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Dynamics and applications of excited cold atoms

Citation for published version (APA):Claessens, B. J. (2006). Dynamics and applications of excited cold atoms. Technische Universiteit Eindhoven.https://doi.org/10.6100/IR609972

DOI:10.6100/IR609972

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https://doi.org/10.6100/IR609972

https://doi.org/10.6100/IR609972

https://research.tue.nl/en/publications/dynamics-and-applications-of-excited-cold-atoms(76f14741-abe0-4341-8cb0-bddbc035f9b8).html

Dynamics and Applications ofExcited Cold Atoms

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan deTechnische Universiteit Eindhoven, op gezag van de

Rector Magnificus, prof.dr.ir. C.J. van Duijn, voor eencommissie aangewezen door het College voor

Promoties in het openbaar te verdedigenop donderdag 6 juli 2006 om 16.00 uur

door

Bert Jan Claessens

geboren te Maaseik, Belgie

Dit proefschrift is goedgekeurd door de promotoren:

prof.dr. H.C.W. Beijerinckenprof.dr. M.J. van der Wiel

Copromotor:dr.ir. E.J.D. Vredenbregt

Druk: Universiteitsdrukkerij Technische Universiteit EindhovenOntwerp Omslag: Jan-Willem Luiten © Nickel/Mills/Ledroit.

CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN

Claessens, Bert Jan

Dynamics and Applications of Excited Cold Atoms/ by Bert JanClaessens. - Eindhoven : Technische Universiteit Eindhoven, 2006. -Proefschrift.ISBN-10:90-386-2531-6ISBN-13:978-90-386-2531-7NUR 926Trefw.: laserkoeling / Magneto-Optische val / deeltjesversnellers /ultra koud plasmaSubject Headings: laser cooling / Magneto-Optical Trap / particleaccelerators / ultracold plasma

voor Robert vanne Kieper en Marie-Jeanne Kelle

Atomic Physics and Quantum TechnologyDepartment of PhysicsEindhoven University of TechnologyP.O. Box 5135600 MB EindhovenThe Netherlands

Cover Requiem chevalier vampire © Nickel/Mills/Ledroit.

The work described in this thesis has been carried out at the PhysicsDepartment of the Eindhoven University of Technology, and is part of the research programof the ‘Stichting voor Fundamenteel Onderzoek der Materie’ (FOM), which is financiallysupported by the ‘Nederlandse Organisatie voor Wetenschappelijk Onderzoek’ (NWO).

Contents

1 Introduction 11.1 Cold atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Laser cooling and trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Laser cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Magneto-Optical Trap . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.3 Rubidium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Ultracold plasmas and cold rydberg atoms . . . . . . . . . . . . . . . . . . 61.4 Cold electron bunches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.1 Brightness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.2 Self-fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 This thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Accurate measurement of the photoionization cross section of the (2p)5(3p)3D3 state of neon 152.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Measurement of the photoionization cross section . . . . . . . . . . . . . . 18

2.3.1 Linear loss rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3.2 Loss rate Vs UV intensity . . . . . . . . . . . . . . . . . . . . . . . 202.3.3 Excited state fraction . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Sources of uncertainty and conclusions . . . . . . . . . . . . . . . . . . . . 242.5 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Dipole-Dipole interactions in a frozen Rydberg gas 273.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 Dipole-Dipole interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3.1 Magneto-Optical Trap and detection . . . . . . . . . . . . . . . . . 313.3.2 Pulsed laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3.3 Microwave setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

i

3.4 From Rydberg gas to ultracold plasma . . . . . . . . . . . . . . . . . . . . 353.5 Dipole-Dipole interactions and plasma formation . . . . . . . . . . . . . . . 373.6 Line broadening due to dipole-dipole interactions . . . . . . . . . . . . . . 39

3.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.6.2 Line shape model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.6.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.7 Ramsey experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.8 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 Ultracold electron source 51Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 First generation pulsed electron source setup 595.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 Loading a Magneto-Optical Trap . . . . . . . . . . . . . . . . . . . . . . . 595.3 2-D MOT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.4 Laser setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.5 Vacuum chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.6 Trapping beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.7 Magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.8 MOT diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.9 Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.10 Plasma diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.11 Second generation accelerator . . . . . . . . . . . . . . . . . . . . . . . . . 725.12 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6 Experimental results 776.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776.2 MOT characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.2.1 Number of trapped atoms . . . . . . . . . . . . . . . . . . . . . . . 776.2.2 Lifetime and loading rate . . . . . . . . . . . . . . . . . . . . . . . . 786.2.3 Temperature and density . . . . . . . . . . . . . . . . . . . . . . . . 80

6.3 Electron and ion beam generation . . . . . . . . . . . . . . . . . . . . . . . 806.3.1 Experimental routine . . . . . . . . . . . . . . . . . . . . . . . . . . 816.3.2 Electron signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.3.3 Ion signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.3.4 Plasma fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846.3.5 Total charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.3.6 Spatial distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 876.3.7 Temperature, emittance and brightness . . . . . . . . . . . . . . . . 89

ii

6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

7 Prospects 95Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Summary 99

Samenvatting 101

Dankwoord 103

cv 105

iii

1

Introduction

1.1 Cold atoms

In 1987, Raab et al. [1] demonstrated the Magneto-Optical trap (MOT) for the first time.In this device, atoms are collected and trapped in a single point-like volume in space andtheir temperature is greatly reduced by the combination of six laser beams and a quadrupolemagnetic field. The MOT provided the experimentalist with a new and practical means tocool and trap dilute samples of neutral atoms [2, 3]. In an MOT, parameters such as atomnumber, density and temperature can be accurately controlled and can be measured withrelative ease. On top of this an MOT is a very pure system where only one atomic isotopeis trapped in a single well-defined state. Furthermore, because of the low temperature incombination with the relatively low density, the corresponding phase-space density is highwith little interaction between the trapped atoms. All these properties caused the MOT toinitiate a cascade of experiments that revolutionized the world of atomic physics. Probablythe most dramatic impact came from the the realization of a Bose Einstein Condensate(BEC) in a magnetostatic trap [4] that was loaded with atoms from an MOT. A BEC wasproduced for the first time in 1995 by Cornell et al. [5] providing exceptional experimen-tal control over a mesoscopic quantum system. The field of degenerate quantum gassesflourished, leading to tantalizing experiments ranging from Bardeen-Cooper-Schrieffer pair-ing [6] to quantum noise correlation measurements [7].All this work has been rewarded with not one, but two Nobel prizes within the time spanof four years. In 1997, Chu, Cohen-Tannoudji and Phillips received the Nobel prize for thedevelopment of methods to cool and trap atoms with laser light. The second Nobel prizewas awarded to Cornell, Ketterle and Wieman in 2001 for the demonstration of a BEC indilute gases of alkali atoms, and for early fundamental studies of their properties.In 1999, Killian et al. succeeded in creating the first Ultracold Plasma (UCP) [8] froman MOT. In their pioneering work, atoms in an MOT were photo-ionized with a pulsedlaser, creating a plasma with very low electron (≈ 10 K) and ion (≈ 1 K) temperatures.These experiments opened the door to laboratory study of a strongly coupled plasma, i.e.a plasma where the potential energy of the ions and the electrons is larger than their ki-

1

Chapter 1.

netic energy. A year after the work performed by Killian et al., Robinson et al. also [9]succeeded in creating a UCP from a gas of cold Rydberg atoms excited from an MOT.Until now, most of the work in this field has been devoted to the fundamental propertiesand the dynamics of a UCP [10, 11]. This thesis on the other hand, deals with using aUCP or gas of cold Rydberg atoms as a source of a very high brightness electron beam [12].The essential idea behind this is that the low electron temperature leads to a low electronbeam divergence and energy spread and therefore to an electron beam with a very highbrightness.

|eñ

|gñ

w+kvw-kv

on resonance

F=- va

A

C

p

Bp- kh

hk

Figure 1.1: Left: (A) Impression of laser cooling showing an atom with momentum p counter-propagated by a light field. (B) The atom absorbs the photon and the corresponding momentum.The resulting momentum of the atom is p− ~k. (C) The atom decays to the ground state andemits a photon in a random direction. The result of many of these cycles is a momentum transferin the opposite direction of the photon. The momenta transferred by all spontaneously emittedphotons add up to zero. Right: The Doppler shift (kv) shifts the laser (ω) counterpropagatingwith respect to the atom closer to resonance. This leads to more absorption of the light counter-propagating the atom. The result is that the atom experiences a damping force.

1.2 Laser cooling and trapping

1.2.1 Laser cooling

The operation of an MOT is based on laser cooling [2]. The physics behind it is that anatom in a laser light field with wavelength λ experiences a force due to the momentumtransfer caused by the absorption and emission of photons with momentum ~k, with kthe wavevector of the laser, |k| = 2π/λ. When a two-level atom in its ground state |g〉moving in a light field absorbs a photon (Fig. 1.1B), it is excited to the excited state |e〉and its momentum changes by ~k in the direction of the light field. After a typical time(τ = 1/Γ with Γ the natural linewidth of the transition) the atom decays to the ground stateagain and emits a photon in a random direction, in which it also gets a momentum kick

2

Introduction

(Fig. 1.1C). After many of these cycles, the recoils of the spontaneously emitted photonsadd up to zero. The momentum kicks of the absorbed photons on the other hand add upin the direction of the light field. For a simple two-level transition, this results in a forceF due to one single-frequency laser beam given by [2]:

F = ~kΓ

2

s

1 + s + (2δ/Γ)2. (1.1)

Here, s = I/I0 is the saturation parameter, I the intensity of the laser, I0 the saturationintensity and δ = ω − ω0 the detuning of the laser frequency ω compared to the atomicresonance frequency ω0.When the atom is moving with a velocity v, it experiences a Doppler shift δD = −k·v. As aresult, when an atom is moving in standing light wave consisting of two counter-propagatinglight fields which are red detuned (δ < 0), the Doppler shift will cause an imbalance in theforces of the two counter-propagating beams. In the reference frame of the moving atom,the Doppler effect will shift the counter-propagating light field closer to resonance, fromwhich the atom will consequently feel a larger force. This is illustrated in the right partof Fig. 1.1. The resulting force on the atom from the light field is approximately linearwith the velocity, i.e. F ∼= −αv (see Fig. 1.1). The atom experiences a damping force, asthough it were moving in a (optical) molasses [13] , this leads to cooling of the atoms.

s-

s+

MJ

0

1

-1

0

-1

1

MJ

w

B B

E

z

J=0

J=1

Figure 1.2: Illustration of the principle of a Magneto-Optical Trap (J=0 to J = 1 transition).The atoms are illuminated by counter-propagating laser beams of circularly-polarized light atthe center of a quadrupole magnetic field. The circularly polarized light field in addition to theposition dependent Zeeman shift causes a position dependent force in addition to the frictionalforce illustrated in Fig. 1.1.

3

Chapter 1.

1.2.2 Magneto-Optical Trap

A three-dimensional optical molasses results in a force that only has a velocity dependence.In order to create a trap, one also needs a spatially dependent force. In an MOT thisis provided by adding a magnetic field gradient to an optical molasses and by carefullychoosing the polarization of the light beams. An MOT consists of three orthogonal pairsof red-detuned, counter-propagating, circularly-polarized laser beams intersecting at thecenter of a magnetic quadrupole field, generated by a pair of anti-Helmholtz coils [2, 3].The trapping principle of the MOT is explained in Fig. 1.2 for a J = 0 (ground state) toJ = 1 (excited state) transition.The counterpropagating laser beams have opposite circularly polarization, i.e. σ+ andσ− light. The magnetic field induces a linear Zeeman shift near the center of the trap.As such the degeneracy of the magnetic sub-levels (MJ = 0,±1) of the excited state islifted. On the right side of the center, the MJ = −1 state is tuned closer to resonanceand correspondingly on the left side the MJ = +1 state. As a consequence, on the rightside the atom interacts with σ− light, while on the left side the atom interacts with theσ+ light (selection rules state that σ± light drives ∆MJ = ±1 transitions). Thus, if thepolarization of the counterpropagating beams is set correctly, the atoms are driven to thecenter of the trap and as such the force becomes spatially dependent in addition to thevelocity dependence coming from the polarization-independent molasses force.In good approximation, the motion of an atom trapped in an MOT can be described byan over-damped harmonic oscillator [3]. Near the center of the MOT the force on a atomwith velocity v at position z can be written as:

FMOT = −αv − αβ

kz. (1.2)

The quantity α is the damping coefficient, given by:

α = −~k2 I

I0

2δ/Γ

(1 + (2δ/Γ)2)2, (1.3)

while β incorporates the effect of the magnetic field and is given by:

β =gµB

~dB

dz. (1.4)

Here g is the Lande factor, µB the Bohr magneton and dB/dz the gradient of the magneticfield in the z-direction (dB/dz = −2dB/dr). As such the motion of a particle entering thetrapping region with a velocity below a certain capture velocity vc can be approximatedby that of a damped harmonic oscillator. Under typical operation conditions the atomundergoes over-damped simple harmonic motion to the center of the trap.A standard MOT contains about 109 atoms at a density of about 1010 cm−3 and a tempera-ture of about 300 µK [14]. The largest MOT reported so far, to our knowledge, in terms ofboth density and number of atoms was reported in 1993 by Ketterle et al.. They succeededin creating an MOT containing 1010 atoms at a density of 1012 cm−3 and a temperature ofapproximately 1.2 mK [15].

