E

(T )1 /N1 as a

function of time. (d) The output probe intensity (solid curves) versus the output in

the absence of atomic dynamics (dashed curves) at revivals times t(2)s corresponding

to the times indicated in (a). The phase of the output pulses are shown as dots.


[30]. It can be shown that the two component system’s ground state will involve

spatial separation of the two components into domains whenever U12 >√U11U22

[87]. While theory and experiment which have studied this phenomenon [87, 30] have

emphasized ground state properties, our system puts the system in a state with a

substantial amount of free energy (and therefore far out of equilibrium) and provides

a possible opportunity to observe the dynamics of this phenomena.

To understand the dynamics in the case shown, we must go beyond the weak

probe limit and include the effects of |2〉 ↔ |2〉 interactions as well as the dynamics

of the ψ1. Both effects make the system highly nonlinear. The wave functions will

evolve according to time-dependent potentials:

V(SP )2−eff ≡ V2 + U12|ψ1|2 + U22|ψ2|2 − µ,

V(SP )1−eff ≡ V1 + U11|ψ1|2 + U12|ψ2|2 − µ. (6.12)

These are plotted (at the times corresponding to Fig. 6.10(a)) in Fig. 6.10(b). We see

initially, the |1〉 atoms are repelled from the location occupied by |2〉 atoms because

U12 > U11. This then creates a small potential well for the |2〉 atoms. These two

processes enhance each other and distinct domains of |2〉 atoms form. The number

of domains depends on the details of the relative interaction strengths and the initial

conditions. In the case shown we get two. Once the majority of the |2〉 atoms are

contained in the small wells, a kind of equilibrium is reached as compressing the wells

further would lead to additional repulsion from |2〉 ↔ |2〉 interactions. At this point

the domains begin moving due to inhomogeneity in the overall condensate density.

At times later than those plotted in the figure, the domains of the two atoms change


directions and go back towards the center, cross each other and continue to move

back and forth in an oscillatory manner. Fig. 6.10(c) shows the kinetic energy per |2〉

particle and |1〉 particle. We see the kinetic energy of the |2〉 atoms initially increases

due to the phase separation and then remains approximately constant at a relatively

low value compared to µTF . The kinetic energy of the |1〉 always remains small.

Because this evolution introduces only a small kinetic energy, the strong probe case

can support good revivals. We observe numerically the transmission of the revivals

decreases for the first 10 ms, but then the kinetic energy holds fairly constant. The

revival transmission remain between 30% and 40% of the input pulse energy after

this. Fig. 6.10(d) shows some examples. The revivals are seen to follow the changes

in ψ2. At 20 ms, the two domains are fairly close and the double pulse structure of the

revival is washed out during the propagation. Note that, in addition to the appearance

of high frequency components in the probe pulse, nonlinearities associated with the

strong probe regime (see Subsection 3.3.1) play a role in the subsequent propagation.

At 50 ms when the two domains are well separated we get a double pulse structure

on the output.

Obviously, the parameter space for the various relative interaction strengths and

the relative number of atoms in each condensate is very large and the richness of the

possible dynamics go beyond the scope of this thesis. However, the phase separation

phenomenon will generally occur when U12 >√U11U22. In terms of revivals, we have

seen that the formation of domains can suppress the quantum reflections and the

kinetic energy gained by the atoms is often small.


Figure 6.11: Losses due to inelastic collisions. (a) The density of |2〉 atoms attimes after the switch-off of 0, 0.36 and 1.16 ms (successively thicker lines) in thepresence of a large inelastic loss rate: U12 = (1 + 0.1i)U11, U22 = U11. We used a weakprobe Ωp0 = Ωc0/4 and other parameters are the same as in previous figures. (b)The total number fraction of |2〉 atoms N2/N as a function of time (solid curve). Ananalytic prediction, based on the loss rate at the center of cloud (described in text),is also shown (dashed curve).

6.2.5 Dissipation: Inelastic loss processes

In any experimental implementation of long revivals, there will be an additional

limit set by losses of atoms due to inelastic collisions, which will be set by ImU12.

The decay constant of |ψ2|2 in the condensate center is Rloss = 2(ImU12/U11)µTF

which provides an estimate for the allowed storage times. In Fig. 6.11 we show the

numerically calculated density N |ψ2|2 when ImU12 = 0.1 U11 and compare it with

this estimate. The significant inelastic collisional loss rate causes the magnitude of

|ψ2|2, and therefore the revival intensity, to be reduced with a time scale ∼ 1 ms. The

analytic estimate is a good estimate of the loss rate (it just slightly overestimates the

total loss because it is assumes the rate due to the central (peak) density everywhere).

This loss mechanism, which can be controlled by controlling the density, effectively

puts a limit on the maximum densities which are feasible for long storage experiments.

In light of this loss mechanism, we note that there is much to gain (about two

orders of magnitude) by using rubidium-87 atoms, as opposed to sodium, when at-


tempting to do long storage experiments in the co-propagating geometry with equal

traps. Long lived (∼ 100 ms) two species condensates in the F = 1 and F = 2 levels

have been observed experimentally at JILA [29]. Also note that the in the stronger

probe regime, the quantity ImU22 can play an important role as well.

6.2.6 Long storage and processing: Trapped |2〉 case with

U12 < U11

The last case we examine is one in which we have no velocity recoil, equal traps,

a small inelastic loss rate, and U12 < U11. This is the case which promises to offer the

largest light storage and processing times in BECs. We will first briefly consider the

evolution and the quality and nature of the revivals in both the weak and strong probe

case. We will then speculate on applying these long storage times for experiments

which investigate and use properties beyond the mean field approximation, such as

phase diffusion, spin squeezing, and quantum processing.

Weak probe case

The potential V2−eff in this case is diagrammed in Fig. 6.12(a). Here the |2〉 atoms

will be attracted to the center of the cloud due to smaller interaction strengths. The

dynamics will be analogous to the V2 > V1 case discussed above in Fig. 6.6. Just as in

that case, the atoms will be trapped and breathe very gently, rather than be pushed

to the border and undergo quantum reflections. Because α(MF ) 1, the dynamics

will again be on a slower time scale than the case in Fig. 6.6.

While the dynamics do not significantly change the density |ψ′2|2 or the intensity


profile of the revival, there is, however, a slow inhomogeneous phase evolution of the

relative phases of ψ′1 and ψ′

2 due to the difference in scattering lengths. We denote

their phases by φ1, φ2, respectively. Because the density is largest at the center,

this relative difference will increase fastest at the cloud center and not at all at the

condensate edge. The rate of relative phase shift ∆φ ≡ φ2 − φ1 at the point of

maximum density (z = 0) will be:

∆φ = −1

(U12 − U11)|ψ(0)

1 (z = 0)|2 = −α(MF )µTF , (6.13)

and at other z follow the same evolution scaled by the density at that point |ψ(0)1 (z)|2.

Because of the rather mild dynamics, this system offers a good way of inducing an

arbitrary phase shift between the input and output probe pulses, without significantly

altering the pulse otherwise. The magnitude of the shift can be controlled by the

storage time. We show an example in Fig. 6.12(b) where a π phase shift occurs after

4.4 ms. In the case shown, the pulse is actually compressed to a rather small region

in the center of the condensate. For this reason, the inhomogeneity of the phase shift

across the pulse is quite small, a fact which may be important in applications. One

could also imagine making the shift completely homogenous by having a flat region

in the center of the trap potential so the density is constant there. The result of a

revival at 4.4 ms is shown in Fig. 6.12(c). There we see an output probe which has

a nearly uniform phase shift of π relative to the output with no storage time (i.e. a

regular USL experiment). The analytic prediction (6.13) is seen in Fig. 6.12(g) to

quite accurately predict the phase shift at the center of the cloud through several

rotations of the phase. In Fig. 6.12(h) we note the important fact that the kinetic


Figure 6.12: Phase shifting a probe pulse. (a) The effective potential V2−eff

when U12 = 0.9 U11, U22 = U11 in a weak probe case (Ωp0 = Ωc0/8). Other parametersas in previous figures. (b) The density N |ψ′

2|2 after the switch-off (solid curve), thephase φ2 (dashed curve). The density N |ψ′

1|2 divided by 100 (dotted curve) and phaseφ1 (dot-dashed curve). The same quantities are then shown after 4.4 ms of evolutionwhen ∆φ = π. (c) The output probe intensity (thick solid curve) and phase (thinsolid curve). These are compared with the output with no atomic evolution (intensityas dashed curve and phase as dotted curve) (Equivalently this can be thought of asthe output of a regular USL pulse but shifted by 4.4 ms so they can be seen on thesame plot.) (d-f) The same plots in a strong probe case (Ωp0 = Ωc0/2). The time6.6 ms is chosen because it is the time that ∆φ is observed numerically to evolve fromπ to 0. In (e) the two densities N |ψ1|2 and N |ψ2|2 are plotted on the same scale.(g) The relative phase ∆φ at the center of the cloud (z = 0) for the weak probecase (filled circles) compared with the analytic prediction (6.14) (solid curve) (whichreduces to (6.13) in the weak probe limit). The same plot for the strong probe caseis shown as open circles and the dashed curve. (h) The kinetic energy per particle of|1〉 (dotted curves) and |2〉 (solid curves) atoms in the weak probe case (thin curves)and strong probe case (thick curves).


energy of the both the |1〉 and |2〉 atoms remains near their original values due to the

very gentle potential in this case (Fig. 6.12(a)).

Strong probe case

In the strong probe case, whenever U12 <√U11U22 it is not energetically favorable

to phase separate and the two components live together for long times. The strong

probe case is then similar to the weak probe case. However, the phase dynamics are

somewhat richer. Because of a the significant density in |2〉, the phase shift at the

z = 0 will be modified:

∆φ = −(α(MF )

( |ψ1|2ψ2

− |ψ2|2ψ2

)+ α

(MF )2

|ψ2|2ψ2

)µ, (6.14)

where α(MF )2 ≡ (U22 − U11)/U11 and ψ2 ≡ |ψ1|2 + |ψ2|2. The densities |ψ1,2|2 are

determined by the relative intensities of the light fields, and we thus see that the

phase evolution will depend on the input intensity of the original probe pulse. In

Fig. 6.12(d-f), we show a case of a π phase shift of the probe pulse after 6.2 ms for

the strong probe case Ωc0 = Ωp0/2. The significant density |ψ2|2 causes the time

for this phase shift to occur to be different than the 4.4 ms in the weak probe case.

In Fig. 6.12(g) we see that a simple analytic prediction, based on the original TF

condensate density and assuming fractional densities in |ψ1,2|2 based on the peak

input Rabi-frequencies, overestimates the effect of the nonlinearity. This is primarily

because, in the strong probe regime, the pulse experiences significant distortion on

its way to the center of the cloud and our light field intensities are thus different

than these analytic estimates. Either numerical calculations, or a more complicated


analytic treatment is needed to accurately predict the phase shift. However, we still

have the important result that we can make the phase shift input dependent, an

aspect which is important in for some applications using the BEC as a processor or

logic gate for probe pulses.

6.2.7 Outlook: Applications beyond the mean field

The set of parameters discussed in the last subsection (k2 = 0, V1 = V2, a12 <

√a11a22, Ima12,22 a11) offer the longest possible storage and processing times and

could be well suited to study and utilize condensate dynamics beyond the mean field

GP equations. In Chapter 4 we developed a formalism which completely neglected

the fluctuations (as defined in (4.8)). Analysis which includes these fluctuations are

beyond the scope of this thesis, but offer exciting potential for the next steps in

harnessing the coherent quantum nature of BECs and the coherent transfer we have

seen between atoms and light fields. We briefly mention three specific possibilities

here.

First, we consider the phase coherence of the BEC. In the limit of large N , the

Heisenberg uncertainty relation between number and phase can often be ignored.

However, it is possible that the dynamics of two component condensates could cause

the phase fluctuations to increase over some characteristic time scale, an effect which

has been called phase diffusion [88]. Understanding this diffusion is important in un-

derstanding the limits of atomic BECs as coherent matter field sources. Theoretically,

there has not been a strong consensus as to how to characterize and calculate this

quantity, though these time scales are generally predicted to be on the order of hun-


dreds of milliseconds. There has been little in the way of experimental observation,

though recently Orzel et al. [36] saw an increase in phase fluctuations by observ-

ing a decrease in the contrast of interference fringes and therefore inferring number

squeezed states.

It is possible our system could address this issue. We note that measurements of

the density in each ground state |ψ1|2, |ψ2|2 (such as absorption images), are really

measurements of the expectations values of the number operators 〈Ψ†1Ψ1〉, 〈Ψ†

2Ψ2〉

in the exact many body state. On the other hand, the amplitude of our revived

pulses are determined by the coherence between the two ground states. The revived

probe amplitude can thus measure the coherence 〈Ψ†2Ψ1〉 independent of 〈Ψ†

1Ψ1〉 and

〈Ψ†2Ψ2〉. Combined, these measurements will tell us the extent to which the mean

field description of the atomic fields breaks down, and therefore the extent to which

fluctuations are present.

Along the same lines, it has been pointed out in the literature [37], that the

nonlinear interaction present in two component BECs could be used to create spin

squeezed states. Assume that a condensate is initially prepared so all atoms oc-

cupy the same state (a product of coherent state) and so the fluctuations are clas-

sical (i.e. Poissonian). Then it can be shown that a non-zero squeezing parameter

χsq ≡ U11|ψ1|4 + U22|ψ2|4 − 2U12|ψ1|2|ψ2|2 will cause the two component sample to

evolve into a spin squeezed state, whereby the fluctuations in certain operators go

below the value for that of a coherent state, while increasing the fluctuations in con-

jugate operators. The problem is usually analyzed in terms of the spin operators

defined as Jz = Ψ†1Ψ1 − Ψ†

2Ψ2, J+ = Ψ†1Ψ2, and J− = Ψ†

2Ψ1. Since our stopped light


technique already creates two component overlapping BECs, this behavior should

occur after long enough storage times. When the coupling field is then switched on,

this squeezed state should then be written onto the probe field. Several recent pa-

pers [14] have recently argued that the transfer of a quantum states (beyond mean

fields), such as squeezed states, should be successfully transferred between atoms and

light fields just as we found robust transfer between the mean fields. Squeezed states

have potential practical applications in precision measurements, as they can reduce

intensity fluctuations below the shot noise limit.

Finally, we note that we saw above how the system could be used to induce phase

shifts on light pulses, acting as a kind of “gate” or processor. In particular we saw

that when the probe pulses are sufficiently intense, the phase shift induced is actually

a function of the probe intensity. This input dependent phase shifting ability could

be used as a the building block for complicated two or more qubit quantum gates and

processors.

Chapter 7

Ultra-compressed light

As we noted at the beginning of Chapter 5, an ultra-slow probe pulse’s suscepti-

bility is a strong function the coupling intensity Wc, an experimental parameter than

we can precisely control. We have not, to this point, considered the possibility of

spatially engineering Wc. One of the advantages offered by cold atoms is the possi-

bility of using an orthogonally-propagating geometry. As we will see in this chapter,

in such a geometry, Wc (and therefore the susceptibility) can be made an arbitrary

function of space.

We implemented this idea earlier this year [4] and made a light roadblock, where the

coupling field was turned off spatially (Wc = 0) in a large region of the condensate.

Both theoretically and experimentally, we saw that when the pulse arrived at the

roadblock it was compressed to extremely small lengths (Lp ∼ 2 µm). This is a

significant compression below that already obtained with regular USL (∼ 50 µm), and

smaller, in fact, than the optical resolution of our system (∼ 5 µm). Furthermore,

it led to unique large amplitude and small wavelength excitations in the BEC. Using

232

Chapter 7: Ultra-compressed light 233

Figure 7.1: The Light Roadblock. The geometry of a light roadblock experiment.A razor blade is used to block the coupling beam from illuminating the z > 0 half ofthe condensate, establishing the roadblock in middle, along the z = 0 plane.

this fact, we made first direct observation of the nucleation of vortices via the snake

instability in BECs [4], which we will discuss in Chapter 8. Here we concentrate on the

technique itself and its ultimate limits. The general technique of spatial engineering of

the coupling intensity offers a number unique and interesting possibilities for creation

of structure in one and two component BECs with a very small length scale.

In Section 7.1 we will give the general arguments which motivated us to consider

this possibility. In terms of pulse propagation, we will see that when it sees a shrinking

Wc in space, we get (1) reduced group velocities, (2) increased probe to coupling

intensities, and (3) a large additional compression of the probe’s spatial length. These

result in the creation of a narrow dense condensate in |2〉 and corresponding large

amplitude, small wavelength excitations in the |1〉 condensates.

The roadblock is both the simplest and most dramatic implementation of this

concept. It is diagrammed in Fig. 7.1. The coupling field is blocked, with a razor

blade, from illuminating the z > 0 region of the BEC. We call this a light roadblock

since the probe pulse sees an halt and compresses upon reaching it. A large part of

the chapter (Section 7.2) will be devoted to presenting our results on this geometry,

both theoretically and experimentally.


We will see that the roadblock eventually absorbs the probe removes some of

the atoms from the condensate, however its interest lies in the excitations it creates

in the |1〉 condensate. More sophisticated versions of this technique, discussed in

Section 7.3, would involve reduce the coupling field to a small, but non-zero value. In

this way, we could achieve an extremely compressed probe and a primarily coherent

small, “defect” condensate of |2〉 atoms inside a condensate of |1〉 atoms.

The methods discussed in this chapter involve compressing pulses to length scales

near the wavelength of light, meaning the assumptions which go into the slowly vary-

ing envelope approximation, which we made in Chapter 3 and have assumed in cal-

culations, begin to break down. We have made a preliminary estimates of corrections

due to the finite wavelength of light but this goes beyond the scope of this thesis. It

is an interesting issue that merits pursuit.

7.1 Motivation and basic picture

We have already established that, in the weak probe limit and under conditions

of EIT, the coupling field in general sees a small polarization and propagates nearly

freely. If we consider an orthogonally propagation geometry (w = x), and make the

input value of Wc (defined in a plane x = xin) a function of z, then this input value

will propagate with a nearly uniform intensity to all x. The situation can then be

analyzed as a z dependent susceptibility.

Suppose we have a probe of length Lp and peak amplitude Ωp0 propagating in the

presence of a coupling Rabi-frequency Ω(0)c0 , and this Rabi frequency changes, at some

position, to√αΩ

(0)c0 . Then the group velocity will change from its original value V

(0)g


to αV(0)g and the spatial length of the pulse will change to αL

(0)p , while the temporal

width τ will remain constant. Furthermore, the probe intensity will remain the same,

and so the peak probe to coupling ratio will change to Ωp0/√αΩ

(0)c0 . As a result, we

drive a much larger fraction of atoms into the |2〉 state in this region. In the case

the coupling field is reduced to values less than or on the order of the probe field

(or completely off) then the weak probe limit breaks down, and we must rely on the

qualitative arguments developed in Section 3.3.1 and numerical results to guide us.

Before we analyze this in detail, we mention the advantages and disadvantages of

this scheme, as compared to the dynamically changing Wc we studied in Chapter 5.

In the case of time-dependentWc, the group velocity Vg varied, however, the intensity

|Ωp|2 and temporal width τ adjusted in way that kept the probe to coupling ratio and

the pulse length Lp constant. So the present method offers an additional advantage

of being able to control these two parameters. And, in analogy to Section 5.3, the

intensity can be reduced only in a small region in the center, and therefore go into the

strong probe regime and highly compress the probe only for a short time, without the

resulting distortion and the attenuation accumulating over the entire optical density

D0. This freedom will be seen to be essential in creating narrow, large amplitude

defects which lead to the observation of quantum shock waves in [4] (discussed in

Chapter 8).

At the same time, this also means this method must be used with some caution.

In particular, unlike the time-dependent Wc, the fields to not automatically adjust

themselves to satisfy the adiabatic requirement (τ W−1c ), so turningWc too low will

lead to a large number of absorption events. This is the case in the light roadblock,


where Wc → 0 for z > 0, and the probe is eventually absorbed, and many atoms

are ejected from the condensate. This was not a concern in the superfluid studies

we performed in [4], where we concentrated primarily on |1〉 condensate and the

excitations created by the removal of a very narrow density defect. In that case,

because both the |2〉 atoms and any atoms which had undergone absorption and

spontaneous emission experienced a large two-photon momentum kick, they quickly

leave the region of the |1〉 condensate on a fast time scale (∼ 1 ms) compared to the

dynamics of the |1〉 condensate itself. We will return to this issue in Chapter 8.

To study the technique of spatial engineering of the coupling field, we performed

1D numerical simulations on large condensates with a large, homogeneous region in

the middle (achieved by using a trap with a flat bottom). We ramped the intensity

down over a distance ∼ 100 µm. In general we assumed an input intensity of the

form:

Wc(z) =Wc(zin)0.5 Erfc

(z

ζ

). (7.1)

In the present case we used ζ = 50 µm. Because the spatial engineering of the

coupling beam requires an orthogonal propagation, which we cannot account for in

1D simulations, we assume in the simulations that the coupling field is constant in

x, y and time, and simply equal to its input at each z (i.e we ignore the adiabatons).

Corrections due to the transverse dynamics of the coupling field are considered later

in the chapter.

Fig. 7.2(a) shows the probe intensity spatial profile for such a simulation at a

series of times. The input pulse was τ0 = 5 µs. The length Lp is seen to compress


Figure 7.2: Ultra-compressing light pulses. Plots are based on numerical 1Dsimulations for a trap containing N = 6× 106 atoms in trap with the same curvatureon the edge as previous cases but a large 323 µm long flat region. The couplingRabi frequency is Ωc0(zin) = (2π)8 MHz where the probe enters but turns down asindicated with the dotted curves in (a) and (b). (a) For a pulse input Rabi frequencyΩp0 = (2π)2.5 MHz and width τ0 = 5 µs, we plot the spatial intensity profile at thetimes indicated. (b) The densities N |ψ1|2 (solid curves) and N |ψ2|2 (dashed curves)at the same times as (a).

as it enters the region of lower coupling intensity (the coupling intensity is indicated

with a dotted curve). From the times indicated in the figure it is also clear the group

velocity is also getting dramatically reduced. Although absorptions are seen to reduce

the probe intensity, the probe to coupling ratio still increases. Fig. 7.2(b) shows the

resulting condensate densities N |ψ1|2, N |ψ2|2 at the same times. The density N |ψ2|2

is indeed seen to increase as the probe to coupling ratio increases until the probe is

significantly absorbed. We note that at t = 94 µs, the number of atoms removed from

the |1〉 condensate is greater than the number of atoms in the |2〉 condensate. This

is because absorptions have removed atoms from the condensates (though strictly

speaking the atoms are still there so the total density of atoms is the same until these

recoiling atoms have left the region).