4

Introduction

F

1

3

2

44

3

2

!"#$

5P%&'

5S()*

Figure 1.3: Level diagram for 85Rb of the hyperfine levels 5S1/2 and 5P3/2, indicating thetrapping laser transition and the repumper transition.

1.2.3 Rubidium

Because of the relative ease with which they can be trapped and the availability of low costdiode lasers to drive the laser cooling transition, alkali atoms are often used for trappingand cooling. In our experiments we mainly use rubidium (85Rb). The optical transitionsused for trapping and cooling are shown in Fig. 1.3. Laser cooling and trapping is donefrom the ground 5S(J = 1/2) to the fine-structure state 5P(J = 3/2). Since 85Rb also hasa nuclear spin (I = 5/2) both the S and the P state have a hyperfine-structure (F = I+J).Trapping and cooling is done with a ”trapping laser” operated at the closed hyperfinetransition 5S1/2,F =3 → 5P3/2,F=4 at a wavelength of 780 nm. Light from this laser alsooff-resonantly excites the 5P3/2,F=3 state, from which the atoms can decay back to the5S1/2,F=2 hyperfine level of the ground state. Once the atoms are in this state, they can nolonger be exited by the trapping light. Because of this, one also has to apply a ”repumpinglaser”, i.e., a laser that pumps the atoms that fall back to the 5S1/2,F=2 state to the5P3/2,F=3 state, from where they can fall back to the 5S1/2,F=3 state. Table 1.1 gives alist of the relevant laser cooling parameters for 85Rb [16].

5

Chapter 1.

Table 1.1: Some important characteristic quantities for 85Rb and the laser cycling transition5S1/2(F=3)↔5P3/2(F=4).

Quantity Symbol ValueAtomic mass m 85 a.m.u.=1.41×10−25 kgWavelength λ 780.24 nm (in vacuum)Natural linewidth Γ 5.98 MHzLifetime of 5P3/2(F=4) τ = 1/(2πΓ) 26.63 nsSaturation intensity I0 1.64 mW cm−2

Recoil velocity vrec = ~k/m 0.602 cm s−1

Doppler limit, velocity vD = (~Γ/2m)1/2 11.85 cm s−1

Doppler limit, temperature TD = ~Γ/2kB 142.41 µK

DE=kT

!"#

$%&'(

)*+,-. /

Figure 1.4: (A) Creation of a UCP by exciting a Rydberg level that evolves spontaneously toa UCP under the right conditions. (B) Creation of a UCP by photo-ionizing the atoms froman MOT to just above the ionization threshold, the excess energy ∆E is mainly transferred tothe electrons as kinetic energy.

1.3 Ultracold plasmas and cold rydberg atoms

A conventional ”cold” and neutral plasma has a temperature of about 10000 K. This is theminimum temperature at which a reasonable fraction (1 %) of the electrons has an energyin excess of 5 eV, which is the lowest possible energy an electron can have to ionize anatom [17]. As mentioned in Section 1.1, the first experimental realization of an UltracoldPlasma (UCP), performed at the National Institute for Standards and Technology (NIST),was reported by Killian et al. [8]. The temperature in a UCP is on the order of 1 K for theions and 10 K for the electrons [10, 18], orders of magnitude less than in a conventionalplasma. This plasma was produced by photoionization of a cloud of laser cooled Xe atomsjust above the ionization threshold (Fig. 1.4B). The excess energy, ∆E, i.e., the difference

6

Introduction

Position

Po

ten

tia

l E

ne

rgy

A

B

Ionisation

Figure 1.5: Impression of correlation heating, (A) initially the atoms in an MOT have nospatial correlation and little energy spread. (B) The atoms are photo-ionized and the electronsand ions are not in the minima of the created potential energy landscape. This potential energyis subsequently converted to kinetic energy.

between the photon energy and the ionization threshold, is transferred to the electrons andthe ions. Due to the large mass ratio, most of it goes to the electrons. As a result, theinitial electron temperature can be adjusted by changing the laser wavelength and is limitedby the bandwidth of the laser. In the experiment performed at NIST, the initial electrontemperature could be varied between 0.1 and 1000 K and the initial ion temperature couldbe as low as 10 µK. The number of atoms ionized could be adjusted by changing the energyof the ionization laser. In this way 105 ions could be produced at a peak density of about109 cm−3.Conventionally, an ionized gas is considered to be a plasma if λD, the Debye screeninglength, is smaller than the size of the sample:

λD =

√ε0kBT

e2n, (1.5)

where e is the elementary charge, ε0 is the electric permittivity of free space, kB the Boltz-mann constant, n the electron density and T the electron temperature. In the NISTexperiment, the Debye screening length was about 500 nm, which is small compared to thesize of the sample which was about 300 µm. Therefore the system created can indeed beregarded as a plasma.The properties of such a UCP are such that the plasma parameters offer practical exper-imental access to the regime of strongly coupled plasmas. These are plasmas where thethermal energy is less then the Coulomb interaction energy. For a neutral plasma withelectron and ion temperatures Te and Ti, this ratio is characterized by the electron and ion

7

Chapter 1.

coupling parameters Γe and Γi:

Γe =e2

4πε0akBTe

, Γi = ΓeTe

Ti

e− a

λD . (1.6)

The exponential term comes from the shielding of ion-ion interactions by electrons, a =(4πn/3)−1/3 is the Wigner-Seitz radius. For densities of about 109 cm−3 and temperaturesmentioned above, the coupling parameters can be as high as Γe = 10 and Γi = 1000respectively, i.e., deep in the strongly coupled regime (Γ >> 1) where one expects orderingeffects such as Coulomb crystallization.On the basis of the results of molecular dynamics simulations [19–21], however, it wassoon realized that immediately after creation of the plasma, a rapid intrinsic heating effectoccurs. Because there is hardly any spatial correlation between the atoms in an MOT,the plasma is initially also completely uncorrelated. As such the electrons and ions ofa UCP are initially not at a potential energy minimum. The subsequent conversion ofpotential energy into kinetic energy rapidly heats both the electrons and ions. This isdepicted in Fig. 1.5. Each subsystem (ions and electrons) heats up until its correspondingcoupling parameter is approximately 1 on a time scale of the inverse plasma frequency1/ωp =

√mε0/ne2, with m the mass of the corresponding subsystem. For the electrons

this is in the ns timescale, for the ions this is in the µs timescale, at a density of 109 cm−3.This, together with other heating mechanisms such as continuum lowering [22] (the electricfield of the ions and electrons results in an effective lowering of the ionization treshold)and three body recombination [10] leads to an effective electron temperature of about 10K. For the ions, effective temperatures of about 1 K have been measured by means ofabsorption imaging [11]. New techniques are being investigated to reduce the heating dueto correlation heating. Suggested techniques are laser cooling [23, 24] of the ions, or addingcorrelation to the initial neutral atoms by placing them in a lattice [25] or by starting witha fermi degenerate gas [26].Only a year after the first UCP produced by photoionization, Robinson et al. [9] reportedthe spontaneous evolution of a gas of cold Rydberg atoms into a UCP as mentioned inSection 1.1. This experiment was similar to the experiment performed at NIST, insteadof tuning the laser just above the ionization threshold the laser was tuned just below theionization threshold as depicted in Fig. 1.4B. Although the parameters of the plasma aremuch the same, the dynamics of the plasma formation are quite different. Chapter 3 gives adetailed description of how a cold gas of Rydberg atoms evolves into a UCP. The advantageof using a Rydberg gas to create a UCP over using photoionization is that using Rydbergatoms can offer more control over the plasma parameters.

1.4 Cold electron bunches

1.4.1 Brightness

Up until now, a UCP has mainly been a subject of research for its own sake, focussed onproperties such as the electron and ion temperatures and understanding its dynamics.

8

Introduction

A

W

z

x

Beam envelope

Electron trajectories

Figure 1.6: Geometry of a electron bunch at a focus. Indicated are the beam envelope andthe electron trajectories in a focus. A is the transverse area of the beam and Ω the solid angle.

The goal of the ultracold electron bunches project at Eindhoven University of Technologyon the other hand is to prove that a UCP or cold Rydberg gas has a enormous potential as abright, pulsed electron source. The figure of merit for electron sources is the brightness [27],i.e. the current density per unit solid angle and per unit energy spread, as indicated inFig. 1.6. The brightness summarizes the complex properties of an electron bunch and givesa good indication about its usefulness for potential applications.Formally the fundamental characterization of a collisionless pulsed charged particle beamis given in terms of the Lorentz-invariant local phase space density distribution f(r,p), forelectrons, p = γmev/

√1− v2/c2, with v the velocity, and me the electron mass, f(r,p) is

normalized to the number of particles N =∫

f(r,p)d3rd3p. As a result a good definitionfor the Lorentz-invariant local 6D brightness is given by [28]:

B(r,p) ≡ em2ec

2f(r,p), (1.7)

with e the elementary charge, and c the speed of light.For a particle bunch with a 6D gaussian phase space distribution the (peak) 6D brightnessB ≡ B(0, 0) at the center of the bunch can be written as:

B =em2

ec2

8π3

N

σxσyσzσpxσpyσpz

, (1.8)

where σA is the 1/√

e value of the quantity A. However in practice, the exact distributionof a beam is not known, and typically it is the variance σA =

√< A2 > − < A >2 that can

be measured, where <> indicates averaging over the distribution.Furthermore, in a beam, degrees of freedom can be coupled. A linear lens for example,results in a linear coupling between x and px. If one would then use Eq. 1.8 as a definitionfor the brightness, this would result in a gross underestimate. As a result, the normalizedroot-mean-square (rms) emittance is commonly used, for the x-direction e.g. the emittanceis given by:

εx =1

mec

√< x2 >< p2

x > − < xpx >2. (1.9)

The emittance ε is a Lorentz-invariant measure for the focusability of the beam. Underthe assumption that the x, y and z directions are decoupled, the following equation is a

9

Chapter 1.

practical estimate for the 6D brightness.

B =1

mec

Ne

(2π)3εxεyεz

. (1.10)

In cases where the longitudinal energy spread is less critical however, one uses the transversenormalized brightness (B⊥) for pulsed electron bunches. This definition is mostly used.Assuming that the distribution in the z − pz space is Gaussian, one defines for a beamtraveling in the z-direction:

B⊥ =Ip

4π2εxεy

, (1.11)

where Ip = vzQ/√

2πσz is the peak current. An important note here is that the normal-ized transverse brightness is not Lorentz invariant, nevertheless for high energy electronbunches it is the figure of merit.The relation between the normalized transverse brightness (B⊥) and the actual 6-D bright-ness (B) is then:

B⊥ = BσE

√2π, (1.12)

with σE the longitudinal energy spread σE = vzσpz .The normalized brightness scales as 1/T (in the case one can speak of a temperature). Inthe best performing sources (see Chapter 4) so far the limiting temperatures are of theorder of 103-104 K, while for a UCP the electron temperature is of the order of 10 K.This would result in a possible normalized brightness three orders of magnitude higherthen current state of the art sources, all other circumstances being equal. This back ofthe envelope calculation shows the huge promise of using a UCP as a electron source. Adetailed analysis of the potential of a UCP as a cold electron source is given in chapter 4.

1.4.2 Self-fields

Although the emittance of a bunch created from a plasma is extremely low immediatelyafter creation, it must also be kept low. The emittance, as defined above, can be regardedas the rms surface area of the projection of a bunch on the x − px, y − py phase spaceplanes. The result is that the emittance corresponding to a curved projection is largerthan the emittance corresponding to a projection with the same actual surface, but witha rectilinear projection. Figure 1.7 illustrates the phase space projections of a linear anda distorted bunch and the corresponding emittance, defined by the area of the ellipsoidalenvelope of the projection.The advantage of using this definition of emittance is that the practical quality of the beamis also included. In a typical accelerator, one uses optics that is as linear as possible, i.e.optics with little aberrations that can only rotate the projections of phase space. As aconsequence, if one wants to focus the beam to a small spot, the non-linear part of theprojection results in a larger spot size, reducing the brightness.Probably the most detrimental effect for the emittance is curvature of phase space, due tothe self-fields, resulting from the space-charge forces [28]. In typical electron accelerators

10

Introduction

x

px

ex

Figure 1.7: The projection on the x − px phase space plane is given for two bunches, as arethe ellipsoids drawn around the projections defining the emittance. One has a linear projectioncorresponding to a low emittance and one has a curved projection corresponding to a largeemittance. The surfaces of both projections are approximately equal.

these space-charge forces are the main limiting factor in the achievable brightness, i.e.the intrinsic brightness of the initial ionization process. Space-charge induced emittancegrowth is mainly important at low energies, because at high beam energies the space-chargeforces are reduced due to relativistic effects.To see this, consider a cylindrically symmetric bunch with velocity v and charge densitydistribution qn(r). The electric field in the radial direction is then derived from Gauss’slaw and given by [29]:

Er =q

ε0r

∫ r

0

n(r′)r′dr′. (1.13)

The azimuthal magnetic field is given by Ampere’s law:

Bθ =qvµ0

r

∫ r

0

n(r′)r′dr′, (1.14)

where µ0 is the permeability of free space. Combining Eq. 1.13 and Eq. 1.14 gives:

Fr = q(Er − vBθ) = qEr/γ2, (1.15)

where γ = [1− (v/c)2]−1/2

. Thus, the radial coulomb forces are suppressed by a factor γ2.Including relativistic effects, the acceleration a of a electron moving with velocity v due toa force F is defined by [30]:

F = mγa + γ2a · v

c2v

. (1.16)

Combining Eq. 1.15 and Eq. 1.16 shows that the acceleration due to space-charge forcesis suppressed by a factor γ3 both in the radial and the longitudinal direction. The conse-quence of the reduction of detrimental self-fields at relativistic velocities is that, in most

11

Chapter 1.

practical accelerators, one wants to accelerate the bunch as fast as possible to relativisticvelocities.A completely new and fundamentally different way to reduce the effect of space chargeforces on the emittance is by carefully shaping the radial distribution of the electron bunch.This is discussed in more detail in Chapter 4.

1.5 This thesis

This thesis presents work done in connection with the ultracold electron bunches (UCEB)project at Eindhoven University of Technology and is a collaboration of the groups Atomicphysics and Quantum Technology and physics and applications of accelerators of the de-partment Applied Physics. The goal of this project is the production of high brightnesselectron bunches from a UCP or cold Rydberg gas.In Chapter 2, we present a study of the effect of an ionizing laser on a sample of trappedatoms, and use this to obtain a precise measurement of the photoionization cross section.In this practical case we have used metastable neon in the 3D3 state, rather then Rb atoms.This value is important for the practical production of a UCP of neon, since it determinesthe power needed to photoionize the atoms. A UCP of metastable neon is relevant forthe production of a continuous ion beam of noble gas atoms, which is an extension of theUCEB project.Chapter 3 presents work performed at the University of Virginia, devoted to the role ofdipole-dipole interactions in a gas of cold Rydberg atoms. These dipole interactions can beused to accurately control the formation of UCP from a gas of cold Rydberg atoms. Po-tentially, dipole-dipole interactions can be used to reduce the initial electron temperaturein a UCP. This could reduce the emittance of cold electron bunches even further.In Chapter 4, we investigate the feasibility of a UCP as a source for an electron acceleratorthrough simulations with the General Particle Tracer Code. We find that using a UCP asan electron source can result in an increase of brightness of over two orders of magnitudecompared to conventional electron sources.Chapter 5 gives a detailed discussion on the first setup designed and constructed for thefabrication of a UCP from rubidium atoms.Chapter 6 presents the results of creating the first UCP from a cold Rydberg gas andimaging them on a phosphor screen. From these images we obtain an upper value for theinitial emittance of our source.Finally, in Chapter 7 we speculate on where the new field of cold atom charged particlesources might lead us.

12

Introduction

Bibliography

[1] E. Raab, M. Prentiss, A. Cable, S. Chu, and D. Pritchard, Phys. Rev. Lett. 59, 2631 (1987).

[2] H.J. Metcalf, P. van der Straten, Laser Cooling and Trapping (Springer, Berlin HeidelbergNew York, 1999).

[3] C.J. Foot, Atomic Physics (Oxford University Press, 2005).

[4] W. H. Wing, Prog. Quant. Electr. 8, 181 (1984).

[5] M. H. Anderson, J. R. Ensher, M. R.Matthews, C. E.Wieman, E. A. Cornell, Science 269,198 (1995).

[6] M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, C. H. Schunck and W. Ketterle, Nature435, 1047 (2005).

[7] S. Folling, F. Gerbier, A. Widera, O. Mandel, T. Gericke and I. Bloch, Nature 434, 481(2005).

[8] T.C. Killian, S. Kulin, S. D. Bergeson, L. A. Orozco, C. Orzel, and S. L. Rolston, Phys.Rev. Lett. 83, 4776 (1999).

[9] M. P. Robinson, B. Laburthe Tolra, Michael W. Noel, T. F. Gallagher and P. Pillet, Phys.Rev. Lett. 85, 4466 (2000).

[10] S. Kulin, T. C. Killian, S. D. Bergeson and S. L. Rolston, Phys. Rev. Lett. 85, 318 (2000).

[11] C. E. Simien, Y. C. Chen, P. Gupta, S. Laha, Y. N. Martinez, P. G. Mickelson, S. B. Nagel,and T. C. Killian, Phys. Rev. Lett. 92, 143001 (2004).

[12] B. J. Claessens, S. B. van der Geer, G. Taban, E. J. D. Vredenbregt, and O. J. Luiten,Phys. Rev. Lett. 95, 164801 (2005).

[13] S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, Phys. Rev. Lett. 57, 314 (1986).

[14] E. W. Streed, A. P Chikkatur, T. L Gustavson, M. Boyd, Y. Torii, D. Schneble, G. K.Campbell, D. E. Pritchard, W. Ketterle, cond-mat 0507348 (2005).

[15] W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, Phys. Rev. Lett.70, 2253 (1993).

[16] C. Hawthorn, Ph.D. thesis (University of Melbourne, 2004).

[17] T. F. Gallagher, P. Pillet, M. P. Robinson, B. Laburthe-Tolra, and M. W. Noel, J. Opt.Soc. Am. B 20, 1091 (2002).

[18] J. L. Roberts, C. D. Fertig, M. J. Lim, and S. L. Rolston, Phys. Rev. Lett. 92, 253003(2004).

[19] S. G. Kuzmin and T. M. O’Neil, Phys. Rev. Lett. 88, 065003 (2002).

[20] F. Robicheaux and James D. Hanson, Phys. Rev. Lett. 88, 055002 (2002).

[21] T. Pohl, T. Pattard, and J. M. Rost, Phys. Rev. A 70, 033416 (2004).

[22] Y. Hahn, Phys. Lett. A 293, 266 (2002).

[23] T. C. Killian, Y. C. Chen, P. Gupta, S. Laha, Y. N. Martinez, P. G. Mickelson, S. B. Nagel,A. D. Saenz, and C. E. Simien, Plasma Phys. Control. Fusion 47, A 297 (2005).

13

Chapter 1.

[24] T. Pohl, T. Pattard, and J. M. Rost, Phys. Rev. Lett. 92, 155003 (2004) .

[25] T. Pohl, T. Pattard, J.M. Rost, J. Phys. B: At. Mol. Opt. Phys. 37 L183 (2004).

[26] M. S. Murillo Phys. Rev. Lett. 87, 115003 (2001) .

[27] For a recent overview, see P. Piot, in The physics and Applications of High BrightnessElectron Beams, edited by J. Rosenzweig, G. Travish, and L. Serafini (World Scientific,Singapore, 2003), p. 127.

[28] S. B. van der Geer, M. J. de Loos, M. J. van der Wiel, and O. J. Luiten, to pe published.

[29] F. Kiewit, Ph.D. thesis (Eindhoven University of Technology, 2003).

[30] M. Fayngol, Special Relativity (Wiley-VCH, 2002).

14

2

Accurate measurement of the photoionization crosssection of the (2p)5(3p) 3D3 state of neon

Abstract. We report a new measurement of the photoionization cross section for the(2p)5(3p) 3D3 state of neon at the wavelengths of 351 and 364 nm a. These data wereobtained by monitoring the decay of the fluorescence of atoms trapped in a magneto-optical atom trap under the presence of a photoionizing laser, a technique developed byDinneen et al [16]. We obtain absolute photoionization cross sections of 2.05 ± 0.25 ×10−18 cm2 at λ = 351 nm and 2.15± 0.25× 10−18 cm2 at λ = 364 nm, an improvementin accuracy of a factor of four over previously published values. These new values arenot consistent with published theoretical data.

aThe work described in this Chapter, is published in Phys. Rev. A 73, 012706 (2006)

15

Chapter 2.

2.1 Introduction

The process of photoionization plays an important role in applied plasmas such as gaslasers or discharge lamps, and also as a technique for producing ionized matter for scientificstudy. Recently, for instance, it has become possible to study a virtually unexplored field inplasma physics experimentally through the production of a so-called ultracold plasma [1]by near-threshold photoionization of a sample of neutral atoms trapped in a magneto-optical trap (MOT). Such ultracold plasmas have ion temperatures around 1 mK andelectron temperatures as low as 10 K at a density of about 109 cm−3, at which pointthe electrostatic interaction energy between nearest neighbors becomes comparable to thekinetic energy. The plasma is then close to entering the strongly coupled regime wherestandard classical plasma physics assumptions may become invalid [2, 3].

Photoionization cross sections furthermore provide fundamental tests of atomic struc-ture calculations. [4–19]. Here, as is often the case, alkali-metal atoms and rare gas atomsare of particular importance as relatively simple test subjects. For ionization out of theground state, absolute cross sections have been determined with great precision (uncer-tainty 1-3 %) over a wide range of photon energies of up to 4000 eV by using synchrotronradiation [5–7]. For excited states, ionization close to threshold has been studied usingdischarge and laser-excited atomic beams [8–13] and, only for alkali-metal atoms, also us-ing atom traps [16–19]. In the latter case, absolute values could be determined with anaccuracy down to 10%.

The atom trap technique was pioneered by Dinneen et al. [16] and later applied byseveral other groups [17–19], with all experiments focusing exclusively on rubidium. Herewe follow this lead to obtain absolute and precise measurements of the photoionizationcross section of the excited (2p)5(3p) 3D3 state of neon for two different ionization wave-lengths. While much has been learnt about photoionization out of this and closely relatedstates from, e.g., photo-electron spectra [10], electron angular distributions [11] and auto-ionization widths [12, 13, 15], so far the absolute value of the corresponding cross sectionshas only been known with a relative accuracy of 50 % [10, 11]. In this paper we present anindependent and direct measurement based on studying the effect of an ionizing laser onthe decay dynamics of laser-excited neon atoms trapped in a magneto-optical trap. Thetechnique used here employs only relative measurements of atom numbers, which allowedthe relative precision to be increased by a factor of four over the measurements of Siegel etal. [11]. At the time, these authors necessarily had to rely on data on collisional ionizationprocesses to obtain an absolute value of their atom flux, which restricted the achievedaccuracy.

This paper is organized as follows: Section 2.2 describes the experimental setup and thecharacteristics of the photoionization laser; Section 2.3 presents the experimental resultsand the corresponding analysis; Section 2.4 discusses the results and the correspondinguncertainties and compares our measured values with previous work.

16

Accurate measurement of the photoionization cross section of the (2p)5(3p) 3D3 state of neon

Figure 2.1: Schematic drawing of the MOT and the UV laser beam. Depicted in the figure isthe telescope used for expanding the PI beam and the glass plate used for moving the PI beamwith respect to the atom cloud.

2.2 Experimental setup

Metastable neon atoms are trapped in an MOT which is loaded from a bright atomic beamas described in [20, 21]. The atomic beam can produce approximately 2×1010 atoms persecond travelling at 100 m/s. These atoms are slowed down further in a second Zeemanslower and trapped in an MOT. The magnetic field gradient of the MOT is about 10 G/cm,and the detuning of the trapping laser was set at -0.5 Γ with Γ = (2π)8.2MHz the naturalwidth of the atomic transition.

The laser light used for all laser cooling stages of both the atomic beam and the MOTwas generated with a frequency stabilized Coherent 899 ring dye laser. This laser pro-duces 700 mW of laser light at a wavelength of 640.224 nm, resonant with the closedNe(2p)5(3s)3P2 →(2p)5(3p)3D3 optical transition. The laser light for the MOT beams wasspatially filtered through a fiber and expanded. The MOT beams have a Gaussian beamprofile with a 1/

√e intensity radius of 1.5±0.1 mm. The combined intensity of all MOT

beams was typically 16 mW/cm2.