In Fig. 7.3 we present quantitative calculations of the propagation and compression

dynamics for a weak (Ωp0 = (2π)0.5 MHz), moderate (Ωp0 = (2π)2.0 MHz), and


Figure 7.3: Comparing ultra-compressing light pulses with analytic formu-lae. Plots are based on simulations for the same cloud and coupling parameters(Ωc0(zin) = (2π)8 MHz) as Fig. 7.2. In each plot, we show results of a simula-tion for Ωp0 = (2π)0.5 MHz (filled circles), Ωp0 = (2π)2.0 MHz (open circles), andΩp0 = (2π)4.0 MHz (squares). The dashed curves shows the coupling intensity pattern(in arbitrary units). (a) The time (delay τd) for the mean position of the intensity ofthe probe to reach a the position z. The solid curve shows the analytic expectationτd = f13σ0

∫ z

zindz′ n(z′)Wc(z

′), which is based on (3.31) but accounting for the chang-ing Wc as the pulse propagates. (b) The numerically calculated length Lp (basedon the second moment of the intensity) compared with the analytic prediction forLp (3.33) (solid curve). (c) The peak probe (normalized) intensities versus propaga-tion distance and the analytic expectation based on (3.31) (solid curve). The dottedcurve shows the non-adiabatic parameter 1/(τ0Wc). (d) The peak of the quantity|ψ2|2 (normalized to the initial density |ψ1|2) versus the mean pulse position. Thesolid curves show the analytic expectation (Ω2

p0/Ω2c0)|ψ1|2 in each of the three cases

(which diverges as Wc → 0).

strong (Ωp0 = (2π)4.0 MHz) probe, as compared to the peak coupling input Rabi

frequency Ωc0(zin) = (2π)8.0 MHz. Fig. 7.3(a) shows the time (delay) it takes to

reach a given position z. In all cases we see agreement with the analytic expectation

based on Eq. (3.31 (solid curve), where we have accounted for the spatially changing

coupling intensity Wc (formula in caption). Fig. 7.3(b) shows that the length of the

probe pulse Lp shrinks according our analytic expectation (3.33) as well, although

there is a noticeable deviation from (3.33) in the strong probe case due to the finite


probe intensity (i.e. the length is determined by Ω2 = |Ωc|2 + |Ωp|2). At high

coupling intensity, this correction is only a small fraction of the total length, however

as the Ωp0 ∼ Ωc0(z) this correction becomes a dominant effect in determining the

Lp. Fig. 7.3(c) then shows how the probe is attenuated as it enters the low coupling

intensity ratio. We see that the analytic expectation based on τsp gives us some

guidance but overestimates the ability of the probe to propagate in the low intensity

region. This is because the binomial expansion (3.29), upon which the result is based,

begins to lose its validity (The dotted curve shows the quantity 1/(Wcτ0) which must

be small for the expansion to be valid.). We also see the strong probe case has more

reduction than the weak probe case.

Now consider the density of |2〉 atoms. We plot the maximum value of the density

N |ψ2|2 (normalized to the initial condensate density) as a function of probe propaga-

tion distance in Fig. 7.3(d) and note that it initially increases then saturates at some

value. While the strong probe case ends with a higher density, the degree to which

it is higher is seen to be less than the naive expectation based on the value Ω2p0 in

the three cases. This is because the high number of absorptions (Fig. 7.3(c)) and less

compression (Fig. 7.3(b)), both of which determine this amplitude, are less favorable

in the strong probe case.

We note the interesting observation that the |2〉 condensate is hardly absorbed

at all in the low coupling intensity region. This is because when we reach a region

where the EIT picture breaks down (that is, where 1/(Wcτ0), plotted in Fig. 7.3(c) is

comparable to unity) the probe photons simply absorb |1〉 atoms and do not interact

with the |2〉 atoms. The coupling field is nearly off in this region and so there is no


mechanism to either absorb |2〉 atoms or induce them to coherently transfer to |1〉.

This observation could be important in designing turn-off profiles which maximize

the density N |ψ2|2, however, this investigation is beyond our scope.

7.2 Light roadblock

The real advantage of spatial engineering of Wc, as far as creating condensate

excitations, is the ability to highly compress the pulse just at the center of the cloud

without propagating it through the entire cloud with a small Wc. Optimal results

for pulse compression will occur for a rapid turn off of the coupling intensity as this

will minimize the optical density D in the region of the low Wc. Thinking of this

in terms of Fig. 7.3(c), the attenuation of the probe at a given z is not determined

by the non-adiabaticity at a given point (τ0Wc(z))−1, but by the cumulative effect

of the non-adiabaticity∫ z

zindz′(τ0Wc(z

′))−1. The razor blade (Fig. 7.1) is the fastest

possible turn off, as the length scale will be determined only by the finite optical

imaging resolution. In our experiments [4] the coupling input intensity went from

95% to 5% of Wc(zin) over a distance of 12 µm. Using this we saw creation of

extremely narrow defects. The defects could have nearly 100% amplitude for sizes as

small as ∼ 2 µm and smaller amplitude defects could be even smaller.

We will first analyze the problem in one dimension. We will present some 2D nu-

merical calculations to check the accuracy of our 1D picture and learn the corrections

which occur due to the probe and coupling field exchanging photons (i.e. adiaba-

tons). We find these corrections are small and easy to understand. We will then

present experimental results of our implementation of a roadblock [4] with a razor


blade and demonstrate its successful creation of sub-resolution sized, large amplitude

defects.

7.2.1 1D picture

In Fig. 7.4 we show two examples with an experimentally realistic 10 µm (95%

to 5%) turn off of the Wc (which corresponds to ζ = 3 µm in (7.1)). Fig. 7.4(a)

shows the propagation of an input pulse with Ωp0 = (2π)2.5 MHz and τ0 = 1.5 µs,

which is characteristic of pulse widths which are completely contained in the cloud

under regular USL conditions (no roadblock) . We see the pulse, upon entering the

roadblock region, is compressed while maintaining a significant intensity. Fig. 7.4(b)

then shows that, in this region, the probe to coupling Rabi frequency ratio becomes

quite large and |ψ2|2 becomes comparable to |ψ1|2. The inset in Fig. 7.4(b) shows

the condensate densities once the probe has been absorbed, but before any significant

atomic dynamics have occurred. We see it has created of a fractional density depres-

sion in the |1〉 atoms of 50 %, with a half-width half-max of 1.5 µm. The density of

condensate |2〉, N |ψ2|2, fills in part of the void but some atoms are lost from both

condensates due to absorptions. By contrast, we show in Fig. 7.4(c-d) a case with a

much longer pulse τ0 = 4.5 µs. In this case the defect is seen to be 100% with and a

2.0 µm half-width.

In all cases we analyzed, we found the number of atoms removed from the |1〉

condensate is almost exactly the number of probe input photons, which we call Nγ.

Each input probe photon ends up either being absorbed and removing a |1〉 atom

or going into the coupling field and coherently transferring a |1〉 atom into the |2〉


Figure 7.4: Creation of defects with a light roadblock. 1D simulations ofa roadblock with a coupling turn off width of ζ = 3 µm according to (7.1) (thiscorresponds to 10 µm to drop from 95% to 5% intensity). The gray scale indicatesthe input coupling intensity. The clouds contain N = 1.2 × 106 atoms, and are ina ωz = (2π)21 Hz trap and Ωc0(zin) = (2π)8 MHz. (a) Inputting a probe withΩp0 = (2π)2.5 MHz and τ0 = 1.5 µs, we plot the probe intensity profile at t = 1.5 µs,4.5 µs, and 7.5 µs (successively thicker curves). The dotted curve indicates the originalcondensate density (arb. units). (b) The atomic densities N |ψ1|2 (solid curves) andN |ψ2|2 (dashed curves) at the same three times. The inset zooms in on the roadblockregion at a time after the probe is absorbed but before significant atomic dynamicshave occurred (t = 11.7 µs). The vertical dotted curves indicate a defect with theminimum possible half-width (7.2). (c) A simulation with a much longer probe pulseτ0 = 4.5 µs at the times t = −0.9 µs, 4.5 µs, and 9.9 µs. (d) The atomic densitiesat these times. The inset shows a 100% defect is created, with a size nearly at thefundamental limit (7.2).

condensate, so the defect in |1〉 must contain Nγ atoms. For a given number of re-

moved atoms, the defect with the smallest possible spatial scale would be rectangular

shaped 100% defect (with infinitely steep walls) and a volume containing the number

of atoms removed, and have a half-width:

δ(min)def =

2Nγ

AN |ψ(0)1 |2, (7.2)

where we have assumed |ψ(0)1 |2 does not significantly vary over the defect region. Ver-


tical dotted lines indicated in the insets of Fig. 7.4(b,d) show these smallest possible

rectangular defects for the Nγ in those cases. We see that the defect is 100% over

a fairly large region in the τ0 = 4.5 µs case, and its total size is almost at this fun-

damental lower limit. The slightly larger size is due to its walls having some finite

spatial scale. The smaller defect created by the τ0 = 1.5 µs pulse is about a factor of

two larger than this limit, due to the its amplitude being only 50%.

It is interesting to examine the ultimate limit on the length scale of defects created

with the roadblock. To investigate this issue, we performed numerical simulations for

a variety of probe input intensities Ω2p0 and widths τ0, keeping the coupling intensity

fixed. We found that the total integrated number of photons Nγ (which scales with

Ω2p0τ0) was the most important quantity in determining both the amplitude and width

of the defects. Fig. 7.5(a) shows the peak fractional defect amplitudes (relative to

the original peak condensate density) versus Nγ (normalized to the initial number of

condensate atomsN). We did a the series varying τ0 while keeping Ωp0 = (2π)2.5 MHz

(filled circles) and another series with one-fourth the intensity Ωp0 = (2π)1.25 MHz

(filled squares). The two curves fall almost on top of each other leading us to conclude

that the Nγ/N is the relevant parameter. We note in either case we get a nearly 100%

defect when Nγ/N > 0.06.

In Fig. 7.5(b) we then plot the half-widths of the defects. The solid line shows

the minimum possible width (7.2) (plotted in Fig. 7.4(b,d) insets). As we would

expect, when we have less than 100% defects, as was the case for the τ0 = 1.5 µs case

plotted in Fig. 7.4(a-b), the defect widths are above this minimum value, but as we

get into the full defect regime (as in the τ0 = 4.5 µs, Fig. 7.4(c-d)), the defect widths


Figure 7.5: Amplitude and width of roadblock created density defects. (a)Simulations of defect creation with varying pulse widths and intensities. Exceptwhere otherwise noted, the coupling intensity Ωc0(zin) = (2π)8 MHz and turns offwith ζ = 3 µm. The horizontal axes plots the total number of input photons Nγ

normalized to the original condensate number N , which determines the total fractionof atoms removed from the |1〉 condensate. The horizontal axis indicates the peakdensity of the defect (relative to the original peak density). (The defect density isdefined to be the difference in the density N |ψ1|2 before and after the probe pulseis input.) The filled circles show a series with Ωp0 = (2π)2.5 MHz and τ0 varyingto adjust the total probe energy. The filled squares show a series with a fourth thepeak intensity (Ωp0 = (2π)1.25 MHz). The triangles show Ωp0 = (2π)1.25 MHz witha more precise turn off ζ = 1 µm. Open circles and squares show the results of cutsthrough x = 0 in the corresponding 2D simulations (discussed in the next section).(b) The measured half widths (second moment), of the defects for the same cases.The solid curve shows the minimum possible width for the given number of removedatoms (7.2).

are increasingly close to this fundamental limit. We also see a somewhat unexpected

trend that the defects for the stronger probe case are slightly but consistently narrower

than the weaker probe case (one might expect the opposite based on the results of

Fig. 7.3).

We also performed simulations with different Ωc0(zin) and found this parameter

did not play a role. The results closely agreed with the ones in plotted in Fig. 7.5.

To test the dependence on the cut-off width ζ of the coupling field, we performed

a series Ωp0 = (2π)1.25 MHz but a faster cut off width ζ = 1 µm, which is shown as


triangles in Fig. 7.5(a-b). We note that we reach a full defect earlier (Nγ/N > 0.04)

though the reduction (from 0.06 to 0.04) does not appear to scale directly with ζ. In

the regime Nγ/N > 0.06, when both ζ = 1 µm and ζ = 3 µm create 100% defects,

the widths are very similar and both near the minimum possible width (compare the

filled squares and triangles).

From these results, we conclude that, to a good approximation, we can approxi-

mate the size of the defect from the minimum width argument once the defect ampli-

tudes are near 100% and this seems to happen once the defects are about 2 µm for our

parameters. The factors which go into determining the point at which we get nearly

100% defects is interesting and merits further investigation. The cut off width ζ plays

some role. One would also expect, since we generally get into an absorptive regime,

that the absorption length (αA)−1 also plays a role. However, the exact dependence

on these two parameters is not clear at present. Furthermore, the absorption length

for our parameters is (αA)−1 = 0.2 µm, which is shorter than the finite wavelength

λ ≈ 0.6 µm so we really must include the effects of the finite wavelength to get a

good quantitative description of defects on this scale.

7.2.2 Effects of the transverse propagation of the coupling

beam

While the above discussion provides the essential intuitive guidelines for under-

standing and optimizing the light roadblock, we must exercise caution because we

have not been treating the coupling field dynamically. As we saw in our discussion

of adiabatons in Section 3.3.1, the fractional variation in the coupling field relative


to its input value scale with Ω2p0/Ω

2c0. Therefore, when we are in a region where the

coupling intensity is significantly reduced and Ωp0 ∼ Ωc0(z), it is conceivable that

they would have an effect on the defect creation.

To test this, we performed 2D simulations in the orthogonal geometry for the

same parameters as our 1D simulations. We plot the results of a simulation with

τ0 = 4.5 µs in Fig. 7.6. In Fig. 7.6(a) we plot the density |ψ1|2 and note that,

because of the smaller densities on the edges of the condensate, the probe reaches the

roadblock there first. It also creates slightly longer defects there (again due to the

smaller density). In Fig. 7.6(b) we plot the probe intensity pattern at the same times

and see the compression at the roadblock, with some leakage very near the condensate

edges. Fig. 7.6(c-d) shows cuts of the probe and coupling intensities along both z and

x as indicated in (b) at t = 7.9 µs, a time where the probe is primarily occupying the

roadblock region and has not been absorbed.

While most of the quantitative effect is due to the varying condensate density

in the x direction, it is interesting to note the small but visible asymmetry between

positive and negative x, especially visible in the light field intensity cuts along x in

Fig. 7.6(d). To understand the behavior, one must examine the propagation of the

coupling field as it propagates transversely across the cloud. We observed that in the

regions where the coupling field input is off, we actually have a reversed adiabaton

situation where the presence of the probe field can drive photons into the coupling

field and the coupling field gains intensity as it propagates across the cloud. As a

result the roadblock is less severe and the probe can propagate for slightly further

distances on the x > 0 side. This mechanism is responsible for the slight asymmetry


Figure 7.6: Transverse effects in the light road block. A 2D simulation of aroadblock. Parameters are the same as in Fig. 7.4(c-d) (τ0 = 4.5 µs) and we use atransverse trapping frequency ωr = 3.8ωz and length in the third dimension (y) ofLy = 2×π(8.8 µ m). (a) The density |ψ1|2 at the times indicated. The original peakdensity of the condensate is 7.1 × 1013cm−3. We see that the lower density on edgesleads to a faster development of the defect and ultimately a larger width. There isalso a slight asymmetry in x due to the adiabatons (discussed in the text). (b) Probeintensity (normalized to the peak input intensity) at the same times. (c) The probe(dashed curves) intensities at cuts along z indicated in the third frame of (b) (7.9 µs),at which point the peak of the probe is in the road block region. The thickest curveshows the center cut (x = 0.0 µm), the thinnest curve shows the x = −4.8 µm (nearestthe coupling input) and the medium thickness curve shows x = +4.7 µm. The solidcurve shows the coupling at these cuts (they are indistinguishable from each other onthis plot). The dotted curve shows the original condensate density (arb. units). (d)Cuts along x at z = 0.0 µm (thin curves) and at z = 6.2 µm (thick curves) (againdashed curves indicate probe intensity and solid curves coupling intensity). (e) Close-up of cuts of the density N |ψ1|2 along z at the same cuts (indicated in the bottomframe of (a)) (again with the x = 0.0 µm (thickest), x = +4.7 µm (medium), andx = −4.8 µm (thinnest)). (f) The solid curve shows the calculated defect amplitudesas a function of x (normalized to the peak density at that particular x, which is plottedas the dotted curve in arb. units) once the probe has been absorbed. The dashedcurve shows the half widths.


seen in the defect in Fig. 7.6(a). In Fig. 7.6(e), cuts of |ψ1|2 along z show that defect

amplitudes are slightly reduced at larger x. In Fig. 7.6(f) we plot the amplitude of

the defect as a function of x (normalized to the peak density at each particular x).

The defect width is also plotted. Note that, because of this small adiabaton effect,

the defect along x = 0 (where one would expect the results to be the same as the

1D case) is not 100%, as it was in the corresponding case for the 1D simulations (see

Fig. 7.4(c-d)). So the net effect of the transverse propagation is to slightly reduce

defect amplitudes.

However, we see that overall the effect of the coupling propagation is generally a

very small correction to the 1D results we obtained above. Most of the important

effects can be obtained by simply accounting for the smaller densities (and therefore

longer and deeper defects) away from the central (x = 0) line. Comparing the central

x = 0 cuts of the 2D simulations with our 1D simulations for a number of cases we

see only very small corrections. In Fig. 7.5 we plot (with open circles and squares)

the defect amplitude and widths along the center cuts (x = 0) for several cases

corresponding to the 1D simulations. We see a slight reduction in defect amplitudes

and slight increase in widths but generally very good agreement with the 1D results.


We have implemented the light roadblock experimentally by using a razor blade

to block the coupling beam on the z > 0 side of the BEC. The best diagnostic to

track the probe propagation spatially is to the monitor the density of |2〉 atoms. In

Fig. 7.7(b) we show several absorption images. In each plot, the coupling beam is


switched off at a different time during the propagation, freezing the |2〉 atoms for

imaging. The 10 µs imaging time is small enough that no significant atomic recoil

occurred during it. We see that as the pulse enters (t = −1.25 µs), there is a small

density sample of |2〉 atoms covering a large region of the z < 0 side of the original

|1〉 condensate (shown in Fig. 7.7(a)). However, at t = −0.5 µs, we begin to see the

build-up of |2〉 atoms at the roadblock, first near the condensate edges. Later, we see

a high density of |2〉 along the entire roadblock. We also see small wings of |2〉 atoms

in the very near small density region near the condensate edge, where the probe leaks

around the roadblock (as in Fig. 7.6(b) at t = 4.5 µs). The finite resolution of the

optical imaging makes quantitative estimates of the densities difficult, but estimates

based on the known cross section at this detuning indicate densities of |2〉 atoms on

the order of the original condensate density. We also see a haze of |2〉 atoms in the

z > 0 region. These are atoms which have undergone absorption and spontaneous

emission into |2〉 or another state near resonant with the imaging beam (we image all

atoms, not just condensed atoms).

Because this creates defects below our optical resolution we are actually unable

to detect the defect in the |1〉 condensate immediately after its creation. Fig. 7.8

shows images of the |1〉 atoms. Immediately after the pulse compression (labelled

0 ms), there is still no visible defect in the density of |1〉 atoms, however, it clearly

must be there since we measured a large density of |2〉 atoms at this time. We then

infer that there is a subresolution defect at this point. By dropping the magnetic

trap and letting the |1〉 atoms expand, we begin to clearly see the defect. The defect

expands due both the dynamics of the defect itself (as discussed in Chapter 8) and the


Figure 7.7: Figure taken from [4]. Experimental implementation of the lightroadblock. (a) An absorption image of the original condensate of |1〉 atoms, consist-ing of N = 1.5× 106 atoms. The imaging was done with a 10 µs laser pulse -30 MHzdetuned from the |1〉 → |3P1/2, F = 2,mF = −1〉 transition and propagating along y.(b) Absorption images of |2〉 atoms at the time indicated. Cuts of the optical densityalong x = 0 are shown on the right hand side. We used Ωc0(zin) = (2π)14.6 MHz,Ωp0 = (2π)2.4 MHz and τ0 = 1.3 µs. Images were obtained with a beam -13 MHzdetuned from the |2〉 → |3P3/2, F = 3,mF = −2〉 transition. The stopped light tech-nique was used to freeze the atomic superpositions at the times indicated, at whichpoint we began the imaging.


Figure 7.8: Experimental observation of a narrow defect. Part of figure takenfrom [4]. Experimental absorption images on |1〉 immediately after the pulse is ab-sorbed in the light roadblock (0 ms) and after 1 ms of free expansion (dropping themagnetic trap). Images were taken with a beam -30 MHz detuned from the transitiondescribed in Fig. 7.7(a).

free expansion of the entire density profile from dropping the trap. The figure shows

an image after 1 ms. We see from the line cut that it is nearly 100% but can not

quantitatively measure how deep because the finite imaging resolution still obscures

the detail at this scale to some extent.

The |1〉 atoms were always imaged by pumping for 10 µs into the F = 2 manifold,

and then imaged for 10 µs. These times were short enough that no significant change

in the atomic density occurred due to recoils from imaging photons.

7.3 More complicated spatial engineering

The light roadblock is in a sense the most of extreme example of spatial engineer-

ing of the coupling input in that the intensity is brought completely to zero. While

this maximizes the technique’s ability to compress probe pulses, it has the disadvan-

tage of destroying some of the coherence of the |2〉 atoms through absorption. This


decoherence can be reduced by turning the coupling intensity to a small but non-zero

value. The decoherence will become quite small if this intensity is large enough to

preserve EIT.

There are a wide variety of possible coupling intensity profiles one can imagine and

the most advantageous is dependent on the goal of a particular experiment. However,

all share the possibility of allowing us to bring the coupling intensity parameter Wc

close to the scale of H2 − µ regardless of the total optical density D0 of the cloud.

Furthermore, we discussed in Section 3.3.1 how the distortion associated with the

strong probe regime increases with D. We can use spatial engineering of Wc to

bring Ωp0 ∼ Ωc0 for only a small fraction of the optical density, thus preventing

this distortion from dominating. This regime is interesting from the standpoint of

inducing more transitions between the ground states and therefore coupling to larger

excitations in the condensate. In fact, an example of this was already shown with the

roadblock.

To illustrate the possibility of turning Wc down to a non-zero intensity, we plot in

Fig. 7.9 a case with a large inhomogeneous Zeeman broadening (α(Z) = −3) and we

turn the coupling beam intensity down to a factor 0.05 its original value for an interval

of 10 µm (dashed curve in Fig 7.9(a)). As a result the Ωp0 = (2π)2.5 MHz, τ0 = 2.0 µs

pulse, which would have a half width of about 15 µm at the cloud center if Ωc0

remained at (2π)8 MHz everywhere, is reduced to about a 3 µm length at the cloud

center, and still has a peak intensity of about half it’s input peak value. At the

center the probe is stronger than the coupling field and the densities |ψ1|2 and |ψ2|2,

plotted in Fig 7.9(b), are seen to be nearly equal in this region. At this moment,


Figure 7.9: Creating two component coherent mixtures with spatial engi-neering. (a) 1D simulation of a Ωp0 = (2π)2.5 MHz, τ0 = 2.0 µs probe pulse beinginput into a medium with the a coupling beam profile as shown with the dashed curveand peak value Ωc0(zin) = (2π)2.5 MHz. The solid curves show the probe intensityat t = 2 µs (thin curve) and t = 10 µs (thick curve). The dotted curve shows theoriginal condensate density in arbitrary units. In the simulation we use α(Z) = −3.(b) The resulting condensate densities |ψ1|2 (dashed curves) and |ψ2|2 (solid curves)at these times (t = 2 µs, thin curves and t = 10 µs, thick curves).

N1 = 0.964N , N2 = 0.024N and 0.012N of the atoms have been lost from the

condensates. Using a coupling intensity this low over the entire condensate would

lead to complete absorption of the input pulse and no coherent atoms in |2〉. We

note that combining this technique with stopped light technique could then freeze

the coherent superposition we create at any point in the propagation, creating novel

two component condensate structures with extremely small length scales.

Chapter 8

Using slow light to probe

superfluid dynamics

In Chapter 7 we looked in detail at how spatial engineering of the coupling beam

could be used to create very narrow, large-amplitude defects in our condensates.

These excitations are of particular interest because they have length scales on the

order of the healing length, a length scale at which excitations in a BEC take on a

very different character. We will see how this will allow us to study issues associated

with superfluidity in BECs.

Superfluidity in Bose condensed systems represents conditions where frictionless

flow occurs because it is energetically impossible to create excitations [20]. When

conditions change so this is no longer the case, the superfluidity breaks down. For

example, when an object moves through a superfluid faster than a critical speed

(which is proportional to the sound speed) vortex rings can nucleate. This has been

seen, for example, in superfluid liquid helium by shooting ions through the fluid

254

Chapter 8: Using slow light to probe superfluid dynamics 255

[16]. Under similar conditions, a shock wave discontinuity would occur in a normal

fluid [89]. Because of the macroscopic wave function associated with a condensate,

discontinuities are not allowed and instead the topological defects, such as solitons

and vortices are nucleated. The reason for this nucleation is that when an object

(or more generally any repulsive potential) moves through a superfluid faster than

the critical velocity, the macroscopic condensate wave function is unable to respond

to density perturbations introduced by the moving potential. Eventually the lack of

smooth response to the perturbation leads to the formation of density defects with a

very small spatial scale. When these density defects reach a spatial scale called the

healing length, the topological defects are nucleated. The healing length is defined to

be the length scale at which the kinetic energy, associated with spatial derivatives of

the macroscopic condensate wave function ψ1, becomes on the order of the atom-atom

interaction energy [5, 27]. Setting these two energy scales equal to each in the single

component GP equation (Eq. (4.21) with ψ2 = 0,Ωp = Ωc = 0) yields the healing

length

ξ =1√

8πN |ψ1|2 a11. (8.1)

This is the minimum length over which the density of a condensate can adjust. In

terms of the condensate response to perturbations, ξ−1 is the cross over between

wavevectors q which induce single particle excitations and many particle collective

excitations.

Atomic Bose-Einstein condensates have recently provided a superfluid system

which is theoretically more tractable than liquid helium, due to the weaker atom-


atom interactions. Early experiments on BECs emphasized the study of macroscopic,

long wavelength, collective excitations [40]. In the past two years, several experiments

have succeeded in creating and observing both solitons and vortices. The techniques

manipulated the phase of the BEC [41, 45, 43], or provided the system with a high

angular momentum which makes vortex formation energetically favorable [42]. How-

ever, a direct observation of the formation of vortices as a result of small wavelength

excitations breaking down the superfluidity had been lacking. Rather, rapid heating

from “stirring” with a focused laser beam above a critical velocity was observed as

indirect evidence of this process [46].

Using our light roadblock, we can simply introduce density defects in condensates

at or near the length scale ξ [4] and observe directly breakdown of superfluidity

which is expected to occur at this scale. In this chapter, we present experiments and

numerical investigations we carried out earlier this year [4] whereby narrow defects

(∼ 2 µm half-width) were observed to shed off a series of solitons. The mechanism for

the soliton formation was a further narrowing of the spatial scale of the defect due to

nonlinearities in the density wave propagation, an effect which occurs in normal fluids

as well. However, rather than continuing to narrow and forming a shock wave, we

showed how in a condensate, this narrowing continued only until it reached a scale

given by ξ, at which point a soliton was shed off. Depending on the total size of

the defect, this process could happen several times, shedding off an array of solitons.

We describe this process in detail theoretically in Section 8.1. The details of the

dynamics are seen to depend on an interplay between the nonlinearity of atom-atom

interactions, which tends to collapse and narrow the density waves, and the effects


associated with the single particle regime of the Bogoliubov dispersion relation, which

tends to disperse narrow density waves.

In Section 8.2 we then discuss how solitons, which are stable in 1D, are unstable

in 2D and 3D. The initially straight wavefront of the solitons begin to curl and

the nonlinearity in the GP equation enhances the process further until vortices are

nucleated. This is known as the snake instability, a process which had been observed

previously in optical systems [54]. It was first observed indirectly in BECs at JILA

[43], and later the process was directly observed for the first time by us [4].

Once the vortices are formed, this is an extremely interesting system to study,

as the numerical results we present here will demonstrate. The roadblock, unlike

previous methods of vortex nucleation, leaves the system with several vortices of

both circulations as well as many sound waves. All these excitations are far out of

equilibrium so they exhibit a rich variety of dynamical effects. The vortices undergo

strong collisions with each other, which suddenly and quickly alter their drift veloc-

ity. Vortices of opposite circulation undergo annihilations. Furthermore, they interact

with the small sound waves in the system, demonstrating the Magnus force [50]. We

expect, based on our numerical results, that these should all be observable experi-

mentally by creating vortices with the light roadblock. The results presented here are

an initial step towards strengthening our understanding of the particle-like nature of

vortices. These types of interactions have not been observed in atomic BECs and are

a subject of current interest [50] as the these interactions are believed to the funda-

mental mechanism by which dissipation is introduced into superfluids. There are also

analogies for this mechanism in superconductors. Most of the analysis in this chapter


is done in 2D, however, we conclude the discussion by briefly considering additional

possibilities present in the true 3D geometry.

Experimental observation of the soliton formation via the quantum shock waves,

and direct imaging of the ensuing nucleation of vortices via the snake instability are

presented in Section 8.3. Using a slicing technique [4, 28] to image only a cross

section of the condensate density, details of the dynamics of the snake instability,

which are not visible with normal absorption imaging, are clearly observed. The

method can also be used to do ’tomography’ to learn the 3D structure of the vortices.

The experiment also observed some of the subsequent vortex dynamics.

This chapter concentrates on single component (|1〉) dynamics. We conclude the

chapter in Section 8.4, with a brief outline of possible extensions of our analysis to

two component condensates.

8.1 1D dynamics: Quantum shock waves and soli-

tons

We first consider the evolution of small defects in a 1D model. We begin by in-

troducing the hydrodynamic picture of condensate evolution [90] in Subsection 8.1.1.

Using it, we analyze how a condensate responds to density perturbations away from

its ground state, and show with a linear hydrodynamic theory that simple shape

preserving sound wave propagation can occur if two assumptions are satisfied: large

spatial scales (compared to ξ) and small amplitude (compared to the total density).

In Subsection 8.1.2, this is then used as a basis to clearly see how the nonlinear-


ity in the GP equation causes distortion and instability in large amplitude, small

wavelength sound waves. The instability leads to the formation of spatial features

near the healing length scale, at which point solitons are shed off because the macro-

scopic wave function cannot support features with sizes below this scale. To obtain

analytic criteria for this to occur, we first consider this in homogenous condensates.

In Subsection 8.1.3 we then turn our attention to the inhomogeneous Thomas-Fermi

condensates which occur in the experiments and see how the sound waves and solitons

reflect off the walls. In 1D, the solitons are seen to be extremely stable and can pass

through each other unaffected. We then show in Subsection 8.1.4 how the defects

created with the light roadblock are well suited to observe the phenomena studied

here.

8.1.1 Linearized hydrodynamics and sound waves

A common and useful method of analyzing excitations in a condensate is the

hydrodynamic description [90], which makes analogies with a normal fluid quite clear.

The single component time-dependent GP equation (Eq. (4.21) with ψ2 = 0,Ωp =

Ωc = 0):

i∂

∂tψ1 =

(−

2∇2

2m+ V1(r) + U11|ψ1|2

)ψ1, (8.2)

which is a complex valued equation, can be rewritten by defining ψ1 in terms of two

real quantities ψ1 =√nce

iθc :


∂

∂tnc = −

m∇ · (nc∇θc),

∂

∂tθc = −µc −

2

2m(∇θc)2;

where µc ≡ V1 + U11nc +

2∇2√nc2m

√nc. (8.3)

In this description, nc represents the condensate density (divided by N due to our

normalization convention of ψi in (4.15)) and the velocity field of the condensate is

given by the quantity vc = (/m)(∇θc). The spatially dependent quantity µc is a

local potential felt by the atoms, and the last term in the definition of µc is called the

quantum pressure. The first equation of (8.3) represents conservation of mass, and

the second equation is the Euler equation, which gives the change in velocity field

due to forces from interactions and the trapping potential. Mathematically (8.3) is

equivalent to the GP formalism and has invoked no additional approximation. In the

ground state, vc = 0, µc is everywhere the chemical potential µ, and the quantum

pressure is the kinetic energy associated with the spatial derivatives of the ground

state profile.

If we define the ground state density as n(0)c (and set θ

(0)c = 0 which can be done

for a stationary condensate without loss of generality) we rewrite the variables as

nc = n(0)c +n

(1)c , θc = θ

(1)c and linearize in the deviations from the ground state to get:

∂

∂tn(1)c = −

m∇ · (n(0)

c ∇θ(1)c ),

∂

∂tθ(1)c = −(U11n

(1)c + V1) (8.4)

where we have ignored the quantum pressure term. This is only valid when the spatial


scale δ of variations in the hydrodynamic quantities satisfy δ ξ (which implies the

kinetic energy is dominated by the interaction energy E(T )1 E

(int)11 ). Thus we have

introduced both linearity and long-wavelength approximations in (8.4).

Consider a condensate in a homogeneous potential V1 = 0 in 1D, with some length

Lz, so the ground state is a given by a homogenous density n(0)c = 1/(LzA), where

A is the cross section area of system. The system then has a chemical potential

µ = U11n(0)c and total energy E(0) = (1/2)Nµ (N is the number of atoms). We then

introduce a density defect of amplitude AD and width δD by removing N (def) atoms

in a region near z = 0 so:

nc = n(0)c + n(1)

c = n(0)c

[1 − AD exp

(− z

2

δ2D

)], (8.5)

so N (def) =√πADδD and we are left with N1 ≡ N − N (def) atoms. Just as in

Subsection 4.3.1, the removal of atoms adds free energy to the system. In the limit

that the defect is much smaller than the total system size (δD Lz), we find:

G(def) = E − Eg = NµδDA

2D

Lz

√π

2

(1 + C(AD)

ξ2

δ2D

), (8.6)

where Eg = E(0)(N1/N)2 is the new ground state energy after the removal of the defect

and C(A) =∑∞

n=0(An/n3/2) is a constant of order unity and has the limits C(A) →

1 asA → 0 and C(A) → ξ(3/2) = 2.612 asA → 1. The first term in parentheses in

(8.6) is the contribution from E(int)11 and the second term is the contribution from the

kinetic energy E(T )1 , and is comparable to the first only for defects near the healing

length size δD ∼ ξ.

We first consider a regime where the linearized hydrodynamic equations (8.4) are


valid. Linearity requires AD 1 and ignoring the quantum pressure is valid if and

only if δD ξ. Under these assumptions we can solve (8.4) by Fourier transforming

in time and space to the variables ˜nc(ω, q),˜θc(ω, q) and find the dispersion relation:

ω = c(0)s q;

c(0)s ≡√U11n

(0)c

m. (8.7)

The speed c(0)s is the speed of sound in a condensate, which is 4.6 mm/s at the center

of condensates we have considered in this thesis. Constructing a solution using (8.7)

and the initial conditions (8.5) we find:

nc(z, t) =1

2n(0)c

[exp

(−(z − c(0)s t)2

δ2D

)+ exp

(−(z + c

(0)s t)2

δ2D

)](8.8)

The original density depression thus splits into positive and negative q components,

and we get a sound wave of amplitude AD/2 propagating in each direction at a speed

c(0)s . Note that the linear dispersion relation (8.7) causes the sound waves to maintain

their shape and width. Numerical 1D simulations corroborating this behavior of nc

are shown in Fig. 8.1(a).

Using the solution (8.8) and plugging back into (8.4) we can solve for the phase

θc to linear order and get:

θ(1)c (z, t) =

√πADδDmcs

2

[1

2Erf

(z + c

(0)s t

δD

)− 1

2Erf

(z − c(0)s t

δD

)]. (8.9)


Figure 8.1: Sound wave formation from a density defect. We plot the resultsof numerical simulations of the condensate response to an imposed defect with AD =0.1, δD = 9 µm in a condensate with a homogenous density of N |ψ1|2 = 4.5×1013cm−3

over a 323 µm length (achieved by using a trap with large flat bottom). (a) Thecondensate density profile at the times indicated. The plots are offset from the scaleon the left hand side of the plot by −3,−2,−1, and 0 × 1013cm−3, respectively, forincreasing times. The positive and negative q components split the original defect intotwo sound waves travelling in opposite directions. They travel at ±c(0)s = ±3.6 mm/s,

each with an amplitude AD/2 (with AD as defined in (8.5)). The positions ±c(0)s t areindicated with vertical dotted lines. (b) The phase, originally homogenous, is plottedat the same times (each plot with an arbitrary offset for graphical clarity). The results

agree with (8.9). (c) The kinetic energy per particle E(T )1 /N1. During the separation

of the left and right moving sound waves, half of the free energy in E(int)11 is converted

into kinetic energy.

There are equal and opposite phase jumps across each sound wave, with a magnitude

given by the front factor. In Fig. 8.1(b), we plot numerical results for θc, which

corroborate (8.9).

In the process of splitting into two sound waves, the defect converts some of its

free energy into kinetic energy. The kinetic energy is plotted in Fig. 8.1(c). In the

limit δD ξ it can be shown that approximately half of the free energy G(def), which

is originally almost all due to the interaction term (see Eq. (8.6)), is converted into

kinetic energy. Mathematically, the new kinetic energy results from the development

of phase gradients. Physically, it is the result of the initially stationary atoms acceler-

ating from hydrodynamic forces associated with the original defect’s density gradient.


The transfer of energy between the interaction and kinetic contributions is a signature

of the condensate being able to respond collectively, via atom-atom interactions, to

the perturbation.

8.1.2 Solitons via Quantum Shock waves

What happens when the assumptions which entered our calculation (linearity

AD 1, and long wavelength δD ξ) break down? We outline a physical argument

as follows. The sound speed c(0)s is proportional to

√n

(0)c . If we use the local sound

speed cs =

√n

(0)c + n

(1)c , we see the lowest density point, at the center of the sound

wave, is moving slower than the front and back edges by a factor√

1 − (AD/2) ≈

1−AD/4. The back edge will then catch up to the center at the time tsol defined by:

c(0)s tsol =2δDAD

(8.10)

(the notation tsol will become clear later). Note the left hand side of (8.10) is the

distance each sound wave travels in time tsol. A more formal analysis can be done by

using our linear solution in the last subsection to calculate the second order correc-

tions. An analytic solution can not be obtained, however, this procedure shows the

second order terms become important after time scale given by (8.10).

Fig. 8.2(a) shows the evolution of the density profile where this effect is visi-

ble. We introduce a AD = 0.35 defect and see (at 5.0 ms) that the trailing edge of

the sound waves becomes steeper while the leading edge becomes shallower. After

trailing edge becomes too steep, the sound wave “sheds” a structure off the back at

11.6 ms and continues to propagate. Examining the density (Fig. 8.2(a)) and phase


Figure 8.2: Sound waves shedding off solitons. The results of numerical simula-tions similar to Fig. 8.1, only now with AD = 0.35, δD = 5 µm. The conventions arethe same as in that figure. The sound waves are seen to shed off small structures. Thephase and density profiles of these structures are consistent with the soliton solutions(8.11).

(Fig. 8.2(b)) profile of this structure reveals that it is a gray soliton [56]. Gray solitons

in condensates are regions where the wave function takes on the form:

ψ1(z, t) = ψ(0)1 (z0, t)

(i√

1 − β2 + β tanh

(β√2 ξ

(z − z0))). (8.11)

Here ψ(0)1 is the slowly varying background wavefunction in the absence of the soliton.

This solution has a density kink of relative depth β2 which travels at a speed vsol =√1 − β2c

(0)s . The dimensionless parameter β (|β| < 1) characterizes the “grayness”

of the soliton. In the limit β → 0 the soliton solution reduces to a small amplitude

(AD 1) sound wave propagating at c(0)s . In the opposite limit β → 1 it becomes

a 100% (AD = 1) stationary structure with a width ∼ ξ, which is 0.56 µm in the

example in the Fig. 8.2. There we see the solitons are travelling at vsol, slower than

the sound speed c(0)s . Unlike sound waves, solitons perfectly maintain their shape as

they propagate, regardless of their amplitude. This is because the spatial length and

depth are such that contributions of the kinetic energy and interaction energy in the

GP equation (8.2) perfectly balance.


In a normal fluid, a similar steepening of nonlinear sound waves occurs [89], and

discontinuities in the pressure, or shock waves, can develop. This causes dissipation.

In contrast, the macroscopic wave of the condensate cannot dissipate energy or contain

discontinuities and solitons form instead. Because of the close analogy with the

normal fluid case, we call the process we see here a quantum shock wave. We note in

Fig. 8.2(c) that the soliton formation process involves no change in the kinetic energy.

Generally, when a sound wave propagates for a sufficiently long distance, the

nonlinearity will steepen the back edge until it reaches a length scale ξ/√AD/2 and

a soliton sheds off. We see numerically that the relative density depression of the

soliton matches the sound wave and therefore β2 = AD/2. The remaining part of the

sound wave continues to propagate at c(0)s with a reduced amplitude A

(1)D /2 which is

determined by keeping the total integrated absence of atoms in the sound wave plus

the soliton constant. Depending on the total energy in the original sound wave, this

process can repeat itself several times. In the example in Fig. 8.2(a) each of the two

original sound waves shed off two solitons before its small remaining energy disperses.

What determines the point at which this dispersion prevents the formation of

additional solitons? To answer this question, we must consider the corrections to

our long wavelength assumption. By neglecting the quantum pressure in (8.4), the

dispersion relation (8.7) does not account for effects which become important when

q ∼ (2π)/ξ. A proper treatment, with the method of Bogoliubov [17] (see article by

Fetter in [5] for a discussion in the context of current experiments), yields a dispersion

relation:


Figure 8.3: Dispersion of a short wavelength defect. A numerical simulationsimilar to the two previous figures, with AD = 0.35, δD = 0.7 µm, which is nearthe healing length ξ = 0.56 µm. The wave is seen to disperse into high frequencycomponents rather than form solitons, and these components travel faster than thesound speed.

ω =

√(c(0)s q)2

+

(q2

2m

)2

. (8.12)

This dispersion relation is valid for all q though it still linearizes with respect to the

amplitude of excitations above the ground state n(1)c /n

(0)c . It reduces to the linear

phonon dispersion (8.7) for q ξ−1 and becomes the quadratic dispersion relation

associated with free single particles for q ξ−1. In this limit, different q components

of the wave travel with different velocities and the wave disperses. An example is

shown in Fig. 8.3 where we see the δD = 0.7 µm (= 1.25 ξ) defect disperses into small

ripples before any solitons form. Fig. 8.3(a) shows these ripples actually travel faster

than the sound speed, as could be expected from (8.12). We also note in Fig. 8.3(c)

that, unlike cases where two clear sound waves formed, the kinetic energy does not

increase as the defect splits into two halves. This is because we are in a range of

q where the dispersion relation is like free single particles, rather than the collective

regime dominated by interactions, and thus energy is not exchanged between E(T )1 and

E(int)11 . This is an extreme example, where the majority of the defect energy disperses


before any soliton formation occurs. For longer wavelength excitations, such as in

Fig. 8.2, one or more solitons will form, and eventually the small remaining energy

disperses by this mechanism.

This dispersive effect spreads the sound waves and works against soliton forma-

tion. We include it phenomenologically in our model by adding a speed ∆cdisp =

(π2ξ2/4δ2D)c(0)s at which the dispersion counteracts the steepening of the back edge.

Combining our two considerations to this point, we can make an improved estimate

for the time of the formation of the first soliton:

c(0)s tsol =2δDAD

1

1 − π2ξ2

ADδ2D

. (8.13)

To check this formula, we performed a series of numerical simulations on BECs,

varying both AD and δD. We plot the position of the first pair of solitons formed

in Fig. 8.4, and see good agreement with the prediction ±c(0)s tsol (8.13). In the limit

δD ξ, (8.10) is valid and we get a linear dependence on δD. As δD → ξ the time

increases and eventually diverges when the denominator of (8.13) vanishes. At δD

lower than this divergence point expect the wave to completely disperse before any

solitons are formed. Numerically, no soliton formation was observed for points with

δD below the data points plotted in the figure.

We can also consider the problem from an energetic point of view. The prediction

for the available free energy G(def) (8.6) is valid in the nonlinear and short -wavelength

regimes. A soliton with an amplitude β has a free energy [57]:

Gsol =4

3NξU11|ψ(0)

1 |4β3A. (8.14)


Figure 8.4: Competing effects of nonlinearity and dispersion in soliton for-mation. The position of the visible separation of the first soliton from the originalsound wave versus δD for several different amplitudes: AD = 0.2 (filled circles),AD = 0.25 (open circles), AD = 0.35 (filled squares), AD = 0.5 (open squares). The

solid curves show the expectation c(0)s tsol (8.13). Simulations with δD lower than those

plotted showed no soliton formation.

From these arguments we see that when δD ≤ (8/3√π)(ξ/A

1/2D ) we do have enough free

energy to create two solitons with an amplitude β2 = AD/2 (two solitons because they

always come in pairs due to the left-right symmetry). This inequality being satisfied

agrees, to within a factor of order unity, to the denominator of (8.13) diverging, where

dispersive effects dominate the nonlinear collapse.

We have concentrated so far on a case where AD is still somewhat smaller than

unity. Because our roadblock technique can create 100% (AD = 1) defects, we con-

sider this regime. An example with δD = 5 µm is shown in Fig. 8.5. Whenever

AD ∼ 1, our analytic estimates for the nonlinear effects break down and the forma-

tion of the solitons can occur before the separation into two sound waves. Importantly,

as seen in the figure, this regime can cause solitons with relative β2 > 1/2, larger than

the β2 of solitons formed once the two sound waves have separated. These large am-

plitude solitons propagate extremely slowly. While most of the free energy goes into


Figure 8.5: Solitons from a 100% defect. (a) An AD = 1, δD = 5 µm defect isseen to form several large amplitude solitons as well as some small ripples. (b) Thekinetic energy, with the same conventions as the previous figures.

the solitons, such a large defect also has significant components at q > ξ−1 and we

therefore observe some high frequency ripples as well.