The spatial profile of the fluorescence emitted by the MOT was imaged with a charge-coupled-device (CCD) camera and had a profile that could be fitted well with a Gaussiandistribution. The 1/

√e radius of the MOT fluorescence in the x (sx), and z (sz) (see

Fig. 2.1) direction was measured to be 80±20 µm. The intensity of the MOT beams wasstabilized to better than 10−2 using an electronic controller connected to the modulationinput of an AOM driver. The stabilization was built such that we could switch between

17

Chapter 2.

two stabilized intensities within 400 µs.The UV light for photoionization (PI beam) was produced with a Coherent Innova

90-5 argon-ion laser which we used in two different modes: (1) a dual wavelength mode or”mixed mode”, in which the laser generates a maximum of 86.5 mW of 364 nm light and53.5 mW of 351 nm light simultaneously, and (2) a ”non-mixed mode”in which the lasergenerates a maximum of 80 mW at 351 nm only.

Both these wavelengths generated by the laser are greater than 290 nm, the maximumwavelength for photoionization from the (2p)5(3s) 3P2 state. This ensures that photoion-ization only occurs from the (2p)5(3p)3D3 state. The Gaussian PI beam was expanded toa 1/

√e intensity radius of 450±20 µm in the x-direction and 530±20 µm in the y-direction

at the position of the MOT (see Fig. 2.1). The beam was passed trough an anti-reflectioncoated glass plate, enabling small, controlled horizontal and vertical movements of the PIbeam relative to the atom cloud, and then sent through a vacuum window into the trapunder a small angle with the z-direction. The transmission of the vacuum window wasmeasured to be 0.73± 0.02 at 351 nm and 0.80± 0.02 at 364 nm.

Relevant information was obtained via the measurement of the decay fluorescence usinga Pulnix TM 1300 CCD camera. The camera measured total fluorescence power emittedby the trapped atoms at a rate of 10 frames/s. The fluorescence of the atoms could also bedetected with an EMI 9862K photomultiplier tube (PMT). This was used to measure fast(≤ 400µs) changes in fluorescence power, resulting from a sudden change in MOT laserintensity. Both the CCD camera and the photomultiplier tube were only used for relativemeasurements so that calibration for absolute detection efficiency was not necessary.

A schematic representation of the MOT setup and the PI beam is given in Fig. 2.1.

2.3 Measurement of the photoionization cross section

2.3.1 Linear loss rate

The rate equation that describes the time evolution of the number of trapped atoms Nt inthe MOT with the presence of the PI beam is

dNt

dt= RL −Nt(ΓBG + ΓPI)− β

∫n(r, t)2 dr, (2.1)

where RL is the loading rate (determined by the beam flux and the capture efficiency of theMOT) and ΓBG the decay rate due to density-independent losses such as background gascollisions (the pressure in the trap chamber during operation is approximately 1.6×10−9

mbar). The rate constant β describes the density-dependent losses and n(r, t) is the atomicdensity distribution of the trapped atoms. Finally, ΓPI is the decay rate due to photoion-ization from which the photoionization cross section can be determined as we show insubsection 2.3.2.

One aspect that makes photoionization experiments with metastable atoms differentcompared to similar experiments with alkali atoms is that the steady-state number in the

18


Flu

ore

scen

ce p

ow

er (

arb. unit

s)

Time (s)

Figure 2.2: Fluorescence power from the atoms trapped in the MOT as a function of timeafter loading is stopped, without (filled circles) and with (open circles) the presence of the PIbeam. The solid lines are fits to Eq. (2.2).

MOT is mainly determined by two body losses [21]. A consequence of this is that it is verydifficult to measure ΓPI by studying the effect of the photoionization laser on the steady-state number of atoms in the MOT. Only its effect on the fill rate or on the decay rate ofthe MOT, and then only at low densities can be used. Here the dynamics are determinedby linear losses, i.e., density independent losses. In this experiment we chose to study thedecay dynamics when the atomic beam is switched off (RL = 0) since then the results donot suffer from fluctuations in the beam flux.

In our trap the spatial distribution is to a good approximation independent of thenumber of atoms [21], i.e. n(r, t) = n(r)g(t) so that the solution to Eq. (2.1) (with RL = 0)can be written as

Nt(t) =Nt(0)exp(−tR)

1 + (βNt(0)/Veff )[1− exp(−tR)], (2.2)

where Veff = (2π)3/2sxsysz is the effective trap volume [21] and R = ΓBG + ΓPI the totaldecay rate. Figure 2.2 shows two typical experimental decay curves fitted with Eq. (2.2),with and without the presence of the photoionizing laser. Fits to the data such as theseenable R to be determined. For both curves the initial part of the decay is dominated bytwo-body losses. The effect of linear losses becomes visible at lower atom numbers, andtherefore lower atomic densities.

19

Chapter 2.

(a)

( )

R (

s )

-1R

(s

)

-1

Figure 2.3: The upper graph (a) shows the measured linear loss rate as a function of intensityof the mixed-mode PI beam. The lower graph (b) shows the fitted loss rate as a function ofintensity of the 351 nm-only PI beam. The solid lines are linear fits to the data.

2.3.2 Loss rate Vs UV intensity

The linear decay rate ΓPI due to photoionization by a monochromatic light source withfrequency ν can be written as

ΓPI =IPIfσ

hν, (2.3)

where IPI is the average photoionizing laser intensity incident on the 3D3 atoms in theMOT, h is Planck’s constant and σ is the photoionization cross section at frequency ν [16].The decay rate is proportional to the excited state fraction f , i.e., the fraction of atoms inthe excited 3D3 state, since only these atoms can be ionized with the UV laser as discussedin section 2.2. As the size of the atom cloud is much smaller than the diameter of thetrapping beams, f is constant over the trapping region. Note that Eq. (2.3) is only validwhen IPI is far from saturation; in the present experiment this is always the case. Bymeasuring R as a function of the average photoionizing laser intensity IPI , a value for fσcan be determined.

20


The value of IPI is determined by the total UV laser power (PUV ) and the spatial profileof the photoionizing laser I(x, y), given by

I(x, y) = I0exp(− x2

2σ2x

)exp(− y2

2σ2y

), (2.4)

with σx and σy the 1/√

e radii of the PI beam in the x and y directions and I0 =PUV /(2πσxσy) the peak intensity. Furthermore, the spatial profile of the (3D3) atomsin the MOT and the alignment of the MOT with respect to the UV laser have to be takeninto account. As mentioned in section 2.2 we measured the spatial profiles of both the 3D3

atoms and the PI-laser by imaging them on a CCD camera and found that both spatialprofiles were fitted well with a Gaussian distribution. Assuming that the MOT and the PIbeam are well aligned with respect to one another (we will discuss alignment of the MOTand PI beam further in section 2.4), the average intensity can be found by averaging thespatial profile I(x, y) given by Eq. (2.4) over the normalized, transverse, spatial distributionn(x, y) of the trapped atoms,

IPI =

∫ ∫I(x, y)n(x, y) dxdy =

I0σxσy√(σ2

x + s2x)(σ

2y + s2

y), (2.5)

where (as before) sx and sy are the 1/√

e radii in the x and y direction of the spatial profileof the 3D3 atoms. Since the size of the PI beam is much larger than the size of the 3D3

atom distribution, the value of IPI/I0 is rather insensitive to fluctuations in the size of the3D3 atom distribution.

Figure (2.3a) shows the measured linear loss rate as a function of intensity seen bythe atoms when the laser was running in mixed mode (both 351 nm and 364 nm light).Each data point is a statistical average of at least five measurements. Figure (2.3b) showsthe measured linear loss rate as a function of intensity seen by the atoms with the laserrunning in non-mixed mode (351 nm only). Once again, each data point is a statisticalaverage of at least five separate measurements. The linear behavior of the data confirmsthe assumption of a loss rate constant that varies linearly with the laser intensity. Theslope of these curves corresponds to the quantity fσ/hν = 0.79±0.05 cm2/J for the 351nm-only beam, and f〈σ/hν〉 = 0.83±0.02 cm2/J for the mixed mode UV beam, where 〈...〉indicates averaging according to the fractional power of the PI beam at the wavelengths of364 and 351 nm.

2.3.3 Excited state fraction

To determine σ, the excited state fraction f must be ascertained. In order to overcomecumbersome calculations to determine the excited state fraction [22], we used an empiricalexpression developed by Townsend et al. [23],

f =1

2

CS

1 + 4δ2 + CS. (2.6)

21

Chapter 2.

Figure 2.4: Measured ratio of fluorescence powers P2/P1 as a function of the ratio of intensitiesIm2 /Im

1 . Every point is a statistical average over at least 8 measurements. The solid line is a fitto the data using Eq. 2.7. The dotted line represents Eq. 2.7 for A=0.

Here, S = Im/Im0 , where Im

0 is the saturation intensity of the 3P2 →3D3 transition, andIm is the laser intensity of all MOT beams combined. The detuning of the MOT beams δis expressed in units of Γ, and for this experiment was set to δ = −0.5. The quantity C isa phenomenological factor which lies between the average of the squared Clebsch-Gordancoefficients of all involved transitions and 1. For the 3P2→3D3 transition, the average ofthe squared Clebsch-Gordan coefficients is 0.46.

The large uncertainty in C and in the effective intensity Im seen by the atoms, dueto laser beam imbalances and alignment uncertainties, makes it necessary to measure theexcited state fraction. For this we adopted a modified version of the technique used byTownsend et al. [23]. We measured the power P1 of the fluorescence scattered by the atomsat a certain trap-laser intensity Im

1 , which is proportional to the excited state fraction f1

at that intensity. We then suddenly changed the intensity and measured the power P2 ofthe fluorescence emitted by the atoms with the new intensity Im

2 (Townsend et al. applieda rapid change in detuning). The intensity was changed fast enough to make sure the lossin atom number and the movement of the atoms may be neglected.

Figure 2.4 shows the ratio P2/P1 as function of the ratio between the two intensitiesIm2 /Im

1 . Our data fits well with the following equation:

f2

f1

=Im2

Im1

1 + A

1 + A(Im2 /Im

1 ), (2.7)

derived from Eq. (2.6). Here, A = CIm1 /Im

0 (1 + 4δ2) is the only fit parameter. Havingdetermined A = 0.77 ± 0.06 in this way from the data in Fig. 2.4, we find the excited

22


state fraction at the MOT laser intensity used in the decay experiments (corresponding toIm2 /Im

1 = 1) to be 21.7±1.5 %. From Eq. (2.6) it follows that f = (1/2)A/(1 + A) so thatuncertainties in the value of the detuning have no influence on this determination.

From the fitted value of A we can extract an estimate for C by inserting the calculatedeffective intensity Im of the MOT beams, which serves as a consistency check. This resultsin a value of 0.4±0.1 for the phenomenological constant C, which is indeed in the expectedrange. We note that this cannot be regarded as an actual measurement of C since whichwe did not independently determine the intensity of the trapping light experienced by theatoms.

R (

s

)-1

R (

s

)-1

Figure 2.5: Linear loss rate as a function of the relative displacement of the PI beam withrespect to the atom cloud in (a) the x-direction, and (b) the y-direction. The curves representparabolic fits to the data.

23

Chapter 2.

2.4 Sources of uncertainty and conclusions

Combining the measurements of the linear loss rate fσ due to photoionization and theexcited state fraction f yields a value of 2.05 ± 0.18 × 10−18 cm2 for σ at 351 nm, and avalue of 2.15± 0.16× 10−18 cm2 at 364 nm. The value for the cross section at 351 nm wasdetermined directly from the measurements with the laser in non-mixed mode. The valueobtained was then used to extract the cross section at 364 nm from the measurementswith the laser in mixed mode. We note that these cross sections correspond to effectivelyunpolarized atoms, due to the presence of trapping beams with various polarizations. Theuncertainties given here correspond only to the statistical standard deviations as deter-mined by the measurements of total loss rate R versus PI intensity IPI and the excitedstate fraction f (which is the dominant source of the uncertainty).

In our measurements there were also several systematic sources of uncertainty; all ofthese are related to the precision with which the average intensity IPI is known, estimatedto be within 8%. This comes mainly from four contributions: (1) The uncertainty of thetransmission through the vacuum window between the point where we measured the powerand where the atoms are (2%); (2) The uncertainty in the power measured with our powermeter (5%); (3) The uncertainty in determining the waists of both the UV laser and theatomic distribution in the MOT. This gives an uncertainty in the determined intensity of3%; (4) The alignment of the UV laser with respect to the atom cloud. Special care wastaken to make sure that the photoionization beam was aligned with respect to the MOT.Figure 2.5 shows the effective decay rate as a function of horizontal (a) and vertical (b)displacements of the PI beam with respect to the MOT, displaced via the rotatable glassplate (see Fig. (2.1)). Based on these measurements we conclude that the MOT was within0.3 σ of the 1/

√e intensity radius σ of the PI beam. This gives an additional uncertainty

of 5% for the average intensity.Taking these various uncertainties into account by adding them quadratically, we con-

clude that we measured the cross section for Ne3D3 atoms to be 2.05 ± 0.25 × 10−18 cm2

at a wavelength of 351 nm and 2.15 ± 0.25 × 10−18 cm2 at a wavelength of 364 nm. Therelative accuracy obtained (≈12%) is quite comparable to the result of Dinneen et al. usingrubidium and is dominated by the contribution from the excited state fraction.