In this section, we have seen that sound waves are unstable and will be distorted

due to both nonlinear effects and dispersion. Solitons play an interesting role as a

structure where these two effects balance and so they perfectly maintain their shape.

When density defects larger than ξ occur, solitons tend to form, where smaller defects

will tend to disperse into small ripples. Thus, most of the free energy introduced in

a condensate will eventually end up in the form of either solitons or ripples.

8.1.3 Sound and solitons in a Thomas-Fermi condensate

The picture we developed above is basically valid in a Thomas-Fermi condensate

since the length scale governing the solitons ξ ∼ 0.4 µm is much smaller than the

condensate size σz ∼ 30 µm. We can apply the analysis and results of the last

subsection by simply using the space dependent TF density profile for n(0)c and using


this to determine a local sound speed c(0)s . Doing this, however, we find several

important effects associated with the spatial variation of the density and the presence

of the condensate boundaries. Further, we consider here effects of the finite size of

the condensate which are present when δD is comparable to σz.

We first note that, unlike in an infinite homogenous medium, the wave will reach

the system edge after propagating√

2σz, which could be smaller or larger than the

time needed for soliton formation c(0)s tsol. When c

(0)s tsol σz the sound waves will

propagate to the edge without exhibiting the nonlinear effects. However, during

the propagation, the density depression of the sound wave remains constant, so the

relative amplitude of the sound wave grows as it propagates into the lower density

region near the boundary. Therefore, as the sound wave approaches the edge both the

nonlinearity leading to soliton formation and the healing length determining soliton

size increasingly favor soliton formation. Fig. 8.6(a) shows an example of an AD =

0.2, δD = 10 µm defect. We see that it splits into two sound waves, however, the

kink-like structures associated with a solitons form when the sound waves near the

condensate border (14.5 ms), and, because their amplitude β2 = 1, the magnitude of

phase jump across these kinks approaches π.

Once the sound wave reaches the border it sees the steep ‘wall’ effective potential

(discussed in Chapter 6), and so the sound wave reflects and propagates back into the

middle of the condensate. At 33.5 ms, we see the energy has returned to the original

defect location. Thus, we see that the reflections effectively make the condensate

an infinite medium from the sound wave’s point of view. The oscillatory behavior

is evident in Fig. 8.6(b), where we see energy is exchanged between the potential


Figure 8.6: Sound waves in a TF condensate. An AD = 0.2, δD = 10 µm defectin a condensate. Originally there is an N = 1.2 × 106 atom sodium condensate ina trap ωz = (2π)21 Hz and with a cross sectional transverse area A = π(9 µm)2

is used. The initial conditions are taken to be the numerically determined groundstate. The defect is then removed, at which point N1 = 0.945N atoms remain.(a) Solid curves show the density at the times indicated and dotted curves show the

phase (with an arbitrary offset). (b) The interaction energy E(int)11 /N1 (dashed curve),

potential energy E(V )1 /N1 (dotted curve), and kinetic energy E

(T )1 /N1 (solid curve) ,

all normalized to the TF chemical potential µTF .

and interaction energy as the sound wave moves from the center to the edge and

back again. Because the background density is smaller near the condensate edge,

c(0)s is shrinking and the sound wave decelerates. Accounting for the varying c

(0)s and

using the TF solution for the density, we calculate the period of oscillation to be

Tsound =√

2(2π)/ωz, which is√

2 higher than the trap period. It turns out this is

also the period of the collective breathing mode in a 1D condensate [57].

However, we also note that the density never exactly reproduces the original defect

shape (compare 0.0 ms and 33.5 ms). The lack of periodicity is a result of the

small nonlinearity, which accumulates during the continued oscillation back and forth.

Eventually, as the sound waves approach their second reflection, we see the formation

of two small soliton structures (41 ms).

Solitons exhibit a similar behavior to sound waves. Just as the relative amplitude


Figure 8.7: Solitons in a TF condensate. (a) An AD = 0.4, δD = 5 µm defect ina condensate with the same conventions as Fig. 8.6(a). This AD, δD gives rise to thesame total number of atoms removed as in that figure. (b) The solid curve shows theposition of the sound wave originally propagating in the z > 0 direction as a functionof time. Various symbols then show the positions of various solitons as they are shedoff the back. Some of the smaller amplitude soliton positions are omitted since theyare not visible when they overlap with a much larger soliton. (c) The contributionsto the energy with the same conventions as Fig. 8.6(c).

of the sound waves varies, the amplitude β2 of solitons varies. We plot an example

in Fig. 8.7(a). There we see two pairs of solitons are quickly formed, and a third

is formed when the sound wave nears the condensate edge. As the solitons move

towards the edge, β2 increases and their velocity vsol decreases. They stop when the

density defect is equal to the background density (β2 = 1) and then reverse directions.

The dynamics of solitons in traps has been analyzed in [57] and the same period of

oscillation Tsound was found. It is worth noting that the sound waves and solitons


of any amplitude all have the same period of oscillation so they will all reach their

point of 100% amplitude at approximately the same time Tsound/4, and will return to

the location of the original defect at the same time Tsound/2. When this occurs, they

cross each other. In the figure they cross each other between the 29.5 ms and 41.0 ms

frames and regain their identity once they propagate through each other. Solitons

are only weakly interacting with sound waves and with each other and therefore are

not effected by the collision.

Just as the accumulation of the nonlinearity caused soliton formation in Fig. 8.6 at

very late times, we observe a fourth pair of solitons in the 41 ms frame of Fig. 8.7(a).

So again, we observe that the nonlinearities prevent perfect periodicity. In Fig. 8.7(b)

we plot with a solid curve the center of the sound wave originally propagating in the

z > 0 direction for one complete period as a solid curve. With different symbols, we

then plot the positions of various solitons that are sheds off. The first soliton (filled

circles) has the largest amplitude and therefore oscillates the least distance. Each

successive solitons is smaller amplitude and more closely follows the propagation of

the sound wave itself. The energetic behavior (Fig. 8.7(c)) is seen to be very similar

to the case in Fig. 8.6.

Lastly we consider effects of the finite size of the TF condensate, which are evident

when δD becomes comparable to σz. When this happens, much of the free energy

of the defect couples into the breathing mode of the condensate, whereby the entire

density expands and shrinks. In Fig. 8.8(a), a δD = 4 µm defect is seen to break

into six solitons, with relatively little effect on the background density. In contrast,

in Fig. 8.8(b), a δD = 10 µm defect breaks into 16 solitons, and, the envelope of


Figure 8.8: Defects coupling to the breathing mode. (a) An AD = 1.0, δD =4 µm defect in a condensate with the same conventions as Fig. 8.6(a). The dottedcurve shows the original ground state density before the defect removal. (b) AnAD = 1.0, δD = 10 µm defect. In this case, the background density varies significantlycompared to its original profile. (c) The contributions to the energy with the sameconventions as Fig. 8.6(c). The thin curves show the 4 µm case from (a) and thickcurves show the 10 µm case from (b).

the background density is seen to be reduced from its original value. Comparison

of the envelope of the density profile between 0.0 and 14.5 ms indicate a collective

breathing motion. We see this energetically in Fig. 8.8(c), where we plot the energy

contributions in the two cases. In the δD = 10 µm there is a large and periodic energy

exchange between the potential and interaction contributions.

It is also interesting to note in the 6.0 ms frame of Fig. 8.8(b) that as we enter this

large defect regime, the behavior looks very much like the interference of two conden-

sates coming together [28]. In fact, the distinction between solitons and interference

fringes is not completely strict. Both phenomena are a result of the macroscopic con-

densate wave function everywhere having a well defined phase. In a noninteracting

or dilute condensate, such as the ones released from the trap in [28], or the low den-

sity samples |2〉 samples we studied in Subsection 6.2.4, the interference description

is more appropriate and the fringe spacing is determined by the relative momenta


Figure 8.9: Soliton formation from large defects. (a) The total number ofsolitons formed, counted at the time they have reached their turning points at ≈ 15 ms(see Fig. 8.8), as a function of δD in the cases AD = 1.0 (filled circles),AD = 0.75(open circles), and AD = 0.5 (squares). (b) The depth of the largest solitons formedin each simulation, relative to the density at the center of the condensate (z = 0). Thesymbol representation is as in (a). The depth was assessed by observing the positionof this soliton when its relative depth was β2 = 1 (at ≈ 15 ms) and calculating theTF density at that point.

of the oppositely travelling matter waves. However when the energy of atom-atom

interactions dominate the kinetic energy from the relative momenta, as is the case

here, the density and phase profiles of the fringes will follow soliton solutions (8.11).

As we increase the sizes of the defects δD we see the number of solitons which can

form eventually get saturated by the total condensate size. In Fig. 8.9(a) we plot the

number of solitons formed versus δD for several different values of AD = 1, 0.75, and

0.5. For small defects δD σz, the macroscopic features of the condensate do not play

a role and we get a linear relationship between the δD and the number of solitons Nsol.

However, as δD ∼ σz some of the energy that would go into forming additional solitons

goes into the breathing mode energy, and the relationship is no longer directly linear.

As might by expected, the number of solitons for lower amplitude defects saturates

earlier, since a higher proportion of the free energy is coupled into the breathing

mode.


We also investigated numerically the relative depth of the largest soliton formed.

In the small AD regime, we expect two sound waves separate and the first soliton

to therefore have β2 = AD/2. For larger AD, solitons form before the sound wave

separation occurs. In Fig. 8.9(b) we plot the depth of the deepest pair of solitons in

each case. Somewhat surprisingly, we observe that the nonlinear effects associated

with these relatively large AD caused solitons with β2 ≈ AD to form whenever δD

was larger than about 4 µm microns. We never saw a soliton with a depth β2 > AD

8.1.4 Solitons from defects created with the light roadblock

Accounting for all the considerations in both inhomogeneous and homogeneous

condensates, the light roadblock is quite well suited to soliton formation. Typical

defects created with the roadblock have AD = 1 and δD = 2 µm at the cloud center,

a regime which gives rise to multiple solitons in the course of very little propagation

distance. By varying the input energy of the probe Ω2p0τ0, we can alter the defect

parameters δD and AD as studied in Chapter 7, and thus vary the number solitons

formed, their relative depth, and the time of formation tsol. We can also vary the

amount of coupling to the breathing mode from almost none to a large amount.

In our theoretical study to this point, we have looked at the response of the con-

densate to density perturbations created by instantaneously removing a piece of the

wave function. We now show how the timescales associated with the roadblock cause

it to be accurately described as the instantaneous removal of a defect. Fig. 8.10 shows

a simulation of the evolution for a roadblock created defect with Ωp0 = (2π)2.5 MHz,

τ0 = 4 µs. We first note that the timescale for removing atoms with the light road-


Figure 8.10: Forming solitons with a light roadblock. Figure taken from [4]. Forthe same condensate parameters as previous figures, we input Ωp0 = (2π)2.5 MHz,τ0 = 4 µs probe pulse. The coupling beam has Ωc0 = (2π)8 MHz in the z < 0 regionand is turned off with a 10 µm scale (ζ = 3 µm) at z = 0. We then plot the densityN |ψ1|2 (solid curves), and N |ψ2|2 (dashed curves) at the times indicated (relative tothe time the probe is completely absorbed at t = 18 µs). The dotted curve shows thephase of ψ1.

block ∼ τ0 is much faster than the atomic dynamics timescale ∼ µ−1TF , so, as we saw in

Chapter 7, the atomic external dynamics play no role during this time. Furthermore,

as we see in the first three frames of the figure, the large momentum kick received by

the atoms remaining in the |2〉 cause the density depression in |1〉 to be completely

void within 0.5 ms. The atoms which have undergone absorption events (which we

do not account for in our formalism) have a similar momentum kick and leave the

defect region on the same time scale. Also, as we see in the first frame of the figure,

the creation of the defect during light field propagation introduces no phase gradient

in ψ1, and so the defect is purely a density perturbation.

The subsequent propagation of the defect is seen to be similar to the results

discussed up to this point. This particular simulation is a short wavelength (δD ∼

2 µm), 100% defect, and we see both the soliton formation as well as some high

frequency components, as in Fig. 8.5. We see a slight asymmetry between z > 0 and

z < 0 in this simulation due to the slightly asymmetric position and shape of created


defect (see Chapter 7 for details).

8.2 2D dynamics: Vortex nucleation and dynamics

We saw in the last section that the solitons, once formed, were remarkably stable

against decay (and could even pass through each other unaffected). In 2D and 3D, this

stability no longer occurs, and they are subject to the snake instability [52, 53, 55],

whereby small inhomogeneities along the transverse front of the solitons are enhanced

by the nonlinearity of the GP equation, and cause them to break up. When this

occurs, vortices form. We will see that the deep solitons which we form in our 3.8:1

aspect ratio traps are in a regime where this instability sets in several millisecond and

gives rise to several vortex pairs. These vortex pairs are then subject to a number

of interesting dynamical effects. While in some ways they play a role analogous to

solitons, in that they are stable structures due to a balance of interaction and kinetic

energy, their dynamics are quite different in that they strongly interact both with

each other and with sound waves. Thus, we will show our system offers a way of

creating and studying the dynamics of a system of stable, interacting, ‘particle-like’

excitations in the condensates.

8.2.1 Vortices in Bose-condensed systems

Vortices are interesting features which arise in fluids whereby the density is de-

pressed at a point and the fluid circles about this point. We mention here several

important results regarding quantized vortices in Bose-condensed samples, which are

discussed in detail in Fetter’s article in [5], to lay a foundation for our discussion of


the vortices formed in our system.

A vortex is characterized by the velocity field pattern near its core position Rcore.

As we move further towards Rcore, the velocity field increases and behaves as:

v(vor)c (R) =

κ

2π|R−Rcore| φ, (8.15)

where φ is a unit vector perpendicular to the line connecting R and Rcore, and κ

is as constant called the circulation. In a Bose-condensed sample, the velocity field

(8.15) gives rise to a rotation of the phase θc in the macroscopic wave function along a

path around the vortex (via the relationship vc = (/m)∇θc). Having θc everywhere

single valued, forces κ = (2π/m)nv where nv is an integer. Thus the circulation in

condensates, unlike classical fluids, is quantized. This flow pattern means a vortex

has a large angular momentum associated with it (the amount of angular momentum

depends on the condensate size and density).

The phase singularity at Rcore requires the condensate density nc to vanish at this

point. The density depression has a characteristic size equal to the healing length

ξ. It can be shown that ξ is the distance at which the magnitude of the velocity

field (8.15) becomes equal to the local sound speed c(0)s , and so the density depression

prevents the contribution to the kinetic energy E(T )1 due to the vortex flow from

ever dominating the ground state chemical potential µ. Similar to the solitons in 1D

discussed above, vortices are stable due to a balance of kinetic and interaction energy.

However, unlike solitons, where the parameter β could vary continuously, vortices

must take on quantized values of circulation and must have a vanishing density at

their core.


In 3D, instead of a vortex core point Rcore, the core occupies a curve in space

running perpendicular to v(vor)c . This curve, called the filament, can have an arbitrary

3D structure and the flow the pattern must change orientations accordingly as we

travel along the filament. We will assume for most of our discussion that the system

is homogenous along y and has some length Ly in this direction, and so the filament

is a straight line of length Ly at (xcore, zcore). We return to more complicated 3D

structures later.

Using the flow pattern (8.15) and ignoring the kinetic energy associated with the

density change (assuming that it is dominated by the phase gradient due to v(vor)c ),

yields an energy due to the vortex of:

Gv = Lπ2n

(0)c n2

v

mln

(Rcyl

ξ

), (8.16)

where Rcyl is some cutoff from the finite size of the system. Thus another important

difference between vortices and solitons is that vortices have a much larger long range

effect on the medium and in fact have a logarithmically diverging energy for infinite

samples. In an inhomogeneous sample, like our TF condensates, the condensate size

will play a role in the vortex energy and some analytic results exist [44] exist, though

numerical analysis must be used to calculate the energy of a vortex with an arbitrary

position and orientation.

Because of the scaling Ev ∝ n2v, it is more energetically favorable for the system to

form two singly quantized vortices rather than one doubly quantized vortex. When

more than one vortex is present, they interact at a distance via their velocity fields.

The nature of the interaction depends on the relative circulation directions. Fig. 8.11


Figure 8.11: Vortex interactions. (a) Two vortices at a distance d with the same

circulation have velocity fields v(vor−1)c (v

(vor−2)c ) (indicated with solid curves) at the

location of the other vortex R(2)(core)(R

(1)(core)) of magnitude v

(vor−1,2)c = /md. This

causes each to a drift velocity in a circular path of diameter d, indicated with thedashed curves. In the presence of dissipation, the radius of the circle increases, asindicated with the dotted curves, to lower the energy. (b) In an interaction betweenvortices with opposite circulation, the velocity fields cause the pair to move alongstraight parallel paths. Energy dissipation in this case causes the vortices to driftcloser together. (c) When the background fluid moves relative to the vortex core

drift velocity, one can add the contributions of the vortex velocity field v(vor)c and the

background fluid v(in)fluid and see a difference in the total velocity above and below the

core. This leads to a pressure imbalance which causes a force on the vortex core. Italso scatters excitations in the background fluid in a manner which conserves angularmomentum.

shows a schematic of possible vortex-vortex interactions. To obtain the effect of one

on the other, one merely needs to calculate the velocity field of one v(vor−1)c (v

(vor−2)c )

at the location of the other’s core R(2)core(R

(1)core). If they are single quantized (nv = 1)

and they are a distance d apart, this velocity has a magnitude /(md). Each core

will then drift with that velocity. If the two have the same circulation they will each

move in along a circular path with a diameter equal to their relative distance (see

Fig. 8.11(a)). This is a constant energy path. In the presence of energy dissipation via

interactions with sound waves or the normal (thermal) part of the fluid, the diameter

of the circle will grow to lower the energy of the configuration.

Because of the symmetry in our system, we will usually not introduce a large


angular moment and therefore we will primarily create vortices in pairs of opposite

circulation (so the total circulation of the pair is zero). When two interacting vortices

have opposite circulation, the velocity fields will cause the two to move along parallel

paths (see Fig. 8.11(b)). In this case the energy is lower when they are closer so

dissipation will cause them to drift towards each other. When the distance between

them reaches their core size, they are both destroyed, or annihilated, and give off

their energy in the form of sound waves. We also note that the velocity fields of the

oppositely circulating vortices cancel at infinite distance and therefore their energy

does not diverge.

In addition to the forces from other vortices, the vortices will also experience

smaller forces due to the background fluid. This is called the Magnus force [50] and

arises from the fact that if the background fluid flows past a vortex core, the vortex

flow pattern will cause a left-right asymmetry. Adding the velocity of the vortex

v(vor)c and the background fluid v

(in)fluid in Fig. 8.11(c), shows that the total velocity

will be different above and below the vortex. This leads to a pressure imbalance

which gives rise to a force on the core. Conservation of momentum requires back-

action on the fluid and so the vortex scatters incoming sound waves in a non-trivial

way, as discussed in detail in [49]. Similar arguments show that inhomogeneities in the

background fluid density contribute a Magnus force which cause the vortices to move

perpendicular to the density gradient. This is obviously an important consideration

in our inhomogeneous TF condensates.

We will now see how multiple vortices are formed from solitons in our system and

how the subsequent dynamics show observable features associated with all three of


the interactions diagrammed in Fig. 8.11.

8.2.2 The snake instability and vortex nucleation

To demonstrate how the solitons in our system decay into vortices via the snake

instability, we show the development of the defect from 2D simulations in Fig. 8.12.

Fig. 8.12(a) shows the density at various times. The dotted lines in the first several

frames of Fig. 8.12(a) indicate the cuts of density which are plotted in Fig. 8.12(c-

d). One sees that an array of solitons forms similar to the 1D case. Fig. 8.12(c)

looks very much like the 1D case (see Fig. 8.10) at these early times. However,

almost immediately, the solitons show inhomogeneities along the transverse front

(1.68 ms) due to inhomogeneities in the original condensate density and, to a lesser

extent, in the defect depth and position, as discussed in Subsection 7.2.2. The defect’s

inhomogeneity is clear in Fig. 8.12(d) at 0.08 ms. This inhomogeneity “seeds” the

snake instability. By looking at the progression in the cuts along x, we see the density

is slowly pushed to zero at certain points. The process looks somewhat analogous to

steepening of the edges of the defects in 1D, only now in the transverse direction. At

1.68 ms, we see the development of two nodes of vanishing density near the condensate

edge along x (Fig. 8.12(d)). By, 3.08 ms, we see another pair of nodes at a different

cut. The density nodes’ spatial scale is ∼ ξ. In Fig. 8.12(a) we see a severe curling

in the deepest soliton at 3.08 ms. Fig. 8.12(b) shows the phase at the same times.

The curling fronts of the solitons causes curling in the curves of constant phase. At

the points where the second pair of vortices form, we see the phase front is getting

“pinched” at 1.68 ms, and by 2.48 ms, this pinching has caused the development


of two phase singularities. These singularities can be recognized by points where

the entire gray scale of phases from −π to π appear to converge. One can trace a

small closed circle of constantly increasing (or decreasing) phase about each of these

singularities. From the phase pattern, we conclude the vortices are clearly singly

quantized and primarily form in pairs of opposite circulation (so their net angular

momentum is approximately zero).

Vortices can form from a soliton whenever the energy of a soliton is larger than the

energy of a vortex pair. In general, these two energies will not be exactly matched so

there will be some remaining energy in the soliton. The additional energy is carried

off by small phonons, between the two vortices, which propagate out of the region

at c(0)s . An example of this is seen in the small density wave between the vortices at

3.88 ms, which then exits the region, leaving the vortex pair behind. This process

means the medium will eventually consist of a several vortices and many small phonon

waves. At later times, we see the formation of additional pairs of vortices.

We note that because the energy associated with a vortex is proportional to the

background density n(0)c (8.16), the system tends to form more vortices near the

condensate edges than near the center. In this region, the healing length is much

larger and there is larger relative inhomogeneity in the background density, so the

circular vortex density profile is less well defined there, but there are clearly phase

singularities present in this region at 15.88 ms and 20.88 ms.

The vortices, unlike the solitons, are stable in a 2D geometry and drift about the

condensate for many milliseconds, interacting with small density waves and, some-

times, with each other, as we will discuss in Subsection 8.2.3. Here we briefly mention


Figure 8.12: Vortex nucleation from a roadblock created defect. (a) Thedensity evolution, at the times indicated, after creation of a defect with the lightroadblock. We used Ωp0 = (2π)2.5 MHz, τ0 = 4.5 µs, Ωc0 = (2π)8 MHz, anda ζ = 3 µm cut off at the roadblock. We originally have N = 1.2 × 106 atomscontained in a trap ωz = (2π)21 Hz, ωx = 3.8ωz, and a length in the y directionof Ly = 2 × 8.8 µm. The original density defect is seen to decay into solitons viaquantum shock waves and then into vortex pairs via the snake instability. The arrowin the last three frames traces the location of a single vortex over time. (b) The phaseat the corresponding times reveals phase singularities at the locations of the vortices.(c) Cuts along z at x = 0 (indicated in (a)) at the times indicated. (d) Cuts alongx at two values of z (also indicated in (a)).


some of the dynamics seen the last three frames in Fig. 8.12(a). The white arrow

tracks the progress of the same vortex core as it drifts towards the edge. It passes close

to another vortex of the same circulation (the one located near z = 0 µm, x = −5µm

at t = 7.88 ms), and is quickly swung around and shot to the other side of the con-

densate by 20.88 ms. Watching animations of the results with equally spaced frames

clearly shows the acceleration as it approaches the other vortex and the deceleration

as it moves away.