When we compare our measurement with previous experimental work done at 351 nmby Siegel et al. [11] using an atomic beam, then we find that our measurements are in perfectagreement with their value but a factor of four more precise. These authors determinedthe photoionization cross section at 351 nm to be (2 ± 1) × 10−18 cm2 by reference todata on collisional ionization on which their absolute flux of excited-state atoms could becalibrated. As the measurements here only rely on relative atom numbers as measured bythe fluorescence yield, in principle they can be made arbitrarily more precise by furtherimprovement of the statistical accuracy and elimination of systematical errors.

For neon there do not seem to exist any recent calculations of absolute cross sectionsfor ionization out of the 3p-state, in contrast to the situation for the metastable (3s)3P0,2

states [14]. To compare the experimental data we have to refer back to somewhat oldertheoretical values from Duzy and Hyman [24], from Chang [25] and from Chang and Kim

24


[26]. Duzy an Hyman used a central-field approximation for the potential experienced bythe outer electron, including a core-polarizability that was adjusted semi-empirically togenerate binding energies that would match experimental data of about 15 excited states.Fine-structure effects were not taken into account. The photoionization cross sections weresubsequently calculated from the resulting wavefunction of the outer electron. In this way,a value of ≈ 5×10−18 cm2 for the cross section at 364 nm was obtained. Chang [25] insteadapplied a Hartree-Fock treatment which included the influence of many-body correctionsto the cross section; these, however, turned out to be quite small (<10%) for ionizationwavelengths greater than 340 nm. In a subsequent paper, therefore, Chang and Kim [26]used a single-configuration Hartree-Fock treatment and found σ ≈ 4.5 × 10−18 cm2 at364 nm, with an about 10% smaller value at 351 nm. All these values are beyond the high3σ-border of our data. This suggests that a refinement of the calculations may be in order,for which the current data may serve as a benchmark.

2.5 Acknowledgements

We would like to thank A. Kemper, R. Rumphorst, W. Kemper, H. van Doorn, V. Mo-gendorff, C. Hawthorn, L. van Moll, J. van de Ven and R. Gommers for technical andexperimental assistance. We also acknowledge useful communications with H. Hotop, andthank Coherent Netherlands B.V. for the loan of the Innova 90 laser used in these exper-iments. This work was financially supported by the Australian Research Council and theNetherlands Foundation for Fundamental Research on Matter (FOM).

Bibliography

[1] T. C. Killian, S. Kulin, S. D. Bergeson, L. A. Orozco, C. Orzel, and S. L. Rolston, Phys.Rev. Lett. 83, 4776-4779 (1999)

[2] T. Pohl, T. Pattard, and J. M. Rost, Phys. Rev. Lett. 92, 155003 (2004)

[3] F. Robicheaux and J. D. Hanson, Phys. Rev. Lett. 88, 055002 (2002)

[4] S. Aloıse, P. OKeeffe, D. Cubaynes, M. Meyer, and A. N. Grum-Grzhimailo, Phys. Rev.Lett. 94, 223002 (2005)

[5] I. H. Suzuki and N. Saito, J. Electron Spectrosc. Relat. Phenom. 129, 71 (2003)

[6] J.A.R. Samson , W.C. Stolte, J. Electron Spectrosc. Relat. Phenom. 123 265 (2002)

[7] A.A. Sorokin, L.A. Shmaenok, S.V. Bobashev, B. Mobus and G. Ulm, Phys. Rev. A 58,2900 (1998).

[8] I.D. Petrov, V.L. Sukhorukov, E. Leber, and H. Hotop, Eur. Phys. J. D 10, 53 (2000)

[9] R. Kau, I. D. Petrov, V. L. Sukhorukov and H Hotop, J. Phys. B: At. Mol. Opt. Phys. 295673 (1996)

[10] J. Ganz, B. Lewandowski, A. Siegel, W. Bussert, H. Waibel, M.-W. Ruf and H. Hotop, J.Phys. B: At. Mol. Phys. 15, L485-489 (1982).

25

Chapter 2.

[11] A. Siegel, J. Ganz, W. Bussert and H. Hotop, J. Phys. B: At. Mol. Phys. 16 2945 (1983)

[12] J. Ganz, M. Raab, H. Hotop and J. Geiger, Phys. Rev. Lett. 53, 1547 (1984)

[13] D. Klar, K. Ueda, J. Ganz, K. Harth, W. Bussert, S. Baier, J.M. Weber, M.-W. Ruf andH. Hotop, J. Phys. B: At. Mol. Opt. Phys. 27, 4897 (1994)

[14] I.D. Petrov, V.L. Sukhorukov and H. Hotop, J. Phys. B: At. Mol. Opt. Phys. 32, 973 (1999).

[15] I. D. Petrov, V. L. Sukhorukov and H. Hotop, J. Phys. B: At. Mol. Opt. Phys. 35, 323(2002)

[16] T. P. Dinneen, C.D. Wallace, K.-Y.N. Tan, and P.L. Gould, Opt. Lett. 17, 1706 (1992).

[17] J. R. Lowell, T. Northup, B. M. Patterson, T. Takekoshi, and R. J. Knize, Phys. Rev. A66, 062704 (2002).

[18] D. N. Madsen, J. W. Thomsen, J. Phys. B: At. Mol. Phys. 35, 2173 (2002).

[19] C. Gabbanini, S. Gozzini, A. Lucchesini, Opt. Comm. 141, 25 (1997).

[20] J.G.C. Tempelaars, R.J.W. Stas, P.G.M. Sebel, H.C.W. Beijerinck, and E.J.D. Vredenbregt,Eur. Phys. J. D 18,113 (2002).

[21] S.J.M. Kuppens, J.G.C. Tempelaars, V.P. Mogendorff, B.J. Claessens, H.C.W. Beijerinck,and E.J.D. Vredenbregt, Phys. Rev. A 65,023410 (2002).

[22] J. Javanainen, J. Opt. Soc. Am. B 10, 572 (1993).

[23] C. G. Townsend, N.H. Edwards, C.J. Cooper, K. P. Zetie, C. J. Foot, A. M. Steane, P.Szriftgiser, H. Perrin, and J. Dalibard, Phys. Rev. A 52, 1423 (1995).

[24] C. Duzy and H.A. Hyman, Phys. Rev. A 22, 1878 (1980).

[25] T. N. Chang, J. Phys. B: At. Mol. Phys. 15, L81 (1982).

[26] T. N. Chang and Y. S. Kim, Phys. Rev. A 26, 2728 (1982).

26

3

Dipole-Dipole interactions in a frozen Rydberg gas

Abstract. Here we studied the effects of dipole-dipole interactions in a gas of coldRydberg atoms b. We show that dipole-dipole interactions affect the formation of anUltracold Plasma and directly and quantitatively show their existence by studying thebroadening of microwave transitions. The measured widths scale proportional to thedensity and scale with the fourth power of n, the principal quantum number, as canbe expected from dipole-dipole interactions. In a last set of experiments we show thatRamsey experiments can be used to measure interactions between Rydberg atoms. Wemeasured the loss of contrast as a function of density which we attribute to the effectof dipole-dipole interactions.

bThe work described in this Chapter, to be submitted for publication, has been performedin the group of Gallagher at the University of Virginia

27

Chapter 3.

3.1 Introduction

A Rydberg atom has at least one highly excited electron and is therefore mainly character-ized by the principal quantum number n of the excited electron (typically n ≥10) [1]. Fora typical Rydberg atom, the excited electron effectively sees a core of one unit of positivecharge. This results in properties very similar to a those of a hydrogen atom. When theatoms in a Magneto-Optical Trap (MOT) (see Chapter 5) are excited to Rydberg statesa gas of cold Rydberg atoms is created with a temperature of ≈ 300 µK and a density of≈ 109 cm−3. On a typical time scale of 1 µs, these atoms move only about 3 % of theinteratomic spacing, and as such the Rydberg gas is considered to be ”frozen” [2]. Re-cently these systems became of interest for mainly two reasons. For one, the exaggeratedproperties such as large dipole moments make the system ideal for studying many-bodyinteractions. These can be of practical interest for quantum computing. For example, anexcited Rydberg atom in a cloud of cold atoms may block excitation of other atoms. Thiswould allow to create a single qubit from a cloud of atoms without the need to addressindividual atoms [3–5].Second, recently it has been shown that dipole-dipole interactions in a cold Rydberg gascan trigger the evolution to an Ultracold Plasma (UCP). Two dipole-coupled Rydbergatoms can collide and ionize, triggering an avalanche ionization that results in a UCP [6].In this Chapter we will report the results of an experimental study of the role of dipole-dipole interactions in a frozen Rydberg gas. First we give a brief description of dipole-dipoleinteractions after which we will discuss the main elements of the setup. In the next part wepresent an example measurement showing that the formation of a plasma can be triggeredby driving dipole-dipole interactions. In the next part we show that the dipole-dipole in-teraction leads to line broadening of microwave transitions. In the last section we reportpreliminary results of a Ramsey experiment aimed at studying the effects of dipole-dipoleinteractions in an alternative way.

3.2 Dipole-Dipole interactions

The fundamental interactions between two neutral atoms are their electric and magneticmultipole interactions; the longest range of these is the dipole-dipole interaction. Classi-cally, two dipoles µ1 and µ2 interact through the dipole-dipole interaction V given by [7]:

V =µ1 · µ2 − 3(µ1 · R)(µ2 · R)

R3, (3.1)

with R the unit vector along the direction connecting the two dipoles and R the distancebetween the two dipoles. This equation can be written in a scalar form, assuming parallelor anti-parallel orientation of µ1 and µ2 along the z direction. This is for example the casefor transition dipoles induced by linearly polarized laser or microwave excitation. One thengets:

V =µ1µ2(1− 3 cos2 θ)

R3. (3.2)

28


Here θ is the angle between z and R as indicated in Fig. 3.1a, where the dipoles are as-sumed to be polarized along the z-axes. The angular dependency of Eq. 3.2 makes thatthe interaction can be either attractive or repulsive, depending on the angle θ. This isindicated in Fig. 3.1b.In our experiments we typically work with Rydberg atoms in the n=40 state (the transi-

m2

m1

Rq

z

a b

Figure 3.1: (a) Two dipoles, µ1 and µ2 are polarized along the z direction and separated by adistance R. The angle between the polarization of the dipole moments and the interatomic axesis referred to as θ. (b) Angular dependence of the dipole-dipole interaction. Three equipotentiallines are depicted. The lobes on the top and on the bottom (with an angle θ ranging from 306

to +54 and from 126 to +216) correspond to an attractive interaction. The left and rightlobe (with a angle ranging from 54 to +126 and from 216 to +306) correspond to a repulsiveinteraction.

tion dipole moment µ for a Rydberg atom in the 35s state is approximately µ=1000 ea0 [1]).At a typical density of ρ =109 cm−3, the average interatomic spacing < R >= (4ρπ/3)−1/3 =6 µm. The resulting interaction energy is of the order of 10 MHz. For a 85Rb Rydbergatom the acceleration a, at this interaction strength (|F| = 3µ2/R4) is then of the order25×103 m/s2. The motion induced by this acceleration results on a typical timescale ofabout 1 µs in a displacement of about 15 nm, small compared to the average interatomicspacing (6 µm). In a typical sample, however, there are always pairs at a smaller inter-atomic spacing. For these pairs the motion becomes important and causes ionization. Apair of Rydberg atoms in the 40 s, p state, starting at a infinite separation and movingtowards eachother on the attractive curve can ionize at an interatomic spacing of approxi-mately 2 µm. In Section 3.5 we will show that this ionization can trigger the formation ofa UCP from a Rydberg gas.In a quantum mechanical treatment, two atoms of opposite parity that are dipole con-nected (e.g. s and p atoms) can form two molecular states sp and ps which are degenerateat infinite interatomic spacing [2]. At finite separation, however, they are coupled by thedipole-dipole interaction V . In a very simplified approximation the Hamiltonian of this

29

Chapter 3.

system is given by [8]:

H =

0 V

V 0

. (3.3)

The eigenvalues of this Hamiltonian are ±V with corresponding repulsive (+) and attrac-tive (-) eigenstates (|sp〉 ± |ps〉)/√2. Figure 3.2 gives the energy shifts due to the dipole-dipole interaction for a pair of 39 s and 39 p atoms as have been used in experiments. The

1 2 3 4 5 6 7 8-505

60

65

70

75

Energ

y[G

Hz]

Interparticle distance [mm]

|sp +|psñ ñ

|sp -|psñ ñ

|ssñ

Figure 3.2: Energy levels of the molecular states of a pair of sp atoms relative to the energyof a pair of s atoms. The upper curve represents a repulsive interaction, the bottom curve anattractive one.

transition dipole moment of the atoms is the matrix element µ = 〈s|µz|p〉 which scales asn2. The exact value has to be calculated by solving the Schrodinger equation, which is, forexample, done with a simple Numerov method [9].Atoms on the attractive curve are pulled towards each other and can collide. This canresult in ionization of one atom, while the other makes a transition to a lower energy state.The time it takes for ionization to occur depends on the value of n and the interatomicspacing R. Typical values for n=40 range from about 1 µs for an interatomic spacing of1 µm to a time of 80 µs for an interatomic spacing of 10 µm. Atoms on the repulsivecurve on the other hand, will repel each other resulting in reduced ionization rates. Thesedipole-dipole interactions have implications for the formation of a plasma from a Rydberggas as will be discussed in Section 3.4, but also have possible applications, e.g., for quantuminformation processing [4].