To study this snake instability process systematically, we imposed artificial defects

of various sizes and depths in 2D condensates. Fig. 8.13(a) shows an example for AD =

1, δD = 4.5 µm (across the entire transverse width). The behavior is qualitatively

similar to the roadblock case though there is now a perfect mirror symmetry in both

x and z. We see that the deepest soliton forms a pair in the high density central

region and also one pair very near the condensate edge. We also see multiple vortices

in smaller density regions from other lower amplitude solitons which propagate to

the edge. We performed a series of simulations, varying δD, and in each case located

the position (x(core), z(core)) where the vortex pair in the region of highest background

density was nucleated. The filled circles in Fig. 8.13(b) indicate the original TF

density at this point, normalized to the peak condensate density. We see that larger

defects form a pair of vortices in the high density region near the center. Larger

δD have more free energy, so their vortices tend to form in higher densities regions,

however the dependence on δD for δD ≥ 3 µm is rather mild. This is because,

regardless of how large δD is, the deepest soliton, which forms this vortex pair, is

more or less the same. Fig. 8.9(b) indeed demonstrates that there is essentially no


Figure 8.13: Positions of vortex nucleation of various defects. (a) We imposea defect with AD = 1.0 and δD = 4.5 µm across the entire transverse width of aground state condensate of 1.2 × 106 atoms in the same trap as Fig. 8.12. The plotsshow the density at the times indicated. The 5.0 ms frame shows the deepest solitonforms two vortex pairs, one near the center, and one near the edge. (b) For a seriesof different defect sizes δD and AD = 1, we locate the location (x(core), z(core)) ofthe largest density vortex pair at the time when they are first clearly visible, Wethen plot (filled circles) the TF density at this core position, normalized to the peakdensity. The open circles and squares show the corresponding results for simulationsAD = 0.9, 0.75, respectively.

difference in the amplitude of the deepest soliton for all δD > 4 µm. Increasing

δD further will simply increase the number of solitons formed (see Fig. 8.9(a)) and

therefore lead to more additional vortices and excitations in the lower density regions.

Once δD < 3 µm, there is dramatic decrease in the density depth of the vortices

formed. This is partially due to the drop-off in soliton amplitude seen in Fig. 8.9(b).

Vortices, since they must be 100% defects, must form nearer to the edge to lower

their density depression. The drop-off is also is probably partially due to the high

q Bogoliubov dispersion dispersing the wave before the nonlinearity can form the


vortices, just as we observed for solitons in 1D in Fig. 8.4.

We also did several simulations with lower AD, plotted in Fig. 8.13(b) as open

circles (AD = 0.9) and squares (AD = 0.75). Surprisingly, these relatively large

amplitude defects only formed vortices very near the edge. We expect some drop off

from the fact that the largest solitons are of slightly smaller amplitude (see the AD

points in Fig. 8.9(b)). However, the dependence on AD observed in Fig. 8.13(b) is

even stronger than one might expect from this argument and is an issue which we

are looking into further. We also note that simple estimates based on energies, such

as (8.16), and even more sophisticated ones which include energy due to interacting

pairs [91], are not good quantitative guides to this behavior. This is because usually

several the vortex pairs are created and due to their strong non-local effects on each

other, energy estimates must include the interactions of all the vortices with all the

others.

In addition, we introduced defects which did not cover the entire transverse width

but instead removed a circle in the center of the cloud. Solitons did not form as

clearly, and less vorticity was observed than for comparably sized defects across the

entire width with the same number of atoms removed.

We also carried out several simulations for traps with several different aspect

ratios, in each case keeping the total number original atoms N = 1.2 × 106 and the

number of atoms removed N (def) constant so N1 = 0.87N remained. The longitudinal

trapping frequency was ωz = (2π)21 Hz in each case while ωx was varied. Fig. 8.14(a)

shows a case with a round (ωx = ωz) trap. The lower density in this case leads to

a slower sound velocity and slower time scale for the snake instability. However, the


Figure 8.14: The snake instability in different shaped traps. (a) Evolutionof an AD = 1, δD = 2.34 µm defect in a trap with N = 1.2 × 106 atoms andωx = ωz = (2π)21 Hz. (b) An AD = 1, δD = 6.92 µm defect with the same numberof atoms and ωx = 15ωz.

large transverse width allows ∼ 10 vortices to develop from the deepest soliton on

each side and thus offers an interesting way to create a “gas” of interacting vortices.

Fig. 8.14(b) goes to the opposite limit (ωx = 15ωz). Here everything happens on

a much faster time scale and only one clear vortex pair emerges from each soliton.

However, more deep solitons are formed and so we have a series of vortex pairs along

the length of the condensate. These demonstrate the variety of situations that could

be created by changing the aspect ratio and also shows how changing the density can

be used to tune the time scale of the snake instability dynamics. A theoretical study


of the number of vortices formed from full β2 = 1 solitons via the snake instability as

a function of aspect ratio was performed in [55]. When the transverse confinement

is made so tight that the healing length is equal harmonic oscillator length in that

direction, the system becomes essentially one dimensional and solitons are expected

to become stable against the snake instability (see Burger, et al., [45]).

8.2.3 Vortex dynamics and vortex-vortex collisions

We have seen a way to create multiple vortices out of equilibrium and it is ex-

tremely interesting to then consider their subsequent dynamics. The vortices are

stable and take on a particle-like nature. They strongly interact with each other and

also with the phonon excitations in the condensate. The presence of vortices is one of

the mechanisms by which dissipation can occur in an otherwise frictionless superfluid

and so is a subject of interest [16].

To demonstrate the myriad of possible vortex-vortex interactions which can occur,

we show a series of snapshots of the density in Fig. 8.15(a). In the figure, at 5 ms, one

pair of vortices is formed on each side (z < 0, z > 0) near the condensate center and

two pairs are formed near the edge. The ones near the edge reflect off the boundaries

and drift towards the pair near the center. A line of three vortices are observed to

spin around in a dramatic way (from 11-15 ms). The circular motion is because the

dominant interaction is with nearby vortices of the same circulation direction, as in

Fig. 8.11(a). At 16.5 ms, all but one pair are thrown to the x = 0 line where they

collide with their oppositely circulating partner and annihilate. The resulting energy

is seen being carried off in form of phonons at 17.5 ms. For the remainder of the


Figure 8.15: Vortex dynamics and collisions. (a) The evolution of an AD =1, δD = 3 µm defect. A description of the dynamics is in the text. (b) The evolutionof the core positions of the two long lived vortex pairs as they undergo motion due tothe condensate inhomogeneity and collisions with each other. The dots indicate thetimes spaced by 4 ms (with the first point at 7 ms).


simulation, only one clearly visibly pair remains on each side of the condensate. This

behavior of one pair eventually solely occupying each side occurred for a wide variety

of defect parameters.

As the remaining snapshots in the figure show, this one pair exists and drifts

around for very long times. Fig. 8.15(b) tracks the behavior of these two pairs. At

27.5 ms they reverse directions and drift towards their partner. They then accelerate

toward the x = 0 line, very close to each other. This is a regime where the interaction

due to the oppositely circulating partner dominates, as in Fig. 8.11(b). However, once

they come upon the pair coming from the opposite z side, they suddenly experience a

strong force from the velocity field of the counter-propagating pair and make a sharp

90 degree turn at 57 ms. When this collision occurs, there is some energy emitted as

a circular ring sound wave (57.5 ms). Each vortex’s motion is then dominated by the

vortex on the opposite z side. Upon reaching the condensate border, they then circle

back away from z = 0. The motion during this time is dominated by the Magnus

force from inhomogeneity of the condensate rather than the interaction with other

vortices. They are seen to follow the same path a second time around. We see that

the vortices are moving more slowly at this time (as seen from the fact that the dots

in Fig. 8.11(b), representing equally spaced times, are closer together there). From

an energetic point of view, the vortex paths are approximately constant energy paths.

An energy analysis was used successfully in [91] to explain the behavior, in numerical

simulations, of single vortex ring in a TF condensate.

At 79 ms, the vortices are again accelerated into the middle and head in for

another collision. Note that the vortices are slightly closer to each other this time


(compare 53 ms and 116 ms). Between 120 ms and 125 ms it is actually hard to

see the phase singularity as the two vortices nearly merge into a single density wave.

Upon the second collision, an even larger sound wave emission occurs (122 ms) and

four outgoing density waves are visible at 123 ms in addition to the vortices. Unlike

the first collision, the vortices, undergo a sudden reversal of direction. They then

circle back about the same path in reverse, as plotted in Fig. 8.15(b).

This simulation hints at the large variety of possible interactions and dynamics

which could be observed. One of the most interesting issues that merits pursuit is

the reason for the completely different behavior after the first and second collisions.

There is dissipation from the emission of large sound waves during the two collisions

as well as small effects from interactions with sound waves in the medium as the

vortices drift between the collisions, and these seem to have a potentially large effect

on the behavior of these collisions.

The simulation just presented focused primarily on a condensate containing four

interacting vortices and few other excitations. Introducing larger defect sizes intro-

duces more sound waves as well as collective breathing motion in the condensate,

both of which can effect the vortex motion. Fig. 8.16(a) shows a simulation with

a relatively large defect, AD = 1, δD = 8 µm. In this regime, we create four vor-

tices near the condensate center and also see a rather significant breathing of the

condensate and a large number phonon-like excitations in the background fluid. The

breathing behavior is evident at 9.75 ms, where a large number of condensate atoms

have filled in the void in the location of the original density defect. There we see a

reduced density in the regions nearer the condensate edge in z and a large density


Figure 8.16: Interaction of vortices with collective breathing and soundwaves. (a) The evolution of a AD = 1, δD = 8 µm defect. The collective breathingbehavior is seen, as described in the text. (b) A zoom in on a sound wave beingscattered by a vortex pair. The area of detail is indicated by the dotted box in (a) (inthe 25 ms frame). The arrow tracks the behavior of one particular density wave as itmoves between the a pair of vortices and eventually scatters into two smaller ones.

in the central region. At 17.25 ms, this dense central region has begun to relax and

expand. The vortices, in a condensate devoid of other excitations would undergo drift

motion due to the inhomogeneity of the condensate as in the previous figure. Here

they are also influenced by the large flow associated with the collective breathing

of the condensate. The dependence of the vortex drift velocity on this breathing is

especially evident when one views animations of the density evolution.

There are many phonon excitations present in the cloud. They originally propa-

gate to the condensate boundary, then reflect and move back towards the vortices at

25 ms. In Fig. 8.16(b) we zoom in on these phonon-vortex interactions. The arrows


track the progress of one of the density waves. It is initially nearly straight in the

transverse direction, and is then bent as it moves between the vortices. Most of the

energy is then split into two smaller waves upon exiting the other side between them.

A series of such events occur in the simulation and can also effect the vortex motion,

in the way described in Fig. 8.11(c). This system could offer a way to test some of

the results originally derived in [49] on sound wave-vortex interactions, and discussed

more recently in [50]. We also observe in Fig. 8.16(a) that at 36 ms, another vortex

pair has drifted in from each side and begins to repel the original pair.

8.2.4 3D considerations

While our 2D numerical simulations capture many of the interesting features as-

sociated with this system, there are still further considerations when the true 3D

geometry is considered. Thus far our results have assumed the y direction is ho-

mogenous and the filament is a straight line along this direction. When one allows

the filaments to curve, there are qualitatively two very different structures which can

occur. In the first, a vortex line, the filament continues along, in either a straight

or somewhat curved path, until it eventually terminates its ends at the condensate

edge, as sketched in Fig. 8.17(a). The filament must always end perpendicular to

the surface. In the other, the vortex ring, the ends circle back and meet, forming a

closed ring which never has to touches the condensate surface. This is sketched in

Fig. 8.17(b). The motion of a vortex ring is qualitatively similar to a vortex pair,

with opposite sides of the ring effecting each other. This was analyzed in [91].

In our numerical simulations, we can consider 3D dynamics when cylindrical sym-


Figure 8.17: Planar cuts of vortex lines and rings in 3D. (a) The structure oftwo pairs of vortex lines in 3D. The dotted lines in the side view diagram indicate theplanar cut which is shown in the end view (and vice versa). Cuts in the y = 0 planelook like the pairs present in the 2D simulations, however, as indicated in the endview, the lines curve so they intersect the condensate surface perpendicularly. Thisleads to a further spacing between the vortices in cuts away from y = 0 and a slightlyelliptical shape. (b) For a ring, the vortices look closer together away from the y = 0plane.

metry is present. The case where the removed defect has very little asymmetry in x in

2D, corresponds to a case of approximate cylindrical symmetry in 3D. We performed

numerical simulations in 3D with cylindrically symmetry disks of atoms removed and

found the vortices formed in qualitatively similar way to the 2D case. A vortex ring

is the only structure consistent with the cylindrical symmetry, so the simulations pre-

dict the formation vortex rings. The behavior of very small vortex rings is a subject

of current interest as the analytic results presented in [91] break down once the ring

diameters becomes comparable to the core size.

However, it is unclear to what the degree a small amount of symmetry breaking

would effect the vortex filament structure. This issue is interesting but will require a

computationally expensive 3D numerical code to address. It is possible, particularly

in traps with high aspect ratio, where a vortex ring structure would require a large


curvature, that a two pairs of vortex lines would form instead.

In addition, the vortex filament itself can have vibrations and support propagation

of waves along its length. It is likely that excitation energy in our system will also

couple to these excitations, another interesting issue that could be addressed with a

full 3D code. Furthermore there are more possible orientations of collisions in 3D due

to the possible filament orientations.

8.3 Experimental observations

Using defects created with the light roadblock, we have directly observed both the

formation of arrays of via quantum shock waves and the subsequent decay of solitons

into vortices via the snake instability.

8.3.1 Observation of soliton arrays

Fig. 8.18 shows a continuation of the series plotted in Fig. 7.8 where we saw

the appearance of a narrow defect in the |1〉 condensate. In that experimental data

Ωp0 = (2π)2.4 MHz and τ0 = 1.3 µs. Taking images at later times showed the

appearance of arrays of solitons. It should be noted that even when the defect can be

seen, as in the 1 ms images, the soliton structures, which have widths ∼ ξ = 0.4 µm in

the trap, are not optically resolvable. It has been shown [92] that when a condensate

containing a vortex is released from the trap, that the vortex core size expands such

that its core size is consistent with the healing length associated with the lower density

during the expansion. A corresponding result holds for solitons, allowing us to use

release of the condensate to expand the soliton structures to the ∼ 5 µm scale, at


Figure 8.18: Observation of soliton formation via quantum shock waves. Wecreate a defect as described in Chapter 7. The parameters are identical to that inFig. 7.8. (a) After a 1 ms expansion only the defect is resolvable. (b) After a 10 msan array of solitons is observable.

which point they become optically resolvable. In Fig. 8.18 we see the appearance of

several solitons over the course of a 10 ms expansion.

We performed the experiment for several different probe input intensities, keeping

τ0 = 1.3 µs and varying (2π)1.0 MHz < Ωp0 < (2π)2.5 MHz found that the number

of solitons formed scaled roughly linearly with probe energy. This is consistent with

our prediction that for δD σz the number of solitons formed scales with the total

defect size (Fig. 8.9) and our findings in Chapter 7 that the defect size scaled directly

with the probe energy.

8.3.2 Observation of the snake instability, vortex nucleation,

and vortex dynamics

To study the dynamics of the solitons and the snake instability we held the con-

densates in the trap for variable amounts of time and then released them from the

trap, for 15 ms, before imaging. This time had to be chosen to be long enough for


Figure 8.19: Imaging a condensate slice. Figure taken from [4]. To take imagesof planes, we first allow free expansion for 15 ms. We then use two razor blades toblock the pump beam selectively, allowing only a 30 µm slice to be pumped. Thepump beam takes atoms from the |1〉 state to the F = 2 manifold, where the are thenimaged with a beam resonant with the F = 2 to F = 3 manifold on the D2 line.

the healing length (and therefore the core size of any vortices formed) to expand [92]

to the size of our imaging resolution, 5 µm, and also short enough that the expanded

cloud was still sufficiently optically dense that the absorption images had enough

contrast. The images in Fig. 8.18 were taken with regular absorption imaging along

y, and the solitons were visible as their density depression initially extends across

the entire width of the condensate. However the snake instability and the resulting

vortices have three dimensional structure, and, since the healing length ξ is much

smaller than the cloud size σx,y,z, a typical column density along y would not show

any significant depression due to these 3D structures. When we attempted absorp-

tion imaging after a several millisecond hold time in the trap and then expansion, we

indeed saw the soliton structures disappear from view.

This motivated us to use a slicing technique, as diagrammed in Fig. 8.19. In


the slicing technique, the pump beam (which pumps the |1〉 atoms into F = 2 for

imaging) was selectively blocked with two additional razor blades, so only a disk of

the condensate in a small range of y, typically of 30 µm height, was imaged. We

first allowed free expansion for 15 ms, during which time the condensate dropped

due to gravity and expanded to ≈ 200 µm along y. At this point the 30 µm slice

corresponded to a relatively small fraction of the total length so we could essentially

image a cut of the density profile in a plane of constant y.

When this was done, the snake instability and the resulting vortices became clearly

visible. Fig. 8.20(a) shows such a series where the hold time in the condensate was

varied according to the times indicated. We see the initially straight density de-

fects of the solitons begin to curl in the first millisecond. Then after t = 1.2 ms,

the deepest soliton is seen to form a vortex pair. The less deep and faster solitons

propagate further towards the edge without forming vortices. We also note there

are some vortex-like structures in the low density regions near the condensate edge,

as in our numerical simulations (see Fig. 8.12). Some preliminary results were also

obtained for the 3D structure of this vortex pair by varying the slice position. Those

results indicated that there was a nontrivial curved 3D structure, but perhaps more

complicated than a simple circular ring.

Fig. 8.20(b) shows another series with slightly different experimental parameters,

and images taken for much larger hold times. We see a clear snake-like curling at

0.5 ms and the formation of two vortices at 2.5 ms. Some features of the dynamics

of the vortices are visible at later times. At 5 ms, there is the development of a high

density near z = 0. We expect that this is related to the atoms on the condensate


Figure 8.20: Observation of the snake instability and vortex dynamics. Figuretaken from [4]. (a) Absorption images of the snake instability, obtained by slicingthe expanded condensates in the y = 0 plane. The times indicate the hold time inthe trap after the probe pulse, before a 15 ms free expansion. The BECs originallycontained N = 1.9 × 106 atoms, and we used Ωp0 = (2π)2.4 MHz, τ0 = 1.3 µs, andΩc0 = (2π)14.6 MHz. The imaging beam was -5 MHz detuned from the F = 2 →F = 3 transition on the D2 line. (b) A similar series with N = 1.4 × 106 atoms,Ωp0 = (2π)2.0 MHz, and the imaging beam directly on resonance.


edge filling in the original void of the defect, as we saw in the numerical simulations

in Fig. 8.16(a). Thus we seem to be in a regime where some of the defect energy

couples into the collective behavior of the condensate. Just as in Fig. 8.16(a), the

vortices move towards the edge partially in response to the relaxation of the high

density central region. At 18 ms they are seen to have come near the edge of the

condensate, at which point we expect them to reflect off the condensate boundary

and begin to drift back towards the center. We are currently investigating this in

more detail.

We also note that in the 8 ms - 11 ms frames, we see, near the condensate edge,

some of the detailed structure associated with small vortices and sound waves re-

flecting off the boundaries as in Fig. 8.16(a). With a more detailed investigation, we

expect to able to observe sound wave-vortex interactions in this way.

We varied the probe energy by varying Ωp0 over a range of values ((2π)1.0 MHz <

Ωp0 < (2π)2.5 MHz). As mentioned above, soliton formation was observed over this

range and the deepest ones always decayed. Clear pairs of vortices, away from the

condensate boundary, were only visible when Ωp0 > (2π)1.4 MHz. Comparison with

the results of Chapter 7 shows these probe parameters should form only a δD =

0.26 µm wide defect for AD = 1. We would expect this to be too small to cause

vortex formation based on our 2D simulations in this chapter. The reason for this

merit further investigation and could be related to differences in the criteria for vortex

formation in 2D and 3D geometries.


8.4 Outlook: Possibilities for two component stud-

ies

We have exclusively discussed issues in one component condensates in this chapter,

using the fact that the |2〉 atoms are ejected on time scales fast compared to the atomic

dynamics. However, it should be noted that other fundamental issues associated with

superfluidity could be addressed by considering varying the experimental parameters

so the two component dynamics play an important role.

8.4.1 One condensate moving through another

As we discussed in Section 7.3, more complicated spatial coupling beam engineer-

ing, combined with the stopped light technique, can create two component conden-

sates with a very small spatial scale in both samples. Thus, in addition to studying

the behavior of small density defects in |1〉, this system could be used to create

condensates in |2〉 with a wave function ψ2 with a total size only a few times the

healing length. Again, one would expect a change in behavior when the sizes become

comparable to the healing length scale.

When the majority of the |2〉 atoms are still coherent, the situation would corre-

spond to one small condensate in |2〉 moving through another larger condensate in

|1〉. Recently it was seen numerically [93] that this could nucleate of vortex rings in

the |1〉 condensate when the |2〉 atoms moved faster than a critical speed proportional

to the sound speed c(0)s . The |2〉 atom acts like a repulsive potential moving through

the medium, leading to the breakdown of the superfluidity. In the simulations pre-


sented in this chapter, the time of interaction between the two condensate was too

short to for vortex or soliton nucleation to occur as a result of |1〉 ↔ |2〉 interactions.

However, increasing the density in the original condensate in |1〉, would speed up the

time scale of the atomic dynamics.

We have performed a preliminary investigation in 1D, where we artificially varied

the velocity Vrecoil with which the |2〉 atoms recoiled. When c(0)s < Vrecoil < 3c

(0)s ,

solitons were formed in the |1〉 atoms. Interestingly, |2〉 atoms filled in the density

defect of the soliton. Thus |2〉 atoms were trapped by an excitation in |1〉, even when

the potential V2 was repulsive.

The lower bound on the velocity for soliton formation is related to the critical

speed (which is proportional to the sound speed) for a potential moving in superfluids.

The soliton formation is related to the |2〉 atoms acting as a moving potential, which

creates steep spatial gradients in ψ1. It would be interesting to study this further and

make more explicit the connection between objects moving faster than the critical

speed and formation of steep spatial gradients in the condensate wave functions. As

for the upper bound for soliton formation (3c(0)s ), it is unclear at present whether

it was primarily determined by the sizes of the condensate wavefunctions, or their

relative velocity Vrecoil/c(0)s . In the experiments and most of the simulations studied

in this chapter Vrecoil/c(0)s ≈ 20, and the effect discussed here would not be expected

to occur. However this ratio could be varied by changing the trap frequencies ωx,y,z

and the number of atoms.