30


Ax

y

Bx

z

W

W

H

I

D

S

M

R

Figure 3.3: (A) Illustration of the top view of the MOT setup showing the MOT beamsM. Depicted is the rod structure R where four rods are pairwise connected. Indicated in thefigure are also the large windows W allowing a practical access of microwave radiation enteringthe vacuum chamber through the horn H. (B) Illustration of the side view of the MOT setup.Depicted is how a voltage pulse is applied to one pair of rods. The other pair is grounded. Theionization laser I enters the chamber from the top and is focussed at the center where the MOTis situated. The field ionization pulse strips the electrons from the Rydberg atoms and pushesthem towards an MCP detector D. The detector signal is visualized on an oscilloscope S.

3.3 Setup

3.3.1 Magneto-Optical Trap and detection

The starting point for the experiments is an MOT (Chapter 5) containing 85Rb atoms. Acommercial vacuum chamber (Kimball physics) is used as trap chamber. The chamber ispumped down by an ion-getter pump to a pressure of approximately 2×10−9 mbar. Onespecial feature of the trap chamber is that it has relatively large side-ports with a diameterof 10 cm, which provide optimum accessibility for microwaves. A schematic drawing of thesetup is given in Fig. 3.3.In the trap, rubidium dispensers are installed which act as a source to create a rubidiumvapor. The lasers for trapping and cooling are home-built diode lasers which typicallyproduce 70 mW at 780 nm. Both lasers are locked to the corresponding atomic resonanceusing saturated absorption [2].The magnetic field gradient necessary to trap the atoms is generated with two coils in aanti-Helmholtz configuration producing a gradient of about 18 Gauss/cm in the y-directionat a current of 12 A. The magnetic field can be switched off in typically 1 ms [2]; this isnecessary since the Zeeman shifts resulting from the MOT magnetic field lead to extrabroadening [10].The number of atoms in the MOT has been determined by fluorescence detection (see

31

Chapter 3.

Chapter 5) to be approximately 5 × 106 atoms at a density of about 3 × 109 cm−3. Thetemperature of the MOT was estimated to be 300 µK.The fraction of atoms in the excited (5P3/2,F=4) state is estimated to be 40 %. From thisstate the atoms are then excited to the appropriate Rydberg state by a pulsed laser thatis described below. The Rydberg atoms could be measured by applying a field-ionizationpulse and detecting the electrons or ions produced with a double MCP detector. The field-ionization pulse is created by applying a voltage pulse of approximately 1 kV (2 µs risetime)to two electrically connected rods as indicated in Fig. 3.3. Opposing these rods are twogrounded rods to make a homogeneous field at the center of the rod structure. The electricfield pulse will state-selectively ionize Rydberg atoms with an ionization threshold belowthe maximum ionization field. The created electrons or ions are pushed towards the MCPdetector where they can be measured. Since the electric field pulse increases approximatelylinearly with time, the different Rydberg states are ionized at different times. This resultsin a difference in arrival time which allows us to distinguish between different states and apossible plasma up to a n of about 50.

3.3.2 Pulsed laser system

To excite 85Rb atoms from the 5P3/2 state to a Rydberg level, pulsed lasers with a wave-length around 480 nm are used. Typically, a pulsed dye laser is used, the advantage beingtheir tunability, low cost and easy operation. However, due to their inherent multi-modebehavior, the fluctuations in the obtained Rydberg state population are 100 %. This makesit unpractical to study processes which scale with density, such as the evolution from acold Rydberg gas to a UCP [2]. To overcome this problem, a single-mode blue pulsedlaser has been built by doubling the frequency of a pulse-amplified continuous wave (cw)infrared laser [11]. A detailed description of the laser system is given by Li [2]. A schematicoverview of the used laser-setup is given in Fig. 3.4.As a single mode cw source, typically a commercial Ti:Sapphire laser (Coherent MBR-110,pumped by a Verdi V-10) is being used. This laser can generate 750 mW at a wavelengthranging between 870-990 nm, our region of interest being around 960 nm. Later on, also acommercial diode laser (Toptica DL100, generating 35 mW at a wavelength around 960 nm)was used. The Toptica laser was less expensive and suffered less from frequency drift, butthe lack of power made pulse-amplification hard in practice.The output of the cw laser is pulse-amplified by using it as a seed light in an array oftwo or three dye amplification stages (depending on the input power and the dye con-centration), where the dye is pumped by the second harmonic of a 10 ns Nd:YAG (532nm) laser. The output of a single amplification stage is a 10 ns laser pulse with the samewavelength as the seed light, albeit it with a factor 1000 more power. Although the systemis conceptually simple to construct, it is very sensitive to alignment. One has to make surethat is effectively the seed light that is being amplified and not the spontaneous emission.Especially when the Toptica diode laser is used, this is a critical issue, because the powerin the spontaneous emission from the first step has to be smaller than the power in theseed light. This was mainly solved by carefully adjusting the amount of green light in the

32


Nd:YAG Verdi Ti:Sapph

Toptica

!"# $

%& '()*& '+

,-. /0

12345637

89: ;:<

Ti:Sapp

Figure 3.4: Schematic representation of the pulsed laser system. The 960 nm cw light of eithera MBR or Toptica laser is pulsed amplified by passing it through two dye cells which are beingpumped by the second harmonic of a pulsed Nd:YAG laser. The pulse amplified 960 nm lightis subsequently doubled by a KNBO3 doubling crystal and sent through a prism to spatiallyseparate the residual amplified spontaneous emission. The result is pulsed 480 nm light with alength of typically 10 ns.

first amplification step. A second step was careful spatial filtering of the seed light fromthe spontaneous emission.The amplified pulse is sent through a potassium niobate (KNbO3) crystal where the fre-quency is doubled [2]. The blue light pulse generated in this way is sent through a prism

33

Chapter 3.

to spatially separate the amplified seed light from the residual amplified spontaneous emis-sion. The light pulse is typically 10 ns long, has an energy of 20 µJ and a linewidth of100 MHz.

Doubler

HP 83620A Switch Outcoupler

TTL pulse Detector

IsolatorTripler

Attenuator

Horn

|gñ

|eñ

Figure 3.5: A schematic representation of the microwave setup. The microwaves coming outa synthesizer are being turned on and off by a microwave switch. The pulses are being sentthrough a directional coupler with a power detector and subsequently sent through an activedoubler. The doubled pulses are then sent throuh a passive trippler and an attenuator andfinally lauched at the atoms through a horn.

3.3.3 Microwave setup

In the experiments, microwaves are often used for transferring populations between Ryd-berg states and for performing spectroscopy on these states. The advantage of microwavesis the high stability and tunability of cw microwave sources. The microwave frequenciesnecessary in the experiments range from from 10 GHz to 200 GHz. A typical setup toproduce microwave pulses with frequencies around 100 GHz is drawn schematically inFig. 3.5. A HP83620A synthesizer is acting as a continuous microwave source, generatingmicrowaves with a frequency ranging between 10 MHz and 20 GHz at a maximum outputpower of 10 dBm. However, for most applications one wants the microwave source to bepulsed. This is achieved by using a microwave switch to chop the continuous microwavescoming out of the HP source. The pulses created in this way can be as short as 80 ns.The microwave switch is followed by an output coupler with a microwave power crystalattached allowing us to measure the pulse shape and relative power. The pulse is subse-quently sent through an active frequency doubler (input frequency ranging between 13.25and 20 GHz, output power approximately 20 dBm) and an isolator to prevent reflectionsfrom damaging the active doubler or the HP source. The doubled pulse is then sent through

34


a passive tripler (input frequency ranging between 25 and 37 GHz, output power around5 dBm) and an attenuator to carefully adjust the output power. Finally, the microwavepulse is launched into the chamber through a horn. The output power is adjusted suchthat the linewidth of the microwave transition due to power broadening of the transitionis as small as possible.

A B

C D

! "#$%

&'()* +,-. /0,1,2 345678 96:578; <=;>8?@ AB6C

Figure 3.6: Spontaneous evolution of a cold Rydberg gas to an Ultracold Plasma. (a) Initialionization processes liberate electrons which leave the cloud. (b) The remaining ions build upa positive potential for the electrons. (c) Enough ions have accumulated to trap the escapingelectrons, which in turn ionize Rydberg atoms. (d) approximately 2/3 of the atoms is ionizedand 1/3 is transferred to lower states.

3.4 From Rydberg gas to ultracold plasma

As mentioned in Section 1.4 until now UCPs have been produced in two different ways:One through means of photoionization [12], and the other by allowing a cold gas of Rydbergatoms to evolve into a UCP spontaneously [13]. In the former case the plasma is created

35

Chapter 3.

Figure 3.7: Example measurement of the electron signal resulting from applying a field ion-ization pulse over a sample of Rydberg atoms and plasma. The field ionization pulse stripsthe electrons from the Rydberg atoms and pushes them towards the MCP detector. Peak Icorresponds to the plasma, the field ionization pulse pulls the plasma apart and pushes theelectrons towards the MCP detector. Peak II represents the electrons stripped from the ini-tial Rydberg state. The signal in III comes from the electrons stripped from the redistributedRydberg atoms.

instantaneously, and as such this process is relatively simple. The evolution from a coldRydberg gas to a UCP, however, is quite complex [14–16]. Understanding the spontaneousevolution dynamics can give a practical handle on the formation of a UCP and as such morecontrol over, e.g., the electron temperature. This might lead to a even higher brightness inthe scheme proposed in Chapter 4. In this Section we will briefly discuss the formation ofa UCP from a cold Rydberg gas and discuss the importance of dipole-dipole interactionsin the process [6].The evolution of a Rydberg gas to a plasma is depicted schematically in Fig. 3.6 and isthought to be as follows:A. In a sample of cold Rydberg atoms, a few atoms are ionized through single-body and two-body mechanisms. Single-body mechanisms include blackbody photoionization, collisionswith the background gas [13] and broadband amplified spontaneous emission (ASE), comingfrom the pulsed laser. In the case of two body processes the main process is consideredto be ionization through dipole-dipole collisions. Independent of how the first ionizationoccurs, almost all the electrons created leave the ion cloud on a timescale of about 50 ns,the heavy and cold ions remain and build a potential well for the electrons.B. If the ionization rate is fast enough (compared to the expansion of the ions) this well willbecome deeper until the freed electrons are being trapped. Once trapped, the electrons willoscillate back and forth through the cloud of Rydberg atoms with which they will collide.The condition for trapping is that the kinetic energy of the electron is smaller than the

36


potential well depth. For a sample of size r containing N ions this can be written as:

kBT ≤ Ne2

4πε0r. (3.4)

Here, kB is Boltzmann’s constant, T is the electron temperature, e is the elementary chargeand ε0 is the electric permittivity of vacuum. Immediately after creation the concept ofa temperature is not strictly valid, it is only after typically 2 times the inverse plasmafre-quency 1/ωp that one can speak of a temperature. Note that Eq. 3.4 can be written asλD ≤ r. For a typical size of 1 mm and 106 ions (corresponding to a density of 109 cm−3)this corresponds to a temperature of about 15000 K for one electron. The electric fieldneeded to extract one electron from this cloud is about 15 V/cm. At this point there is aUCP which will continue to grow.C. The trapped electrons will collide with the remaining Rydberg atoms, ionizing themin turn, starting an ”avalanche” process in which more Rydberg atoms are ionized andredistributed.D, Finally, approximately 2/3 of the initial Rydberg atoms is ionized and 1/3 (these num-bers have been determined empirically [2]) is transferred to lower states.An example measurement of the electron signal coming from a cloud existing of a Ryd-berg gas and a UCP is given in Fig. 3.7. Depicted here is the electron signal measuredwith an MCP when a electric field ramp is applied to a UCP as described in 3.3.1. Theelectric field strips the electrons of the Rydberg atoms and pushes them towards the MCPdetector. Also the plasma electrons are pushed towards the MCP detector. If a plasmahas been formed this can be seen as an early peak (I), in this case at a delay of about0.5 µs. After about 1 µs a large peak appears (II). This represents the original d-state, theatoms producing this peak have not been converted to plasma. The third, broader peakappearing after 1.5 µs (III) comes from the redistributed Rydberg atoms; since these aremore tightly bound they appear at a later time.