8.4.2 Light propagation at the sound speed

Another interesting extension of the methods and analysis in this chapter and

in Chapter 4 would be to study light propagation near the sound speed. The lower

limits on Vg derived in Chapter 4 can be minimized for a co-propagating geometry

with V1 = V2. Furthermore, Section 5.3 outlined a way of using dynamical control of

the coupling field to further relax these limitations, so Vg ∼ c(0)S is an experimental

possibility. At some point, the light field propagation will be affected by the differences

in H2ψ2 and H1ψ1 even if the only difference is due to the difference in the spatial

gradient terms ∇2. Whether propagation in the condensate will show some signature

related due to increased light pulse-sound wave coupling when Vg ∼ c(0)S would be an

interesting issue to pursue.

Appendix A

The Optical Bloch Equations

versus amplitude equations.

In Chapter 2 we introduced a formalism, used throughout the thesis, which de-

scribed the internal states of the atom with the amplitudes ci. In that formalism we

treat the light fields as classical fields. An alternative approach is to treat radiation

field (both the lasing modes and vacuum modes) quantum mechanically and then

trace over the degrees of freedom associated with it. To do this, one must describe

the atomic state as a density matrix. The result is a description of the evolution of

this atomic density matrix called the Optical Bloch Equations (OBEs).

Here we briefly outline a derivation of the OBEs in three-level atoms. In the

derivation, the non-Hermitian excited state decay term Γ, which had to be introduced

phenomenologically in (2.4), appears naturally. Thus, this puts our introduction of

this term on firmer ground.

We then compare the two descriptions and see that the 3×3 atomic density matrix

307

Appendix A: The Optical Bloch Equations versus amplitude equations. 308

contains more information about the state of the atom than the three-component

amplitude description with the ci. Thus, there are situations where the density matrix

is needed for accurate calculations. However, we will argue that under conditions of

EIT, the differences between the two are negligible and therefore, our amplitude

approach provides an accurate description of the dynamics.

A.1 Outline of derivation

A discussion of the density matrix approach and a derivation of the OBEs in

two-level optical systems is supplied in Chapter 8 of [7]. A more general approach

which can be applied to multi-level systems is presented in [64]. We do not rederive

the OBEs here, but simply outline some of the steps in a way that will clarify the

comparison between it and the amplitude approach that we use.

The full many-body density matrix ρ0, representing a single atom plus all the

modes of the radiation field can be decomposed as

ρ0 = ρ(atom)0 ⊗ ρ(rad)0 + ρ

(c)0 ;

where ρ(atom)0 ≡ Tr(rad)ρ0,

ρ(rad)0 ≡ Tr(atom)ρ0, (A.1)

where Tr(A) indicates tracing over the degrees of freedom associated with subsystem

A. The first term ρ(atom)0 ⊗ ρ(rad)0 is the factorizable part of the density matrix. This

part will describe the true density matrix ρ0 if and only if the atom and the radiation

field are completely uncorrelated. The second term ρ(c) is the correlated part and


is the error introduced by neglecting these correlations. For example, if a lasing

photon is absorbed and spontaneously emitted into a vacuum mode, the atom and

the radiation field have interacted in way that correlates their new state.

In our case, we describe the internal degrees of freedom of a three-level atom ρ(atom)0

with a 3×3 matrix, with the diagonal terms (ρ(atom)0 )ii representing the population in

state |i〉, and (ρ(atom)0 )ij the coherence between the states |i〉 and |j〉. The radiation

density matrix ρ(rad)0 is n × n, with n → ∞ being the number of modes k we are

including. The diagonal elements of this matrix are the number of photons in each

mode. We assume that the two lasing modes (coupling and probe) are macroscopically

occupied and in coherent states, and all the other modes are in vacuum. One can

show the thermal occupation of them is negligible [7] so this is initially true. Treating

these modes as vacuum modes at later times amounts to ignoring multiple scattering

events.

To calculate the evolution of the density matrix, one uses

i∂ρ0∂t

= [H, ρ0], (A.2)

where H is in (2.2), only now E is the quantum mechanical electric field operator [7]:

E =∑k,εK

εk

√ωk2ε0V

(akeik·R + a†ke

−ik·R). (A.3)

Here εk are unit polarization vectors, V is the quantization volume, and ωk = ck.

The ak(a†k) are destruction (creation) operators for photons in the mode k. Taking

the quantum mechanical average of (A.3) yields the classical electric fields (2.1) from

the modes in the summation associated with the lasing modes, and zero from the


vacuum modes.

We once again go into the interaction picture and take ρ0 → ρ via (2.3). The

calculation outlined in [64, 7] then calculates the evolution (A.2) in a course grained

way that takes steps dτ such that Γ−1,Ω−1p,c dτ ω−1

13,23. The idea is that the

evolution of ρ on the scale of the latter frequencies washes out due to the fast rotating

phases at all the different frequencies ωk. Furthermore, it is shown that to second

order in dτ we can ignore ρ(c). This is an instance of a Markov approximation [64],

whereby coherence between modes in a large reservoir (in this case the non-lasing

modes) are assumed to decohere quickly compared to the dynamics of interest. We

carry out a second order calculation and take the trace over the radiation degrees of

freedom:

ρ(atom)(t+ dτ) = ρ(atom)(t) − i

∫ t+dτ

t

dt′ Tr(rad)[H(int)(t′), ρ(atom)(t) ⊗ ρ(rad)(t)]

− 1

2

∫ t′

t

dt′′ Tr(rad)[H(int)(t′), [H(int)(t′′), ρ(atom)(t′′) ⊗ ρ(rad)(t′′)]]. (A.4)

Doing this and again making a transformation accounting for the laser detunings

analogous to the ones before (2.4) eventually yields the OBE:


ρ11 = Γ31ρ33 − i2

Ω∗pρ31 +

i

2Ωpρ13,

ρ22 = Γ32ρ33 − i2

Ω∗cρ32 +

i

2Ωcρ23,

ρ33 = −Γρ33 +i

2Ωpρ31 − i

2Ω∗

pρ13 +i

2Ωcρ32 − i

2Ω∗

cρ23,

ρ12 = −i(∆p − ∆c)ρ12 − i2

Ω∗pρ32 +

i

2Ωcρ13,

ρ13 = (−i∆p − Γ

2)ρ13 +

i

2Ω∗

p(ρ11 − ρ33) +i

2Ω∗

cρ12,

ρ23 = (−i∆c − Γ

2)ρ23 +

i

2Ω∗

c(ρ22 − ρ33) +i

2Ω∗

pρ21,

ρ21 = ρ∗12, ρ31 = ρ∗13, ρ32 = ρ∗23, (A.5)

where Γ is defined in (2.5). Thus this method gives us a quantum mechanical deriva-

tion of this decay rate which we put in phenomenologically in Chapter 2. The non-

Hermiticity is a result of tracing over the quantum-mechanical degrees of freedom

associated with the radiation field. We have also seen more formally how the decay

of the population of |3〉 at a rate Γ will decay the coherences ρ13, ρ23 at half this rate.

The constants Γj3 = fj3Γ, where the oscillator strengths fj3 characterize the fraction

of atoms decaying from |3〉 which go into state |j〉. They are related to the dipole

matrix elements fj3 ∝ |rj3|2, which must be obtained for the particular atom’s energy

level structure with the Clebsch-Gordon coefficients [63].

Note that (A.5) contains population exchanges which increase the populations

ρ11, ρ22 without effecting the coherence ρ12. These are due to incoherent exchanges

of the population. The incoherence arises from the fact that the process occurs from

spontaneous emission into a random direction k and we average over all possible k.

An open system then has f13 = f23 = 0, and a completely closed system has


f13 + f23 = 1. In the experimental systems studied in this thesis, the systems were

partially open, with some fraction of the atoms going to other levels outside the

three-level system of interest (see Fig. 2.4).

In Section 2.6 we saw how to account for off-resonant transitions with an effective

dephasing rate γ and in Section 4.2 we also saw how collisional losses could be included

in this way. A similar method can be used in the OBE. If we assume that the |1〉 and

|2〉 have loss rates γ1, γ2, respectively, then the OBEs are modified as:

ρ11 = −γ1ρ11 + Γ31ρ33 − i2

Ω∗pρ31 +

i

2Ωpρ13,

ρ22 = −γ2ρ22 + Γ32ρ33 − i2

Ω∗cρ32 +

i

2Ωcρ23,

ρ33 = −Γρ33 +i

2Ω∗

pρ31 −i

2Ωpρ13 +

i

2Ω∗

cρ32 −i

2Ωcρ23,

ρ12 =

(−1

2(γ1 + γ2) − i(∆p − ∆c)

)ρ12 − i

2Ω∗

pρ32 +i

2Ωcρ13,

ρ13 = (−i∆p − Γ

2)ρ13 +

i

2Ω∗

p(ρ11 − ρ33) +i

2Ω∗

cρ12,

ρ23 = (−i∆c − Γ

2)ρ23 +

i

2Ω∗

c(ρ22 − ρ33) +i

2Ω∗

pρ21. (A.6)

In particular, note the decay of the coherence ρ12 is again half the rate of the popu-

lation losses of the states |1〉, |2〉 with which it is associated. Strictly speaking γ1, γ2

enter into ρ13, ρ23 but in the limit γ1, γ2 Γ this is a negligible contribution.

A.2 Comparison with amplitude equations

The atomic density matrix ρ corresponds to the amplitudes as :

ρij ↔ cic∗j . (A.7)


For an open system, this correspondence is an equality and the equations for the

density matrix (A.5) and amplitudes (2.4) are mathematically equivalent. For a

closed or partially closed system, it is impossible to reproduce (A.5) with amplitude

equations, since the density matrix ρ has six independent degrees of freedom and

the amplitude vector (c1, c2, c3) contains only three. In that case the correspondence

(A.7) is not a strict equality. If one simply uses (2.4) and calculates the corresponding

density matrix equations, one finds that we lack the terms Γ31,Γ32, which represent

the incoherent feeding of the decay from |3〉 into |2〉 and |1〉. The amplitude equations

have perfect coherence between the states implicitly built in and cannot describe this

incoherent exchange of populations. It can, however, describe loss out of the system,

because in that case the coherences and populations decay in a way that preserves

(A.7).

If we take the case of a closed two-level atom (i.e. disregard |2〉 and assume

Γ31 = Γ) and calculate what happens when we apply a c.w. laser field Ωp, the

amplitude approach will predict that the total population ρ11 + ρ33 = 1 decays at the

pumping rate R = (Γ/2)|Ωp|2/(Γ2+4∆2p). The cross-section seen by the laser will thus

exponentially decay with time as the atoms leave. However, the OBE approach will

accurately predict that the atoms which are excited in |3〉 return to |1〉 so ρ11+ρ33 = 1

at all times, and the cross-section will, after a short time, reach a steady state value.

In EIT systems, however, ignoring these feeding terms do not lead to these un-

physical results. This is because, in good EIT, the process of absorption into |3〉

and subsequent spontaneous emission always plays a secondary role to the coherent

exchanged between |1〉 and |2〉, as discussed in Section 2.4. This process perfectly


preserves the coherence of |1〉 and |2〉 and so ρ12 ≈ c1c∗2 holds throughout the process.

To confirm that the amplitude approach indeed accurately describes our three-

level system, we have performed numerical calculations with the code described

in Appendix D, which propagates the OBE equations (A.6). By varying between

Γ13 = Γ23 = 0 and Γ13 + Γ23 = Γ (in various combinations), we simulated from a

completely open to completely closed system. The results for different choices were

indistinguishable. We would expect a disagreement only in cases which involved a lot

of absorption and subsequent spontaneous emission. However, when this was true,

the probe pulse was completely attenuated from absorptions during the propagation

before significant deviations were seen.

Appendix B

Adiabatic elimination

We invoke the adiabatic elimination method [61] to simplify equations many times

throughout this thesis. In Chapter 2 it was used to eliminate c3 as a dynamical

variable, and later it is further used to eliminate the absorbing amplitude cA. In

Chapter 4 we use it to eliminate ψ3 for the purposes of our numerical code, and it

also used in analytic results in that chapter, including eliminating the absorbing wave

function ψA in the strong probe regime. Here we justify the approximation and derive

the limits of its validity by comparing it with the exact solution for a time-independent

two-level case. We then consider the issues in the more complicated systems in which

adiabatic elimination is used practice, including time-dependent evolution terms and

three-level systems.

B.1 The adiabatic elimination solution

Adiabatic elimination can be used in sets of equations of the following form:

315

Appendix B: Adiabatic elimination 316

x1

x2

=

g f

f −G

x1

x2

, (B.1)

where G > 0 is assumed to be real and G |g|, |f |. The sign of G is chosen so

it corresponds to a large damping rate (such as the G = Γ/2 damping of c3, or the

G = Ω2/2Γ damping of cA). Adiabatic elimination is also applicable when there is

a large detuning G = i∆, but here we concentrate the real G case. Typically f is a

Rabi-frequency coupling f = −iΩ/2 and g is some small term due to a detuning or

a ground-state dephasing rate. All quantities can be time-dependent. In general, the

two off diagonal elements f do not have to be equal, but they often are in practice.

We set them equal here to make the problem more tractable and because this does

not effect the qualitative results.

The adiabatic elimination approximation is to assume that |x2| |Gx2| and thus

solve the second equation by setting x2 = 0:

x2 =f

Gx1, (B.2)

which reduces (B.1) to one dynamical variable:

x1 =

(g +f 2

G

)x1. (B.3)


B.2 Comparison with the exact solution for time

independent terms

In the case that the matrix elements f, g,G are time-independent, (B.1) can be

solved exactly:

Ex = α1Eu1eλ1t + α2Eu2e

λ2t, (B.4)

where λi, ui are the eigenvalues and eigenvectors of the matrix in (B.1):

λ1 =1

2

(g −G−

√4f 2 + (g +G)2

), Eu1 =

(g+G−

√4f2+(g+G)2

)√

8f2+2(g+G)(g+G−

√4f2+(g+G)2

)2

2f√4f2+

(g+G+

√4f2+(g+G)2

)2

;

λ2 =1

2

(g −G+

√4f 2 + (g +G)2

), Eu2 =

(g+G+

√4f2+(g+G)2

)√

8f2+2(g+G)(g+G+

√4f2+(g+G)2

)√

2f√4f2+

(g+G−

√4f2+(g+G)2

)2

;(B.5)

and αi are constants determined by the initial conditions. In the limit G |g|, |f |

we have:

λ1 → −G,

λ2 → g + f 2/G (B.6)

so the u1 term in (B.4) is quickly damped and the state Ex is purely determined by u2.

Adiabatic elimination requires that u2 is the dominant contribution and thus requires

Reλ1 to be negative and |λ2| |λ1|. In this same limit:


u2 →

1

fG

, (B.7)

which is the adiabatic elimination solution (B.2). The term “adiabatic elimination” is

clear in this context. The large negative λ1 always drives the Ex into the u2 eigenvector

meaning there is only a slow “adiabatic” change of Ex at a rate λ2. The adiabatic

solution is a quasi-steady solution and is an exact steady state solution in the limit

λ2 → 0.

In the simple two-level atom case, we have f = iΩ/2, G = Γ/2, x1 = c1, x2 = c3.

We let g = −γ1 be some dephasing rate of |1〉. Then the adiabatic solution is

c3 = (iΩ/Γ)c1 is the quasi-steady state solution. The limiting value of λ2 (B.6)

is −Ω2/Γ − γ1. Physically, these are the rates at which the amplitudes c1, c3 change

due to loss from the laser pumping atoms out of the system or from the dephasing

rate γ1.

Fig. B.1 shows the real parts of λ1, λ2 according to (B.4), as well as the components

of u2, as a function of f/G. In the case g = 0 (solid curves) Fig. B.1(a) shows λ2 = 0

and λ1 = −G at f = 0 and that they only gradually deviate from this value through

f/G = 0.45. We see |λ1| |λ2| is maintained and so u2 is the dominant contribution

to Ex. Fig. B.1(b) then shows the x1 and x2 components of the dominant eigenvector

Eu2, and compares with the x2 value obtained with the adiabatic solution (B.2). Fairly

good agreement is seen through f/G = 0.45. The dashed curves in Fig. B.1(a-b)

show the case with the fairly large value of g = 0.25G. As expected, it modifies λ2

according to (B.6) but adds only a small correction to λ1 and we see |λ1| |λ2| is

still well satisfied.


Figure B.1: Eigenvectors and eigenvalues of the two-level system. Plots arebased on (B.5). (a) Solid curves show real parts of λ1 (thick curve) and λ2 (thincurve) for the case g = 0 and varying f as a real number, plotted on the horizontalaxis. The dashed curves show the case g = 0.25 G. (b) The x1 (thin curves) andx2 (thick curves) components of Eu2. The adiabatic solution (B.2) for x2 (assumingx1 = 1) is shown as a dotted curve. (c-d) The corresponding plots to (a) and (b)only now with f varied as a purely imaginary number. In this case the x1 componentof u2 is purely real and the x2 component is purely imaginary.

In Fig. B.1(c-d) we plot cases where f is imaginary. The g = 0.25G case (solid

curves) shows good agreement through f/G = 0.45i. In Fig. B.1(d), however, we

notice in the g = 0 case there is a somewhat pronounced deviation from the adiabatic

solution near f = 0.45i. This is because at the f = 0.5iG the discriminant 4f 2 +

(g+G)2 disappears and there is a resonance in the expression for u2 (B.5). Anytime

we are near a resonance such as this, the adiabatic solution is not accurate. Besides

the eigenvector u2 no longer corresponding to (B.2), the adiabatic solutions also loses

validity because the λ1 and λ2 become on the same order of magnitude when this


discriminant vanishes. In the g = 0.25 G case this resonance occurs at a larger value

of f not near any values on the plot. Note that the discriminant can only vanish

when |f |, |g| ∼ G.

In a simple two-level case this resonance occurs when Ω → Γ/2. In this case both

c1 and c3 decay on a very rapid time scale Ω2/Γ ∼ Γ, and our adiabatic solution loses

validity.

From this discussion, we conclude that indeed when |f |, |g| G, adiabatic elimi-

nations can be performed and that the relative error introduces scales as |λ2|/|λ1| =

Max|g|/G, |f |2/G2.

B.3 Considerations for time-dependent terms

The system (B.1) with time-independent coefficients allowed comparison with an

exact solution. The real power of the adiabatic elimination approximation is that it

can be used it when the terms are time-dependent and a general analytic solution is

impossible. For example, in the two-level problem, a time-dependent Ω would give

rise to a time-dependent f . The above analysis provides a framework to see the

validity of the adiabatic elimination. When time dependence is introduced, it simply

causes a coupling between u1 and u2, in much the same way time-dependent light

fields caused a coupling ΩNA between cA and cD (see Eq. 2.10). Namely, if the terms

are time-dependent with some time scale T , then a rate of coupling ∼ T−1 between

u1 and u2 will be introduced. This prevents the system from solely occupying the u2

eigenvector, even when u1 is being damped at a large rate |λ1| ≈ G. Instead we have

a fraction ∼ 1/(GT ) of the total probability in the u1 component. Our adiabatic


elimination solution is then still valid only if T G−1, giving us one additional

criteria to check whenever applying it.

B.4 Considerations for three-level systems

Eliminating c3 in a three-level Λ system is qualitatively no different than eliminat-

ing it in a two-level system. One must have a situation where the adiabatic elimination

criteria are satisfied for both the |1〉, |3〉 and |2〉, |3〉 systems separately. However, this

actually underestimates the range of parameters for which adiabatic elimination can

be applied. Throughout this thesis, we often use Ωc ∼ Γ which would seem to imply

the adiabatic elimination procedure would not be valid. But one must also consider

that this term is always multiplied by c2 ∼ Ωp/Ωc. So under conditions of EIT with

Ωp Γ/2 , or any situation in which the product c2Ωc Γ is maintained, adiabatic

elimination can be applied.

In addition to this, we note that under EIT conditions, we often do even better,

as the two terms driving c3 mostly cancel. To get a quantitative sense of this effect,

recall that in Section 2.3, where we worked in the dark/absorbing basis. There we

saw the coupling to c3 was proportional to cA (see Eq. 2.11), which is much smaller

than either amplitude c1, c2 under conditions of EIT, due to quantum interference of

the two terms.

For numerical evidence of the validity of adiabatic elimination in our systems,

we refer you to Fig. 2.3. For further analytic justification, we also note that the

results obtained for c2, c3 in Subsection 3.2.7, which solves the three-level system by

Fourier transformation without eliminating c3, agree with those that were obtained


with adiabatic elimination in Subsection 2.3.

B.5 Imaginary G

In this thesis, we primarily rely on the large negative damping due to G. In

systems where we are instead adiabatically eliminating levels which are far detuned

(e.g. a far off-resonant Raman transition, or our elimination of c4 in Section 2.6.2), a

similar analysis can be carried out. However, the mathematical argument is slightly

different. Rather than relying on a large damping rate from G, we must make use

of the fact that the coupling into the “bad” state is much reduced because a quickly

varying phase evolution washes it out. One can show, using this argument, than

adiabatic elimination can be applied even for |f | comparable to ReG, so long as

ImG |f |.

Appendix C

Matrix elements and light field

couplings

The purpose of this Appendix is two-fold. First we list the matrix elements for

the transitions on the D1 (with the signs included) so all the relevant information

in choosing three-level systems in atoms is listed. As we saw in Section 2.6, the

relative signs could play a role in double-Λ systems. Second, we show how to calculate

the coupling of an atomic transition from a laser with an arbitrary direction and

polarization.

C.1 Matrix elements

The following table lists the matrix elements for all the possible transitions on

the D1 line in Sodium. Rubidium-87 has the same hyperfine structure so this table

applies to it as well. All the numbers listed are proportional to εl ·rij where εl is a unit

323

Appendix C: Matrix elements and light field couplings 324

polarization vector of a laser with a polarization vector aligned with polarization of

the |i〉 → |j〉 transition (i.e. εl ‖ rij). The numbers are normalized such that taking

the square of absolute value of the number in the table would give the oscillator

strength fij for the transition. Multiplying fij by 60 would then give the numbers

listed in Fig. 2.4(b). The states listed across the top are the excited levels in 3P1/2

and the states listed down the left are the ground states in 3S1/2.

3P1/2

F = 1 F = 2−1 0 +1 −2 −1 0 +1 +2

−1 − 12√

3− 1

2√

3- − 1√

212

12√

3- -

F = 1 0 − 12√

30 − 1

2√

3- −1

21√3

12

-

+1 - − 12√

31

2√

3- - − 1

2√

312

1√2

3S1/2 −2 1√2

- - − 1√3

− 1√6

- - -

−1 −12

12

0 − 1√6

− 12√

3−1

2- -

F = 2 0 − 12√

3− 1√

31

2√

3- −1

20 −1

2-

+1 - −12

−12

- - −12

12√

3− 1√

6

+2 - - − 1√2

- - - − 1√6

1√3

C.2 Light field couplings

The question then becomes what the matrix elements εl · rij are when εl is not

parallel to rij. This is important in determining the coupling to other levels in the

presence of bad polarizations βpol and misalignment between the quantization axis

and the beam propagation directions, both discussed in Section 2.6.