T

MicrowaveFrequency n

Electric Field pulse

Gate

Time

delay t1 delay t2

Excitation

Figure 3.8: Experimental procedure to measure the plasma signal as a function of the fre-quency ν of a microwave pulse.

3.5 Dipole-Dipole interactions and plasma formation

In this Section we present the results of a measurement that shows that dipole-dipole in-teractions are important for the formation of a plasma.

37

Chapter 3.

Figure 3.9: Plasma signal obtained as a function of the frequency of a microwave pulse. The leftside corresponds to attractive interactions, the right side corresponds to repulsive interactions.

A cloud of cold Rydberg atoms in a 37s state is prepared. Without any further manip-ulation the cloud does not evolve into a plasma for another 15 µs. In a second set ofexperiments, the procedure above is repeated, but 200 ns after excitation a 500 ns longmicrowave pulse has been added to the atoms driving a s to p transition, about 1 µs afterthe pulse a field ionization pulse is applied. On the red side of the resonance, which isdefined as the transition frequency for one isolated atom, one expects to populate pairs ofatoms on the attractive dipole-dipole curve. These atoms will attract each other which canlead to a collision and subsequent ionization. This is then the initial ionization necessaryto trigger the plasma formation. On the blue side of the resonance, the atom pair is onthe repulsive curve. The atoms repel each other, subsequently there should be no extraplasma formation.Figure 3.9 gives the result of such a experiment. Depicted here is the number of plasmaelectrons as a function of the microwave pulse frequency. The plasma signals have beenmeasured by setting the gate of a boxcar integrator over the plasma signal. For large de-tuning, there is no plasma formation, neither on the blue nor the red side. The closer onegets to the resonance via the red side the more plasma forms until a maximum forms nearthe resonance. On the blue side of the resonance the plasma signal drops rapidly. Thisshows that attractive dipole-dipole interactions lead to plasma formation.Li et al. performed a very detailed set of experiments, proving that for large values ofn, it are mainly dipole-dipole interactions driving the plasma formation. The experimentsperformed by Li et al. [6] show that the dipole-dipole interaction can give a practical handleon the formation of a plasma from a Rydberg gas. It can be used to create well-controlledsamples of plasma from a pure s-state cloud of atoms. This is otherwise hard in practicesince it requires a high density. Another, more promising application, is that it can be used

38


to reduce the plasma temperatures. By sweeping a microwave pulse on the blue side of theresonance atom pairs are repelled from each other. This diminishes the lack of correlationin the Rydberg atom cloud. As such, the correlation heating as discussed in Section 1.3can be reduced, resulting in lower electron and ion temperatures and subsequently largervalues for Γ, the coupling parameter.

3.6 Line broadening due to dipole-dipole interactions

3.6.1 Introduction

In the previous Section we showed that dipole-dipole interactions between Rydberg atomscan be large enough to trigger plasma formation in a gas of cold Rydberg atoms. Recentlyit has been argued that dipole-dipole interactions in a gas of cold Rydberg atoms can alsobe relevant for quantum computing [4]. In a standard recipe for a quantum computer,a well defined system of qubits with strong interactions is needed. Ryabtser et al. [3],amongst many, propose to use the hyperfine sub-levels of the S ground state of a alkaliatom (e.g. sodium or rubidium) as qubit states. Following Lukin et al. [4] they explainhow, using dipole-dipole interactions as a mediator, a conditional quantum phase gatecan be constructed. This quantum logic gate is of importance since it can be convertedinto a conditional not gate, from which any other unitary operation can be constructed.Lukin et al. also discuss how the so-called ”dipole blockade” can be used to store qubits inatomic ensembles exceeding many optical wavelengths. This would facilitate the practicalconstruction of a quantum computer, removing the necessity to address single atoms.In this Section we present work that is a extension of the pioneering work performed byMartin et al. [5]. They studied the effect of dipole-dipole interactions between Rb atoms onradiative transitions. In their experiments they observed line broadening in the microwavespectra of cold Rydberg atoms due to dipole-dipole interactions. They created a sampleof cold 45d5/2 (50%) and 46d3/2 (50%) (these states are strongly dipole coupled) Rydbergatoms and measured the width of the 45d5/2→46d5/2 microwave transition as a functionof the Rydberg atom density. In their experiment they observed a broadening of up to 55kHz at a maximum density of about 107 cm−3.In our experiments we performed similar experiments on s states which are dipole coupledto their nearest p state. We measure broadening of up to 24 MHz at a maximum Rydbergdensity of up to a few times 109 cm−3. We measured the effect of density dependentbroadening as a function of the principal quantum number n to verify the n4 scaling thatis expected from dipole-dipole interactions. As a last extension, we present a simple modelgiving an appreciable agreement with our measurements.

39

Chapter 3.

Figure 3.10: Modeled line shapes due to broadening of microwave transitions due to dipole-dipole interactions (39 s→p, 109cm−3). The solid line represent the direct result (Eq. 3.7)showing a dip at the center. The dashed line is the result of convoluting the direct result witha sinc2 function of width 2 MHz.

Figure 3.11: The dashed line is the result of the model convoluted with a 2 MHz sinc2 function(39 s→p, 109cm−3). The solid line is a Lorentzian fit to the modeled curve.

3.6.2 Line shape model

As discussed in Section 3.2 the effect of dipole-dipole interactions results in an effectivefrequency shift ∆ν given by Eq. 3.5:

∆ν =µ1µ2

4πε0

(1− 3 cos2 θ)

hR3. (3.5)

40


Figure 3.12: Widths (FWHM) derived from fitting modeled lineshapes (convoluted with a 2MHz wide sinc2 function) with a Lorentzian. The widths are derived for different values of theprincipal quantum number n and density ρ. The widths increase approximately linearly withdensity and the slopes scale as n4.

This energy shift is dependent on the angle θ between the dipole moments and the inter-atomic axis, and R gives the distance between the two dipole moments. A representation ofthe dipole-dipole interaction and the corresponding equipotential lines is given in Fig. 3.1.It is exactly this frequency shift that results in the observed line broadening. Atom pairsat different interatomic spacings R feel different frequency shifts. By changing the densityρ of the Rydberg atoms we change the average interatomic spacing < R >= (4πρ/3)−1/3.In a first order approximation, this results in a linear dependence of the frequency shift onthe density ρ. Martin et al. measured the widths of microwave transitions as a function ofdensity and used a Lorentzian convoluted with a sinc2(πfT ) function to fit the data. Thesinc2 represents the transform limited linewidth of their microwave pulse. This functionmatched their data well, and the Lorentzian line shape was supported by numerical simu-lations.Here we present a simple model for the observed line shapes based upon work done byNiemax and Pichler [17] on the effect of dipole-dipole interactions on principal Cs-lines.The probability distribution P (R)dR for finding an atom at separation R in a cloud withdensity ρ in the nearest neighbor approximation is given by [17]:

P (R)dR = 4πR2ρ exp−4ρπR3

3dR. (3.6)

Inserting Eq. 3.5 in Eq. 3.6 results in:

P (∆ν)d∆ν =ρ

3∆ν2

(µ1µ2(1− 3 cos2 θ))

ε0hexp

−ρ(µ1µ2(1− 3 cos2 θ))

3∆νε0h

d∆ν. (3.7)

41

Chapter 3.

Strictly speaking Eq. 3.7 is only valid in the wings of the distribution where the binaryapproximation is valid. This means that the interaction with only one neighboring atom isincluded. Note that in the wings of the distribution Eq. 3.7 has the form of a Lorentzianwing (P (ν) ∝ 1/(∆ν)2).The next step is to integrate over the angular part of the distribution. In Fig. 3.10 one cansee the modeled line shape as a function of detuning ∆ν assuming a density ρ = 109 cm−3

and Rb Rydberg atoms in the n =39s state. Note that the distribution is asymmetric. Thiscomes from the fact that the shape of the dipole-dipole equipotential lines are asymmetricfor positive and negative detunings. The result from integrating Eq. 3.7 has Lorentzianwings and a sharp dip at the center. This dip was also observed in the numerical simu-lations of Martin et al., and comes from the fact that in the model there are no particlesat an infinite separation, i.e. at zero frequency shift. We see that at moderate broadening(up to 20 MHz) there is only a small effect of this central dip. Also depicted in Fig. 3.10is the result of integrating Eq. 3.7, convoluted with a 2 MHz broad sinc2 function, similarto Martin et al.. The convolution has been performed to include the effect of the finitelength of the microwave pulse and power-broadening. The resulting function fits well witha Lorentzian as indicated in Fig. 3.11.In Fig. 3.12 we plot the results of fitting the modeled line shapes with a Lorentzian, show-ing the full width half maximum (FWHM) as a function of density and for different valuesof the principal quantum number n. From these plots it is clear that the width scales ap-proximately linearly with density. The slopes of these linear parts scale as n4 as expected.

3.6.3 Experiment

The experimental apparatus has been described earlier in this Chapter. The experimentalroutine is about the same as indicated in Fig. 3.8. The center part of an MOT containingapproximately 2×106 85Rb atoms in the 5P3/2 state is excited to a Rydberg state by thepulsed laser setup described in Section 3.3; about 200 ns after excitation a 500 ns longmicrowave pulse is launched at the atoms with the microwave-setup described in Section3.3; 200 ns later we apply a field ionization pulse and the signal detected on the MCP ismeasured. We place the gate of a boxcar integrator around the signal that results fromfield ionization of atoms in the p-state, and scan the frequency of the microwave pulse. Inour experiment we used four different n states, i.e. 30, 34, 39 and 44. We changed thedensity by reducing the intensity in the MOT repumper laser. The relative density wasmeasured by integrating the total signal on the MCP. The absolute Rydberg atom densityis estimated from the MOT density but is uncertain within a factor 2 and ranges up to3×109 cm−3.

3.6.4 Results

In Fig. 3.13 we depict two traces measured for a 39 s → p transition, both at a low densityand a high density. It is clear from this figure that the signal is broader at a high density.

42


Figure 3.13: Measured lineshapes of a 39 s →p transition for a high (upper trace) and a lowdensity (lower trace). The top curve is fitted with a Lorentzian (dashed curve).

0.1 1

0.0

0.2

0.4

0.6

0.8

1.0

Energ

y[a

.u.]

interatomic distance [a.u.]

|sp + ps> | >

|sp - ps> | > |ss>

|pp>

Figure 3.14: Relevant energy levels for the line broadening experiments. Atoms from thes state are excited to a molecular state depending on the interatomic distance. For a largeinteratomic distance the atoms are virtually isolated, and both atoms are excited to the p state.

In our analysis, we fitted the measured signals with the sum of two Lorentzian functions.One, broad, Lorentzian function represents the signal due to dipole-dipole interactionsas described above. A second, narrow Lorentz function represents atom pairs that areseparated by a large interatomic distance and, and as a result, are virtually isolated. Forthese atom pairs, both atoms are excited to the p state. This is depicted schematically inFig. 3.14. In Fig. 3.15 we plotted the fitted widths of the broad Lorentzian (full width half

43

Chapter 3.

Figure 3.15: Fitted widths as a function of density for four different values of n, i.e.n=30, 34, 39, 44. The dashed lines are linear fits to the data with a fixed value of 2 MHzat zero density.

Figure 3.16: The broadening per unit of density as a function of n. The line is a linear fitthat is fixed to go through the origin.

maximum) as a function of the density for the 4 different n states. For low densities, thesignals fit better with a single Lorentzian, the fitted amplitude of the narrow Lorentzianis about ten times smaller than the fitted amplitude of the broad Lorentzian. It is onlyfor the high density points that the fits with two Lorentzians give better results. For the

44


Figure 3.17: Comparison between the measured line shape and the modeled line shape for a39 s→p transition at a density of 109cm−3.

broadest curves, the amplitude of the broader Lorentzian was of the same order as theamplitude of the narrow Lorentzian. The width of the narrow peak was typically 3 MHz.The fitted width of the transitions goes up with increasing Rydberg density. Extrapolationof the curves to low densities leads to a width of approximately 2 MHz for all four curves.Depicted are also linear fits to the widths versus density-curves (with a fixed value of 2MHz at zero density) giving satisfactory agreement. The higher the principal quantumnumber n the broader the distribution gets at the same density. In Fig. 3.16 we plotted theslopes obtained from fitting the curves in Fig. 3.15, as a function of n4. These points showa clear linear behavior as expected for dipole-dipole interactions. In Fig. 3.17 we plotteda measured curved and compared it to the modeled curve that gave the best comparison,i.e. at a density of almost 2×109 cm−3, which is within a factor 2 of what we expected.