The procedure is to write z-circularly polarized light (which propagates in the

direction z) εl = (x ± iy)/√2. The analogous expressions for x, y are obtained by

circular permutation of the x, y, z. Linearly polarized light in a direction j is just j

(and must be propagating in a direction orthogonal to j).

For the atomic states we assume z as our quantization axis since that is direction

Appendix C: Matrix elements and light field couplings 325

of the magnetic trap’s bias field. We write polarization vectors εij for the matrix

elements corresponding to ∆mF = ±1 as εij = (x± iy)/√2 and ∆mF = 0 as εij = z.

One then calculates εl · εij. and multiplies this number (which can be complex)

by the matrix elements in the table above to determine the actual coupling matrix

element. Carrying out this calculation for x, y-circularly polarized light shows the it

couples the ∆mF = ±1 with 1/4 of the full strength and ∆mF = 0 with 1/2 the full

strength. Similarly linearly polarized light εl = x, y couple ∆mF = ±1 transitions

with 1/2 the full strength. They do not couple the ∆mF = 0 transition (since x, y

obviously has no component linearly polarized along z).

This procedure allows us to obtain the coupling matrix elements for light field

propagating in an arbitrary direction. To do this, we decompose it into x, y, z compo-

nents, calculate the matrix elements for each component, and then sum them together.

Similarly, if the B field is slightly off the z axis, one can rotate the light fields to a

basis where the actual B field corresponds to z and do the same calculation.

Appendix D

Numerical algorithm to solve

Amplitude-Bloch Equations

In Chapter 3 we used a numerical code which propagated the equations governing

the internal atomic amplitudes ci and the light field Rabi-frequencies Ωp,c to check our

analytic results and extend them beyond the weak probe approximation. In Chapter 4

we added atomic dynamics and superseded this code with a more advanced one,

described in Appendix E. This latter one generally has a larger range of applications

and validity, however, the one described here still includes two possibilities which are

absent in the more advanced code: first, it does not adiabatically eliminate c3 and

so can be applied to light field dynamics on the order of or faster than the natural

linewidth Γ−1; second, it propagates the full Optical-Bloch Equations (OBE) and so

can check the errors which are introduced by using amplitudes rather than the full

density matrix to describe the states of the atoms.

In this Appendix, we briefly outline the numerical technique. It solves the light

326

Appendix D: Numerical algorithm to solve Amplitude-Bloch Equations 327

field propagation in steps of dz with a central differencing algorithm accurate to

O(dz2), described in Section D.3. We evolve the atomic density matrix ρ in time in

steps of dt, with a forward differencing algorithm accurate to O(dt2), as described in

Section D.2. Because the equations we solve are coupled, we must apply these two

procedures in a very specific order to solve them self-consistently to the same accuracy.

This is described in Section D.4. All programs were written as C code. This which

was used in conjunction with Mathlink, so the user interface and subsequent data

analysis and graphing could be done in Mathematica.

D.1 Grid representation

All quantities (Ωp,c, ρ were defined on an equally spaced time grid

tinit, tinit + dt, · · · , tinit +mdt, · · · , tfin − dt, tfin, (D.1)

and an equally spaced space grid:

zin, zin + dz, · · · , zin + j dz, · · · zout − dz, zout. (D.2)

We consider only 1D dynamics. With these conventions, all points in time and space

are labelled with an integer pair (m, j). A schematic of the grid is shown in Fig. D.1.

D.2 Atomic density matrix evolution in time

The OBE equations for the 3 × 3 atomic density matrix ρ are given in Eq. (A.6).

In the limit that very few atoms spontaneously decay into |1〉 or |2〉 they reduce to the


Figure D.1: Schematic of grid representation and numerical solution of thej = 1 row. All quantities are defined on equally spaced grids in z and t. Thepoints known from the given boundary conditions for the atoms ρ(0,j) are indicatedwith the vertical dotted line (labelled with ‘A’) and known values of the light fieldsare F (m,0), G(m,0) are indicated with a horizontal dashed line (labelled with ‘L’). Theinitial step is to integrate (D.4) along the j = 0 row (dashed arrow). At that point,we alternately apply (D.7) and (D.4) in the order indicated until the j = 1 row iscalculated (the numbering continues in the same pattern across the row). Then wediscard the j = 0 row from memory and use the known j = 1 row to solve the j = 2row in the same way. The gray shaded area indicates the points kept in memoryduring the computation of calculation of the j = 1 row. This entire area is thenshifted up one row after calculation of each j.

amplitude equations (2.22) with the correspondence ρij = cic∗j . Appendix A contains

a discussion of this correspondence. For generality, we allow arbitrary ground state

decay rates γ2, γ1 and an arbitrary relationship between the decay rate of the excited

level Γ and the incoherent feeding of the excited level to the ground states Γ31,Γ32. At

each (m, j), ρ(m,j) was represented as a nine element array, storing the real quantities

ρ(m,j)11 , ρ

(m,j)22 , ρ

(m,j)33 , and the real and imaginary parts of ρ

(m,j)12 , ρ

(m,j)13 , ρ

(m,j)23 . At any

given point, the program stored in memory the values of ρ(m,j) at one particular

spatial point j and all times m and also stored the value one space back ρ(m,j−1). In

Fig. D.1, we diagram the calculation of j = 1 row, so at this point ρ(m,0) and ρ(m,1)


(indicated with gray shading) are stored in memory.

The evolution equation (A.5) can be written in the form:

ρ =Mρ, (D.3)

where M is a 9 × 9 matrix representing Eq. (A.6) and is time- and space-dependent

via its dependence on the light fields. We take it as a given for the moment and

discuss how it is updated below. Since the equation is local, i.e. it only depends the

values at one j, we propagate in time to second order in dt by computing the solution:

ρ(m+1,j) =

(I +M (m,j)dt+

1

2(M (m,j)dt)2

)ρ(m,j), (D.4)

where I is the 9 × 9 identity matrix.

D.3 Light propagation in space

The light fields Ωp,c were normalized to their peak values and called:

F ≡ Ωp

Ωp0

, G ≡ Ωc

Ωc0

. (D.5)

We assume a co-propagating geometry. Modifying the propagation equations (3.6) in

terms of the density matrices and these normalized quantities we get:

∂

∂zF = −iαpn ρ31,∂

∂zG = −iαcn ρ32. (D.6)


where αp,c ≡ f13,23σ0Γ/(2Ωp0,c0) and n is the possibly inhomogeneous density. The

right-hand side of these equations do not depend on F,G but only on ρ.

We stored an array of four numbers, the real and imaginary parts of F (m,j) and

G(m,j), at all times m at one particular position j, just as we did for ρ. The density

n was input as a grid with values on the spatial grid as an array n(j).

To propagate forward in z (D.6) to second order in dz we used:

F (m,j+1) = F (m,j) − i dz αp1

2

(n(j)ρ

(m,j)31 + n(j+1)ρ

(m,j+1)31

),

G(m,j+1) = G(m,j) − i dz αc1

2

(n(j)ρ

(m,j)32 + n(j+1)ρ

(m,j+1)32

). (D.7)

D.4 Self consistent procedure

We then must carry out (D.7) and (D.4) in a self-consistent manner which pre-

serves the accuracy O(dz2, dt2). Fig. D.1 indicates how this is done. The initial

condition for the atoms are ρ(0,j)11 = 1 and all other ρ

(0,j)ab = 0. These points corre-

spond to the first column of Fig. D.1 indicated with a vertical dashed line under the

label “A”. We used (D.4) iteratively from the m = 1 through all m at j = 0, as

indicated with the horizontal dashed arrow. This was possible because the light fields

are already known at zin = z(0) from their boundary conditions. These boundary con-

ditions are given time-dependent functions at zin and thus determine F (m,0), G(m,0) as

indicated with the horizontal line labelled with the “L”.

We could then alternatively apply (D.7) and (D.4), in the order indicated with the

numbers in the figure, to solve the j = 1 row. Careful consideration of the equations


shows that when done in the order indicated, all quantities on the right hand sides

of (D.7) and (D.4) are known (to see this, we must recall the matrix M (m,j), which

enters (D.4) is dependent on F (m,j) and G(m,j) but not ρ(m,j)). M (m,j) was updated

after each application of (D.7).

After computing each row j, we discarded the j−1 row from memory and assigned

the values for the j row into this array. This then freed up an array to store our

computation of the j + 1 row. We selected certain subsets of z(j), t(m) where the

quantities ρ(m,j), F (m,j), G(m,j) were saved to disk before they were discarded.

D.5 Diagnostics and extensions

We checked for accuracy by changing the grid spacing in both time and space to

assure this did not effect the result to the accuracy we were considering.

We also mention that when we performed calculations of extremely fast switching

in Chapter 5 we implemented a straightforward extension of the above method with

a variably spaced time grid. By using a very fine spacing at and near the switch

time, and a large spacing where the dynamics were much slower, a big improvement

in computation time was achieved.

Appendix E

Numerical algorithm to solve

Maxwell-GP Equations

In Chapter 4 we introduced a formalism which was used throughout the remainder

of the thesis to analyze slow light in BECs, stopped light, and compressed light. In

situations where the light fields were off, it was also used to investigate one and

two component dynamics in the atom laser, quantum reflections, processing of light

pulses, and the formation of solitons and vortices. In all these investigations, we relied

on numerical integration of the equations to corroborate and extend analytic results.

Here we outline our numerical technique used to simultaneously and self-consistently

propagate the atomic dynamics in time and the light propagation in space. We have

versions of the code which work for 1D and 2D as well as a 3D version applicable

when there is cylindrical symmetry.

We will first cast the equations in a dimensionless form convenient for numerical

calculations, and specify our grid representation and notation (Section E.1). There

332

Appendix E: Numerical algorithm to solve Maxwell-GP Equations 333

we also outline the order in which we carry out the various steps. We then outline our

method to propagate the light fields Ωc,Ωp according to (4.22) with a Runge-Kutta

algorithm [81] (Section E.2). In Section E.3 we outline how to evolve the atomic fields

ψ1, ψ2 according to (4.21). We split the atomic evolution into external (trap, kinetic,

interaction energy) and internal (light field coupling) parts. The two are combined

using a split-operator approach. We use a Crank-Nicolson (CN) algorithm for the ex-

ternal dynamics in 1D [82] and a variation of this known as the Alternating-Direction

Implicit (ADI) algorithm in 2D. We also outline our extension of this method to

3D systems with cylindrical symmetry. We then use a separate central differencing

method to integrate the internal evolution. Both the internal and external methods

are accurate to second order in the spatial grid spacing O(dz2, dx2) and the time step

size O(dt2). In Section E.5 we then discuss ways in which our accuracy were assessed.

E.1 Fundamental equations and grid representa-

tion

Our starting point is GP equations for the two ground state wave functions ψ1, ψ2

(4.21) and the spatial propagation of the two light fields Ωc,Ωp (4.22). To minimize

the need for constants and quantities which are many orders of magnitude different

than unity, we introduce characteristic length, time, and energy scales by which all

quantities are normalized:


Length : a(HO)z =

√

2mωz;

Time : ω−1z ;

Energy : ωz. (E.1)

We will use the notation of a bar f to represent the dimensionless quantity corre-

sponding to some f . We will here present the equations in 2D. The reduction to the

1D case is straightforward. We will discuss additional terms necessary in 3D later.

In these dimensionless units, the equations (4.21) are:

∂

∂tψ1 = − (iH1 + Lpp

)ψ1 − Lcpψ2,

∂

∂tψ2 = − (iH2 + Lcc

)ψ2 − Lpcψ1,

where

H1 = − ∂2

∂z2− ∂2

∂x2+

1

4(z2 + εx2) + U11|ψ1|2 + U12|ψ2|2 − ∆p,

H2 = − ∂2

∂z2− ∂2

∂x2− 2i(k

(z)2

∂

∂z+ k

(x)2

∂

∂x) +

M2

4(z2 + εx2) + U22|ψ2|2 + U12|ψ1|2 − ∆c

Lij = ¯Ωi¯Ω∗j ,

¯Ωp,c ≡ Ωp,c√Γ,

Uij =8πNaijLy

ε ≡ ωxωz, M2 ≡ V2

V1

= 1 + α(Z). (E.2)

Note the additional scaling of Ωp,c → ¯Ωp,c to normalize out the constant Γ. The light

propagation equations (4.22) in these units are


∂

∂zΩp = −αp( ¯Ωp|ψ1|2 + ¯Ωcψ

∗1ψ2),

∂

∂wΩc = −αc( ¯Ωc| ¯ψ2|2 + ¯Ωpψ

∗2ψ1);

where αp,c ≡ Nf13,23σ0

2Ly

. (E.3)

All four dynamical quantities ψ1, ψ2,¯Ωp,

¯Ωc were represented on cartesian grids

spaced by dz in the one dimension case, and dz, dx in two dimensions. We use

integers j, k to label the grid points via the convention

z(j) = z(0) + dz j, j = 0, 1, 2, · · · , nz

x(k) = x(0) + dx k, k = 0, 1, 2, · · · , nx (E.4)

We label times t(m) = tinit + dtm so the set (j, k,m) represents a point in space and

time. Unlike in the code in Appendix D, the arrays stored the dynamical quantities at

all positions j, k and one particular time m. We generally stored the wave functions

arrays at the current time m and several past times, ψ(j,k,m−2)1,2 , ψ

(j,k,m−1)1,2 , ψ

(j,k,m)1,2 , in

memory. This allowed us to project the wavefunctions forward in time to second

order to via:

ψ(j,k,m+1)1,2 = 3ψ

(j,k,m)1,2 − 3ψ

(j,k,m−1)1,2 + ψ

(j,k,m−2)1,2 + O(dt3) (E.5)

to use in calculating the light field propagation at time m + 1 before we actually

calculate the values ψ(j,k,m+1)1,2 with (E.2). For the light grids we stored only ¯Ω

(j,k,m)p,c

since we never needed to project them forward in time. To initialize these values we


assumed stationary initial conditions (usually all the atoms in |1〉 in the ground state)

and simply assigned ψ(j,k,m−2)1,2 = ψ

(j,k,m−1)1,2 = ψ

(j,k,m)1,2 to this initial state. The light

field initial conditions always had the probe field off ( ¯Ω(j,k,0)p = 0) and the coupling

field on at some homogeneous value ( ¯Ω(j,k,0)c = ¯Ω

(init)c ).

Assume we have the values of the wave functions at m,m− 1,m− 2 and the light

fields at m. We then want to calculate all these quantities at m+ 1, thus propagation

the system forward time by an incremental step dt. The procedure was as follows:

• Find the light fields at the new time ¯Ω(j,k,m+1)p,c by propagating (E.3), as described

below, using the projected values ψ(j,k,m+1)1,2 from (E.5).

• Using the newly calculated ¯Ω(j,k,m+1)p,c and the old values ¯Ω

(j,k,m)p,c calculate the

evolution ψ(j,k,m)1,2 → ψ(j,k,m+1)

1,2 as described below.

• Rearrange our pointers in memory and discard old results, then repeat the

procedure.

For each incremental step dt, we accumulate an error Estep = O(dt3). Since the

number of steps is Nt = T/dt, where T is the total time of integration, we expect our

error over the entire integration to scale as EstepNt = O(dt2). An analogous result

holds for spatial steps.

At specified times m we saved the dynamical quantities to disk before freeing

up these arrays for new points. The methods we outline below rely on an equally

spaced Cartesian grid of space points, but at specified times, we could change the time

progression step dt so long as this did not introduce a large error in our projection

scheme (E.5).


E.2 Light propagation in space: Runge-Kutta

For concreteness, let us assume from now on the orthogonal geometry so w = x.

Thus the probe propagates from j = 0 → nz and the coupling field from k = 0 → nx.

The generalizations to w = z,−z are straightforward. The boundary conditions for

the light fields are given functions of time, defined along the edge of the grid over the

plane in which they enter. They thus determine all ¯Ω(0,k,m)p , ¯Ω

(j,0,m)c which we then

propagate with (E.3). We employ the Runge-Kutta (RK) method [81]. The following

algorithm gives the method to take ¯Ω(j,k,m+1)p → ¯Ω

(j+1,k,m+1)p . Initially the ψ

(j,k,m+1)1,2

are known from the forward projections (E.5). However, the ¯Ω(j,k,m+1)c are not, so

initially we must use the values at the already known time m. Therefore set n = m

in the following procedure to obtain ¯Ω(j+1,k,m+1)p :

K1P = −αP dz(

¯Ω(j,k,m+1)p |ψ(j,k,m+1)

1 |2 + ¯Ω(j,k,n)c ψ

(j,k,m+1)∗1 ψ

(j,k,m+1)2

),

¯Ω(1/2)p = ¯Ω(j,k,m+1)

p +1

2K1P ,

¯Ω(1/2)c =

9

16

(¯Ω(j,k,n)c + ¯Ω(j+1,k,n)

c

)− 1

16

(¯Ω(j−1,k,n)c + ¯Ω(j+2,k,n)

c

),

ψ(1/2)1,2 =

9

16

(ψ

(j,k,m+1)1,2 + ψ

(j+1,k,m+1)1,2

)− 1

16

(ψ

(j−1,k,m+1)1,2 + ψ

(j+2,k,m+1)1,2

),

K2P = −αP dz(

¯Ω(1/2)p |ψ(1/2)

1 |2 + ¯Ω(1/2)c ψ

(1/2)∗1 ψ

(1/2)2

),

¯Ω(j+1,k,m+1)p = ¯Ω(j,k,m+1)

p +K2P . (E.6)

We apply this procedure iteratively from j = 0 → nz. We use an analogous algorithm

to take ¯Ω(j,k,m+1)c → ¯Ω

(j,k+1,m+1)c .

Nominally, the method is accurate to O(dz2). However, there is a problem because

we have had to use the quantities ¯Ω(j,k,m)c rather than ¯Ω

(j,k,m+1)c (i.e. we set n = m). To


circumvent this problem, we did the procedure iteratively, calculating first ¯Ω(j,k,m+1)c

using the old values ¯Ω(j,k,m)p (i.e. applying the ¯Ωc analogue of (E.6) with n = m). We

then calculate ¯Ω(j,k,m+1)p by applying (E.6) with n = m + 1. To assure our solution

is self-consistent, we then calculate both ¯Ω(j,k,m+1)p and ¯Ω

(j,k,m+1)c once more with

n = m+ 1.

E.3 Atomic dynamics

The atomic dynamics are more difficult to solve because their evolution is governed

by second-order partial differential equations. They have two spatial boundary condi-

tions in each spatial dimension, which we take in all our simulations to be ψ1, ψ2 = 0

at all boundaries of the grid. In time, our boundary condition is the initial state,

which we take to be all atoms in ψ1 in the ground state. The method to obtain this

ground state is described below.

We describe below a method to propagate ψ1,2 according to H1,2 a time dτ accurate

to O(dτ)2 with the CN and ADI methods. We then describe a method to propagate

the Lij parts for a time dτ accurate to O(dτ)2. To combine these so we obtain

evolution for dt accurate to O(dt)2, we then must apply these in a particular order

and use smaller steps dτ < dt as we now describe.

E.3.1 Split operator approach

To treat the external and internal (light coupling) dynamics simultaneously and

maintain second order accuracy, we must use a split-operator approach [81]. The split

operator approach relies on the operator identity


exp ((A1 + A2)t) = exp (A1t/2) exp (A2t) exp (A1t/2) + O(dt3). (E.7)

Consider (E.2). Using the identity (E.7), we can see:

ψ(m+1) = exp

(−dt

4L

)exp

(−i dt

2H)

exp

(−dt

4L

)ψ(m) + O(dt3);

where ψ =

ψ1

ψ2

, H =

H1 0

0 H2

, L =

Lpp Lcp

Lpc Lcc

. (E.8)

So we see that we can obtain a solution of the full problem, accurate to order O(dt2),

by evolving ψ according to L for dτ = dt/4, then according to H for dτ = dt/2, and

then doing the L evolution once more for dτ = dt/4. In situations where light fields

were off, we could perform the entire CN step (dτ = dt) at a time.

We now discuss how to calculate the evolution of ψ for dτ due to each these

individual parts H and L.

E.3.2 1D: Crank-Nicolson

The CN method is a central differencing method of propagating a partial differ-

ential equation of second order in one variable and first order in another. It has the

nice features of preserving the norm of the wavefunction for a Hermitian Hamiltonian,

being second order accurate in the time step, and being unconditionally stable [82].

The norm preserving feature of CN is quite valuable when implementing propagation

with losses, as we do, since then can be assured that even relatively small fractional

losses which do occur are due to the true propagation of the equations rather than

the small errors accumulating in the numerical integration.


The external part of (E.2) in 1D can be written in the form:

∂ψi∂t

= −iHiψi,

where Hi ≡ − ∂2

∂z2ψi − iκ ∂

∂zψ1 + Veff (z), (E.9)

where Veff (z) includes all terms in the trap potential, the detuning and the interac-

tions, and κ is due to possible momentum kicks from k2 in the case i = 2. Because of

the nonlinearity, Veff (z) depends on the wavefunctions through the interaction terms

but we ignore that for the time being and assume that Veff is independent of ψ1,2.

We now show how to evolve ψ(j,m)i → ψ(j,m+n)

i (n = 1/2 in the case dτ = dt/2 and so

forth). The Crank-Nicolson finite difference solution to (E.9) is:

(1 +i

2dτH†(fd)

i )ψ(j,m+n)i = (1 − i

2dτH(fd)

i )ψ(j,m)i , (E.10)

which is accurate to accurate to O(dτ 2). In this equation we discretize the Hamilto-

nian Hi with its finite differencing expression H(fd)i by making the substitutions for

the derivatives:

∂2

∂z2ψ

(j,m)i → 1

dz2

(ψ

(j+1,m)i + ψ

(j−1,m)i − 2ψ

(j,m)i

)+ O(dz3),

∂

∂zψ

(j,m)i → 1

2 dz

(ψ

(j+1,m)i − ψ(j−1,m)

i

)+ O(dz3). (E.11)

Note that the left hand side of (E.10) consists of the yet to be computed values

ψ(j,m+n)i , which we call the implicit part of the solution, and the right hand side

contains the already known values ψ(j,m)i , which we call the explicit part of the solution.

In general explicit methods, while easier to solve, are unstable [82] meaning that for


certain step sizes dτ , the solution will diverge in an unphysical way after many steps,

even when the error of individual steps is not particularly large. Implicit methods are

unconditionally stable. The symmetric solution (E.10), which employs half of each, is

also stable. It has the added advantage of having the second order corrections O(dτ 2)

on each side cancel and so its error is O(dτ 3).

Plugging these expressions into (E.10), discretizing V(j)eff and multiplying through

by (2i dz2/dτ) gives:

(1 + iκ dz) ψ(j+1,m+n)i + (1 − iκ dz) ψ(j−1,m+n)

i

+

(−2 +

2idz2

dτ− V (j)

effdz2

)ψ

(j,m+n)i = S(j,m);

where S(j,m) ≡ − (1 + iκ dz) ψ(j+1,m)i + − (1 − iκ dz) ψ(j−1,m)

i

+

(2 +

2i dz2

dτ+ V

(j)effdz

2

)ψ

(j,m+n)i (E.12)

Due to our partially implicit representation, we have three unknown quantities on the

left hand side: ψ(j+1,m+n)i , ψ

(j,m+n)i , ψ

(j−1,m+n)i . We have nz − 2 equations (E.12), one

for each j except the boundaries j = 1, j = nz. However our boundary conditions

fix ψ(1,m+n)i = ψ

(nz ,m+n)i = 0 so there are nz − 2 unknowns. The problem is then a

tridiagonal matrix, which can be solved in order O(nz) steps. We write:

ψ(j+1,m+n)i = α(j)ψ

(j,m+n)i + β(j) (E.13)

and plug this back into (E.12) to get the solution:


α(j−1) = −(1 − iκ dz)γ(j);

β(j−1) =[S(j,m) − (1 + iκ dz)

]β(j)γ(j);

where γ(j) ≡ 1

α(j)(1 + iκ dz) − 2 − V (j)eff dz

2 + 2i dz2

dτ

, (E.14)

with the boundary conditions giving us α(nz) = β(nz) = 0.

As mentioned above, Veff (z) actually depends on the wave functions ψ1, ψ2. We

thus had to use an iterative procedure similar to the one we used for the light field

propagation. We first propagated each ψ(j,m)i → ψ

(j,m+n)i using the values |ψ(j,m)

1,2 |2 in

the nonlinear density terms in Veff (z). We then did a second iteration, using both

the old and new values in the nonlinear density terms: (|ψ(j,m)1,2 |2 + |ψ(j,m+n)

1,2 |2)/2. This

central differencing expression preserves the O(dτ 2) accuracy.

In implementing the algorithm, we noticed that the dominant source of noise was

from reflection of small parts of the ψi off the boundaries of our grid since our bound-

ary conditions made them like hard walls. This problem was avoided by introducing

a “gobbler”, a large imaginary potential near the boundary to absorb components

before they could reflect and reenter the region of interest.

We note that making the replacement dτ → −i dτ in the above can implement

propagation in imaginary time which preferentially decays components of ψi in excited

eigenstates [78]. Therefore, we repeatedly applied this to ψ1, each time renormalizing

the values to preserve the original norm, until we converged to the ground state. We

took this to be the initial conditions for our real time propagation.


E.3.3 2D: Alternating-Direction Implicit method

When implementing the 2D version of the code, the analogous procedure would

replace the tridiagonal matrix with a pentagonal matrix, which is somewhat messier

to solve. Fortunately, a more straightforward extension of the 1D procedure, the ADI

method is possible. Details can be found in [81]. In the regular CN method, we

used a half explicit, half implicit expression (E.10). In the ADI method we retain

this representation for the Veff terms, but represent the derivatives in z in a purely

implicit manner (i.e. evaluated at the time we are propagating to) and derivatives in

x in a purely explicit manner (i.e. evaluated at the time we are propagating from)

and propagate ψ(m)i → ψ

(m+n/2)i . We then reverse the roles of x and z to propagate

ψ(m+n/2)i → ψ(n)

i .

E.3.4 3D: Special treatment at the origin

When cylindrical symmetry is present, we can let r =√x2 + y2 and the equa-

tions (E.2) are valid if we make the replacement x→ r and add the additional term

(1/r)(∂/∂r) in the kinetic energy to account for the ∇2 operator in cylindrical coordi-

nates. Note that in 3D, the assumption k(x)2 = 0 is necessary for cylindrical symmetry,

meaning we can not calculate orthogonal propagation of the light fields in 3D.

The difficulty in implementing the 3D equation numerically is that the analogous

finite differencing expressions to (E.11) for (1/r)(∂/∂r) are not accurate near r = 0.

Holland et al. [94] have published a method to treat the origin. We define the wave-

function φ(r, z, t) = rψ(r, z, t) and rewrite the equations for φ. This representation

allows us to continue to have the boundary condition of a vanishing wave function at


the grid boundary, which we now take to be at r = 0. The trick is then to use a more

complicated finite differencing scheme:

∂2

∂r2φi → Wk

1

dr2

(φ

(j,k+1,m)i + φ

(j,k−1,m)i − 2φ

(j,k,m)i

)+(1 −Wk)

1

dr2

(φ

(j,k+2,m)i + φ

(j,k,m)i − 2φ

(j,k+1,m)i

),

1

r

∂

∂rφi → Wk

1

2 k dr2

(φ

(j,k+1,m)i − φ(j,k−1,m)

i

)+(1 −Wk)

1

2 (k + 1) dz2

(φ

(j,k+2,m)i − φ(j,k,m)

i

);

where Wk ≡ k(4k + 3)

(2k + 1)2. (E.15)

which reduces to the normal differencing scheme at large r (where Wk → 1), but

contains no contribution at r = 0 from φ(j,0,m)i (W0 = 0). This representation again

is accurate at to O(dr2) at all r.

E.3.5 Internal Dynamics: Central differencing propagation

The internal part of the propagation equation (E.2) is of the form

∂

∂t

ψ1

ψ2

= L

ψ1

ψ2

, (E.16)

If we invert L we can immediately write down the solution:

ψ(j,m+n)

1

ψ(j,m+n)2

=

(1 − dτ

2L−1

)(1 +dτ

2L

) ψ(j,m)1

ψ(j,m)2

, (E.17)

accurate to O(dτ 2).


E.4 Typical numerical parameters

Typically, we needed between 600− 1200 points in z and 300− 600 points in x for

2D simulations. A total grid size of ∼ 5σz × 5σr was usually wide enough to avoid

edge effects. For 1D simulations, we usually used a ∼ 5σz grid with 4000 points.

Typically the gobbler had height ∼ 25µTF at the border and exponentially decayed

as it moved in with a width about 2.5% of the total grid size.

We used dt < 0.025 ms for the vortex dynamics when the light fields were off

in Chapter 8, the time scale being governed by the atomic dynamics ∼ µ−1. These

were the most computationally expensive simulations. The dt for during light field

propagation needed to be much smaller and was governed by the probe pulse width

τ0 (or the switching time τs in the case of fast switching). Generally we found dt <

Minτ0, τs/80 was sufficient. Just as in the other numerical code (Appendix D),

we often changed the time grid spacing dt several times during the calculation. For

example, after the light fields were switched off or on, the maximum possible dt

changes by several orders magnitude.

E.5 Diagnostics

We briefly mention here the methods we used to assess our accuracy. The CN

and ADI methods, while they preserve the norm by construction, do not conserve

energy when the grid spacing is too course. Thus when we have an energy conserving

Hermitian Hamiltonian (which only occurs when no light fields or collisional losses are

present), energy conservation was a good diagnostic of our algorithm being physically


accurate. Indeed, we found for extremely long complicated simulations of dynamics,

such as the vortex dynamics simulations in Chapter 8, that eventually energy con-

servation would cease to occur for dt too large. Also when there were no lossy terms

except the gobbler, we used a loss of norm as a signature a significant probability

hitting the boundaries of a the grid and so we increased our grid size.

When light field coupling was present, there were no conserved quantities available.

In this case, we simply had to change the grid spacing and assure that this did not

affect the results to the accuracy we were considering. In the course of varying dt, we

calculated the differences in the results and confirmed that the error scaled as ∝ dt2,

assuring us that our method was as accurate as expected.

In practice, the 1D simulations presented in this thesis were extremely fast (< 1

minute on a PC machine with a 1 GHz processor), while the 2D simulations were

very time consuming (∼ several hours). Because the grid spacing requirements for

good accuracy and stability criteria were very similar in either case, the 1D code was

a good tool by which we see the dependence of the accuracy on dt, dz. In this way,

we could fine tune these parameters for the 2D simulations in a way that they were

small enough to be accurate but also not so small that the computation time was not

much longer than necessary.

Bibliography

[1] L.V. Hau, S.E. Harris, Z. Dutton, and C.H. Behroozi, Nature 397, 594 (1999).

[2] C. Liu, Z. Dutton, C.H. Behroozi, and L.V. Hau, Nature 409, 490 (2001).

[3] S.E. Harris and L.V. Hau, Phys. Rev. Lett. 82, 4611 (1999).

[4] Z. Dutton, M. Budde, C. Slowe, and L.V. Hau, Science 293, 663 (2001); Science

Express, published online 28 June 2001, 10.1126/science.1062527.

[5] For a review, see M. Inguscio, S. Stringari, and C. Wieman, eds., Bose-

Einstein Condensates in Atomic Gases, Proceedings of the International School

of Physics Enrico Fermi, Course CXL, (International Organisations Services

B.V., Amsterdam, 1999).

[6] S.E. Harris, Physics Today 50, 36 (1997).

[7] M.O. Scully and M.S. Zubairy, Quantum Optics, (Cambridge Univ. Press, Cam-

brdige, UK, 1997).

[8] K.J. Boller, A. Imamoglu, and S.E. Harris, Phys. Rev. Lett. 66, 2593 (1991).

[9] G. Alzetta, A. Gozzini, L. Moi, G. Orriols, Nuovo Cimento B 36, 5 (1976).

347

Bibliography 348

[10] E. Arimondo, in Progress in Optics (ed. E. Wolf), 257 (Elsivier Science, Ams-

terdam, 1996).

[11] M. Xiao, Y.-Q. Li, S.-Z. Jin, and J. Geo-Banacloche, Phys. Rev. Lett. 74, 666

(1995); A. Kasapi, M. Jain, G.Y. Yin, and S.E. Harris, Phys. Rev. Lett. 74,

2447-2450 (1995).

[12] M.M. Kash, et. al., Phys. Rev. Lett. 82 5229 (1999); D. Budker, D.F. Kimball,

S.M. Rochester, and V.V. Yaschuk, Phys. Rev. Lett. 83, 1767 (1999).

[13] D.F. Phillips, A. Fleischhauer, A. Mair, R.L. Walsworth, and M.D. Lukin, Phys.

Rev. Lett. 86, 783 (2001).

[14] M.D. Lukin, S.F. Yelin, and M. Fleischhauer, Phys. Rev. Lett. 84, 4232 (2000);

U.V. Poulson and K. Molmer, Phys. Rev. Lett. 87, 123601 (2001).

[15] S.N. Bose, Z. Phys. 26, 178 (1924); A. Einstein, Sitzber. Kgl. Preuss. Akad.

Wiss., p. 261 (1924) and p.3 (1925).

[16] For reviews, see R.J. Donnelly, Quantized vortices in Helium II (Cambridge

Univ. Press, Cambridge, 1991); R.M. Bowley, J. Low Temp. Phys. 87, 137

(1992).

[17] N. Bogoliubov, J. Phys. USSR 11, 23 (1947).

[18] E.P. Gross, Nuovo Cimento 20, 454 (1961); L.P. Pitaevskii, Zh. Eksp. Teor.

Fiz. 40, 646 (1961) [Sov. Phys. JETP 13, 451 (1961)].

[19] P.C. Hohenberg and P.C. Martin, Annals of Physics 34, 291 (1965).

Bibliography 349

[20] P. Nozieres and D. Pines, The Theory of Quantum Liquids, vol. II: Superfluid

Bose Liquids (Addison-Wesley Publishing Company, Inc., Reading, MA, 1990).

[21] W.D. Phillips and H. Metcalf, Phys. Rev. Lett. 48, 596 (1982); T.E. Barrett,

et al., Phys. Rev. Lett. 67, 3483 (1991).

[22] E.L. Raab, M. Prentiss, A. Cable, S. Chu, and D.E. Pritchard, Phys. Rev. Lett.

59, 2631 (1987).

[23] D.S. Weiss, E. Riis, Y. Shevy, P.J. Ungar, and S. Chu, J. Opt. Soc. Am. B 6,

2072 (1989).

[24] P.D. Lett, W.D. Phillips, S.L. Rolston, C.E. Tanner, R.N. Watts, and C.I.

Westbrook, J. Opt. Soc. Am. B 6, 2084 (1989).

[25] N. Masuhara, et al., Phys. Rev. Lett. 74, 935 (1988); W. Petrich, M.H. Ander-

son, J.R. Ensher, and E. A. Cornell, Phys. Rev. Lett. 74, 3352 (1995).

[26] M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell,

Science 269, 198 (1995); C.C. Bradley, C.A. Sackett, J.J. Tollet, and R.G.

Hulet, Phys. Rev. Lett. 75, 1687 (1995); K.B. Davis, et al., Phys. Rev. Lett.

75, 3969 (1995).

[27] For a theoretical review, see F. Dalfovo, S. Giorgini, L.P. Pitaevskii, and S.

Stringari, Rev. Mod. Phys. 71, 463 (1999).

[28] M.R. Andrews, et al., Science 275, 637 (1997).

[29] D.S. Hall, M.R. Matthews, J.R. Ensher, C.E. Wieman, and E.A. Cornell, Phys.

Rev. Lett. 81, 1539 (1998); D.S. Hall, M.R. Matthews, C.E. Wieman, and E.A.

Bibliography 350

Cornell, Phys. Rev. Lett. 81, 1543 (1998); D.S. Hall, M.R. Matthews, J.R.

Enscher, C.E. Wieman, and E.A. Cornell, Phys. Rev. Lett. 83, 3358 (1999).

[30] H.-J. Miesner et. al, Phys. Rev. Lett. 82, 2228 (1999).

[31] W. Hansel, P. Hommelhoff, T. W. Hansch, and J. Reichel, Nature 413, 498

(2001). H. Ott, G. Schlotterbeck, A. Grossmann, and C. Zimmermann, Phys.

Rev. Lett. 87, 230401 (2001).

[32] I. Bloch, M. Kohl, M. Greiner, T. Hansch, and T. Esslinger, Phys. Rev. Lett.

87, 030401 (2001).

[33] M.-O. Mewes, et al., Phys. Rev. Lett. 78, 582 (1997); E.W. Hagley, et al.,

Science 283, 1706 (1999); I. Bloch, T.W. Hansch, and T. Esslinger, Phys. Rev.

Lett. 82, 3008 (1999).

[34] R.J. Ballagh, K. Burnett, and T.F. Scott, Phys. Rev. Lett. 78, 1607 (1997);

M. Naraschewski, A. Schenzle, and H. Wallis, Phys. Rev. A 56, 603 (1997); M.

Naraschewski, A. Schenzle, and H. Wallis, Phys. Rev. Lett. 80, 1 (1998).

[35] L. Deng, et. al. Nature 398, 218 (1999).

[36] C. Orzel, A.K. Tuchman, M.L. Fenselau, M. Yasuda, and M.A. Kasevich, Sci-

ence 291, 2386 (2001).

[37] M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138 (1993); A. Sorensen, L.-M.

Duan, J.I. Cirac, and P. Zoller, Nature 409, 63 (2001); U.V. Poulson and K.

Molmer, Phys. Rev. A, 64, 013616 (2001).

Bibliography 351

[38] H.-J. Briegel, T. Calarco, D. Jacksch, J.I. Cirac, P. Zoller, quant-ph/9904010.

[39] L.V. Hau, et al., Phys. Rev. A 58, R54 (1998).

[40] D.S. Jin, et al., Phys. Rev. Lett. 77, 420 (1996). M.-O. Mewes, et. al, Phys.

Rev. Lett. 77, 988 (1996).

[41] M.R. Matthews, et al., Phys. Rev. Lett. 83, 2498 (1999).

[42] K.W. Madison, F. Chevy, W. Wohlleben, and J.Dalibard, Phys. Rev. Lett. 84,

806 (2000); J.R. Abo-Shaeer, C. Raman, J.M. Vogels, and W. Ketterle, Science

292, 476 (2001).

[43] B.P. Anderson, et al., Phys. Rev. Lett. 86, 2926 (2001).

[44] For a review, see A.L. Fetter and A Svidzinsky J. Phys. B 13, 135 (2001).

[45] S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, Phys. Rev. Lett.

83, 5198 (1999); J. Denschlag, et al., Science 287, 97 (2000).

[46] C. Raman, et al., Phys. Rev. Lett. 83, 2502 (1999); B. Jackson, J.F. McCann,

C.S. Adams, Phys. Rev. A 61, 051603(R) (2000).

[47] V.N. Popov, Functional integrals in Quantum Field Theory and Statistical

Physics (D. Reidel Publishing Company, Boston, 1993).

[48] Y. Kagan and B.V. Svistunov, Zh. Eksp. Teor. Fiz. 74, 279 (1992) [Sov. Phys.

JETP 75, 387 (1992)]; Y. Kagan and B.V. Svistunov, Zh. Eksp. Teor. Fiz. 105,

353 (1994) [Sov. Phys. JETP 78, 187 (1994)].

[49] A.L. Fetter, Phys. Rev. 136, 1488 (1964).

Bibliography 352

[50] E.B. Sonin, Phys. Rev. B 55, 485 (1997). E.B. Sonin, quant-ph/0104221.

[51] S. Burger, F.S. Cataliotti, and C. Fort, Phys. Rev. Lett. 86, 4447 (2001).

[52] B.B. Kadomtsev and V.I. Petviashvili, Sov. Phys. Dokl. 15, 539 (1970); C.A.

Jones, S.J. Putterman, and P.H. Roberts, J. Phys. A 19, 2991 (1986).

[53] C. Josserand and Y. Pomeau, Europhys. Lett. 30, 43 (1995).

[54] A.V. Mamaev, M. Saffman, A.A. Zozulya, Phys. Rev. Lett. 76, 2262 (1996).

[55] D.L. Feder, et al., Phys. Rev. A 62, 053606 (2000).

[56] S.A. Morgan, R.J. Ballagh, K. Burnett, Phys. Rev. A 55, 4338 (1997); W.P.

Reinhardt and C.W. Clark, J. Phys. B 30, L785 (1997).

[57] Th. Busch and J.R. Anglin, Phys. Rev. Lett. 84, 2298 (2000).

[58] S.E. Harris, J.E. Field, and A. Kasapi, Phys. Rev. A 46, R29 (1992).

[59] R. Grobe, F.T. Hioe, and J.H. Eberly, Phys. Rev. Lett. 73, 3183 (1994).

[60] R. Loudon, The Quantum Theory of Light, 3rd ed. (Clarendon Press, Oxford,

2000).

[61] J. Javanainen and J. Ruostekoski, Phys. Rev. A 52, 3033 (1995).

[62] M. Fleischhauer and A.S. Manka, Phys. Rev. A 54, 794 (1996).

[63] B.W. Shore and D.H. Menzel, Principles of Atomic Spectra (John Wiley & Sons,

Inc., New York, 1968).

Bibliography 353

[64] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interac-

tions, (John Wiley & Sons, Inc., New York, 1992).

[65] J.D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, Inc., New

York, 1999).

[66] A.B. Matrsko, I. Novikova, M.O. Scully, and G.R. Welch, quant-ph/0101147.

[67] Matthew Eisaman, unpublished.

[68] B.D. Busch, PhD. Thesis, (Harvard University, 2000).

[69] L.V. Hau, J.A. Golovchenko, and M.M. Burns, Rev. Sci. Instrum. 65, 3746

(1994).

[70] S. Giorgini, L.P. Pitaevskii, and S. Stringari, Phys. Rev. A 54, 4633 (1996).

[71] A. Kasapi, G.Y. Yin, M. Jain, and S.E. Harris, Phys. Rev. A 53, 4547 (1996).

[72] K. Huang, Statistical Mechanics, 2nd ed., (John Wiley & Sons, New York,

1986).

[73] E. Tiesinga, et al., J. Res. Natl. Inst. Stand. Techol. 101, 505 (1996).

[74] J.P. Burke, C.H. Greene, and J.L Bohn, Phys. Rev. Lett.

81, 3355 (1998). A Mathematica notebook is available at

http://fermion.colorado.edu/∼chg/Collisions/.

[75] A.L. Fetter and J.D. Walecha, Quantum Theory of Many-Particle Systems

(McGraw-Hill, Inc., New York, 1971).

Bibliography 354

[76] Y. Castin and R. Dum, Phys. Rev. A 57, 3008 (1998).

[77] S. Giorgini, L.P. Pitaevskii, and S. Stringari, J. Low Temp. Phys. 109, 309

(1997).

[78] J.E. Williams, (PhD. thesis, University of Colorado, 1997).

[79] G. Baym and C. Pethick, Phys. Rev. Lett. 76, 6 (1996).

[80] Y.B. Band, M.Trippenbach, J.P. Burke, P.S. Julienne, Phys. Rev. Lett. 84,

5462 (2000).

[81] W. H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical

Recipes in C, 2nd Edition (Cambridge University Press, Cambridge, 1992).

[82] S.E. Koonin and D.C. Merideth, Computational Physics, (Addison-Wesley,

Reading, MA, 1990).

[83] L.V. Hau, “BEC and Light Speeds of 38 miles/hour”, Feb. 10, 1999, Work-

shop on Bose-Einstein Condensation and Degenerate Fermi Gases, Center

for Theoretical Atomic, Molecular, and Optical Physics, (Boulder, CO),

http://fermion.colorado.edu/∼chg/Talks/Hau; L.V. Hau, “Bose-Einstein Con-

densation and light speeds of 38 miles per hour”, Harvard University Physics

Dept. Colloquium, Feb. 22, 1999, videocasette (Harvard University, Cambridge,

MA, 1999).

[84] M. Fleischhauer and M.D. Lukin, Phys. Rev. Lett. 84, 5094 (2000).

[85] A. Mair, J. Hager, D.F. Phillips, R.L. Walsworth, and M.D. Lukin, quant-

ph/0108046.

Bibliography 355

[86] Q-Han Park and J.H. Eberly, Phys. Rev. Lett. 85, 4195 (2000).

[87] T-L. Ho and V.B. Shenoy, Phys. Rev. Lett. 77, 3297 (1996); H. Pu and N.P.

Bigelow, Phys. Rev. Lett. 80, 1130 and 1134 (1998); P. Ao and S.T. Chui, Phys.

Rev. A 58, 4836 (1999).

[88] P. Villian, et. al, J. of Modern Optics 44, 1775 (1997); M.J. Steel, et. al, Phys.

Rev. A 58, 4824 (1998).

[89] L.D. Landau and E.M. Lifshitz, Fluid Mechanics (Pergamon Press, New York,

1959).

[90] S. Stringari, Phys. Rev. Lett. 77, 2360 (1996). E. Zarremba, A. Griffin, and N.

Nikuni, Phys. Rev. A, 57, 4695 (1998).

[91] B. Jackson, J.F. McCann, and C.S. Adams, Phys. Rev. A 61, 0103604 (2000).

[92] F. Dalfavo and M. Modugno, Phys. Rev. A 61, 023605 (2000).

[93] B. Jackson, J.F. McCann, and C.S. Adams, Phys. Rev. A 60, 4882 (1999).

[94] M.J. Holland, D.S. Jin, M.L. Chiofalo, and J. Cooper, Phys. Rev. Lett. 78,

3801 (1997).

ultra-slow, stopped, and compressed light in bose-einstein condensates

Documents