3.7 Ramsey experiments

As an alternative way to study the effect of (dipole-dipole) interactions in a gas of coldRydberg atoms we also performed a Ramsey experiment [18]. This is a sensitive techniqueto measure interactions. When a two level atom with a ground state |g〉, an excited state|e〉) and a resonance frequency ω0 interacts with a square pulse of radiation with frequencyω, the probability of excitation |c(t)|2 is given by [18]:

|c(t)|2 =Ω2

W 2sin2

(Wt

2

), (3.8)

where Ω is the Rabi frequency and W 2 = Ω2 + (ω − ω0)2. This function represents a sinc2

function. A pulse of length τp has a frequency width of 1/τp. When such a pulse transfers

45

Chapter 3.

tp

Microwave pulse Electric Field pulse

Gate

Time

T

Excitation

tp

Microwave pulse

Figure 3.18: Experimental procedure to perform a Ramsey experiment. Two π/2 pulses oflength τp are given separated by a time T . As a diagnostic the gate of a boxcar integrator isset around the p-state signal. The lower left trace represents the signal for a single π/2 pulse(Eq. 3.8), the lower right trace represents the single for two π/2 pulses (Eq. 3.9).

the atom to a superposition of its ground and excited state with equal amplitudes it iscalled a π/2 pulse. In a typical Ramsey experiment, two π/2 pulses separated by a time Tare given (we assume the pulse to have a length τp). If there is no interaction in an atomicsample, the amplitude of the excited state after this pulse sequence is given by:

|c|2 =2Ω2

W 2sin2

(Wτp

2

)cos2

(δT

2

), (3.9)

where δ = ω − ω0 is the frequency detuning.When there is interaction between the atoms in the sample, this leads to a phase shiftduring the interaction time, which has an effect on the signal. The amplitude of theexcited state population can be written as:

|c|2 =2Ω2

W 2sin2

(Wτp

2

) ∫P (φ)

∣∣∣∣cos2

(δT + φ

2

)∣∣∣∣ dφ (3.10)

Here P (φ) is the probability that a phase φ has been picked up in the time T , Eq. 3.10 isthen written as:

|c|2 =Ω2

W 2sin2

(Wτp

2

)(1 + C cos(δT )). (3.11)

Here C =∫

P (φ) cos(φ)dφ (the term containing∫

P (φ) sin(φ)dφ equals zero since we as-sume P (φ) to be a symmetric function around zero). We call C the contrast, since itdetermines the actual contrast of the measured Ramsey curves. Note that if C=1, Eq. 3.11equals Eq. 3.9. In case all the atoms feel the same interaction, and consequently pickup the same phase shift φ0, the effect of the interaction is merely an overall phase shift(P (φ) = δ(φ− φ0)). However, when there is a spread in the picked-up phase, the result isthat the contrast of the signal diminished.

46


In our experiment we measured the contrast as a function of Rydberg density. For everypossible interatomic distance R, the atoms pick up a different phase φ = 2π∆ν(R)T duringthe time T (see Eq. 3.1), due to dipole-dipole interactions. When changing the density ofthe sample, we increase the spread in the possible values for the interatomic distance R,resulting in a larger spread in the picked-up phase, which in turn reduces the contrast.In our experiment we applied two 150 ns long microwave pulses, separated by about 500ns, with frequency ω. These pulses couple the s to the p state. After the second pulse wemeasured the population of the excited (p state) state by setting the gate of our boxcarintegrator around the p state signal. In Fig. 3.19 we plotted two measured curves for a lowand a high density which show clear Ramsey patterns, together with the fits according toEq. 3.11. For the low density the contrast is higher than for the high density. To obtainthe contrast from the measurements we fitted the signals as in Fig. 3.19 with Eq. 3.11,with C,W, T and a scaling factor as free parameters. The fitted contrast as a function ofdensity is given in Fig. 3.20, where the density is given in terms of frequency width.We compare the fitted values with a model assuming loss of contrast due to dipole-dipoleinteractions. In the previous section we measured broadening of microwave transitions andshowed that the broadening was due to dipole-dipole interactions. The measured curvescould be well fitted with a Lorentzian. As a consequence, here we may take the probabilityP (ν) that an atom undergoes a certain frequency to be a normalized Lorentzian of widthw. We then get for the contrast:

C =

∫P (ν) cos(2πνT )dν. (3.12)

In our analysis we calculate the contrast as a function of width. In our measurements weknow the density within a factor of two to three. We translated the density to frequencywidth following the analysis from the previous section. We estimate the maximum densityto be in the range of 1-2×109cm−3, resulting in a width of about 7-10 MHz, according tothe analysis in Section 3.6. The relative densities are known with an accuracy of about10 %. The results of the analysis are plotted in Fig. 3.20. The agreement between themodel and the experimental data is satisfactory, indicating that indeed the dipole-dipoleinteraction is mainly responsible for the loss of contrast. Both the measurements on dipole-dipole broadening and the Ramsey experiment are consistent in the sense that they havethe same density to frequency width conversion within a factor of two.

3.8 Discussion and conclusions

We presented an extensive set of measurements studying the density dependent broad-ening of microwave transitions for cold Rydberg atoms. We found that the width of thetransitions scales approximately linearly with density. We measured the density dependentbroadening for different values of n, the principal quantum number and found that thisscales as n4. Following Martin et al. [5] we explain this broadening as being due to dipole-dipole interactions. A simple model assuming only binary interactions gives a reasonable

47

Chapter 3.

Figure 3.19: Measured Ramsey curves for a high (lower trace) and a low (top trace) density.The contrast of the high density curve is lower than that of the low density one. The dashedlines are the results of fitting the data with Equation 3.11.

comparison, we measured widths of up to 24 MHz.In a second set of experiments, we performed Ramsey experiments where we measured theloss of contrast as a function of density. A simple model explaining the loss of contrast dueto dipole-dipole interactions gives a good agreement. Both data sets are consistent witheach other within a factor of two.

48


Figure 3.20: The fitted contrast as a function of atomic density. The grey line is the result ofour model.

49

Chapter 3.

Bibliography

[1] T.F. Gallagher, Rydberg Atoms (Cambridge University Press, 1994).

[2] W. Li, Ph.D. Thesis (University of Virginia 2005).

[3] I. I. Ryabtsev, D. B. Tretyakov and I. I. Beterov, J. Phys. B: At. Mol. Opt. Phys. 38, 421(2005).

[4] D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Cote, and M. D. Lukin, Phys. Rev. Lett.85, 2208-2211 (2000).

[5] K. Afrousheh, P. Bohlouli-Zanjani, D. Vagale, A. Mugford, M. Fedorov, and J. D. D. Martin,Phys. Rev. Lett. 93, 233001 (2004).

[6] W. Li, P. J. Tanner, and T. F. Gallagher, Phys. Rev. Lett. 94, 173001 (2005).

[7] J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids(John Wiley & Sons, New York, 1954), pp. 955−964.

[8] M. R. Doery, E. J. D. Vredenbregt, J. G. C. Tempelaars, H. C. W. Beijerinck, and B. J.Verhaar, Phys. Rev. A 57, 3603 (1998).

[9] M. L. Zimmerman, M. G. Littman, M. M. Kash, and D. Kleppner, Phys. Rev. A 20, 2251(1979).

[10] W. Li, I. Mourachko, M. W. Noel, and T. F. Gallagher, Phys. Rev. A 67, 052502 (2003).

[11] O. Svelto, Principles of Lasers (Plenum Press, New York and London, 1998).

[12] T.C. Killian, S. Kulin, S. D. Bergeson, L. A. Orozco, C. Orzel, and S. L. Rolston, Phys.Rev. Lett. 83, 4776 (1999).

[13] M. P. Robinson, B. Laburthe Tolra, Michael W. Noel, T. F. Gallagher, and P. Pillet, Phys.Rev. Lett. 85, 4466 (2000).

[14] T. Pohl, T. Pattard, and J. M. Rost, Phys. Rev. A 70, 033416 (2004).

[15] F. Robicheaux, and James D. Hanson, Phys. Rev. Lett. 88, 055002 (2002).

[16] W. Li, M. W. Noel, M. P. Robinson, P. J. Tanner, T. F. Gallagher, D. Comparat, B.Laburthe Tolra, N. Vanhaecke, T. Vogt, N. Zahzam, P. Pillet, and D. A. Tate, Phys. Rev.A 70, 042713 (2004).

[17] K Niemax, and G. Pichler, J. Phys. B: At. Mol. Opt. Phys. 7, 1204 (1974).

[18] C.J. Foot Atomic Physics (Oxford University Press, 2005).

50

4

Ultracold electron source

Abstract. We propose a technique for producing electron bunches that has the poten-tial for advancing the state-of-the-art in brightness of pulsed electron sources by ordersof magnitude c. In addition, this method leads to femtosecond bunch lengths withoutthe use of ultra-fast lasers or magnetic compression. The electron source we propose isan ultracold plasma with electron temperatures down to 10 K, which can be fashionedfrom a cloud of laser-cooled atoms by photo-ionization just above threshold. Here wepresent results of simulations in a realistic setting, showing that an ultracold plasmahas an enormous potential as a bright electron source.

cThe work described in this Chapter, is published in Phys. Rev. Lett. 95, 164801 (2005)

51

Chapter 4.

Ultra-short, high-brightness electron bunches find application in many areas of sci-ence and technology. For instance, they are used as time-resolved probes for the solid-liquid phase transition of surfaces heated by ultra-fast lasers [1] and for observing transientstructure in femtosecond chemistry [2]. Such electron bunches are furthermore a sine quanon for the realization of high-brightness X-ray sources [3], in particular the hard X-rayfree-electron laser [4], enabling time-resolved studies in new parameter regimes in physics,chemistry, and biology [5, 6]. An extremely exciting prospect is single-shot, sub-picosecondtime-resolved electron microscopy [7], which may become possible with continuing advancesin pulsed electron sources with ever higher brightness.

The brightness B, i.e., the current density per unit solid angle and per unit energyspread, is a comprehensive figure of merit for particle beam quality. It is proportional tothe 6-dimensional (6D) phase space density of an accelerated particle bunch. The highestbrightness measured at present is produced by carbon nanotube (CNT) field-emitters,recently developed for electron microscopy [8]. CNTs can produce up to 1 µA of continuouscurrent from a few nm2 source area. The more established, pulsed, picoseond photo-emission-based guns [9], on the other hand, produce peak currents up to a few 100 Afrom a few mm2 source area, and are typically 2 − 3 orders of magnitude less bright. Ina recent hybrid approach, 10 ns electron pulses of 0.1 A current are extracted by pulsedphoto-emission from micron-sized needle cathodes, suggesting a brightness comparable tocontinuous CNT emitters [10].

The strategy for improving the brightness that both photo and field-emission-basedsources have in common up to now, is to increase the current density at the source. Inall cases the angular spread, which is determined by the effective electron temperatureof the source T (typically 103 − 104 K), is kept constant. In this Letter we discuss acompletely different approach, which aims instead at reducing the electron temperature T .This new source concept is based on pulsed extraction of electrons from an Ultracold Plasma(UCP), which is created from a laser-cooled cloud of neutral atoms by photo-ionization justabove threshold [11]. Such plasmas are characterized by electron temperatures down to10 K [12], i.e. 2− 3 orders of magnitude lower than photo or field-emission based sources,implying a potentially enormous gain in brightness. Here we present results of numericalsimulations based on realistic and well-established experimental conditions which showthat ultracold, high-current, sub-picosecond electron bunches can be produced, orders ofmagnitude brighter than the best ultra-short pulsed electron sources today.

In practice, the quality of pulsed electron beams is usually expressed in terms of thetransverse brightness B⊥, which is a measure for the current density per unit solid angle.The full 6D brightness B is proportional to the ratio of B⊥ and the energy spread. For abeam traveling in the z-direction [13]

B⊥ =I

4π2εxεy

, (4.1)

where I is the peak current and εx and εy the normalized root-mean-squared (rms) emit-

52


tance in, respectively, the x and y-direction,

εx =1

mc

√< x2 >< p2

x > − < xpx >2. (4.2)

Here <> indicates averaging over the distribution, m is the electron mass, c the speed of

light, and px = γmvx, with vx the x-velocity and γ = (1− (v/c)2)−1/2

. The emittance ε isa Lorentz-invariant measure for the focusability of the beam. For a beam with a gaussiandistribution in transverse (x, px, y, py) phase space, (γ2 − 1)B⊥ is equal to the peak valueof the current density per unit solid angle. In the field of electron microscopy the beamquality is usually expressed in terms of the reduced brightnes Br = eB⊥/mc2, with e theelementary charge [14]. For a thermal electron source with peak current density J one mayshow that [14]

B⊥ = mc2J/πkT, (4.3)

which clearly illustrates the advantage of reduced electron temperatures.The best performing pulsed picosecond sources are radio-frequency (rf) photoguns,

in which electron bunches are created by pulsed-laser photoemission and subsequentlyaccelerated in rf fields of, typically, 100 MV/m. The rf photogun of the Accelerator TestFacility at Brookhaven National Lab can produce 0.5 nC electron bunches with I ≈ 120A and εx ≈ 0.8 µm, corresponding to a beam brightness B⊥ = 5×1012 A/(rad2m2), Br =1× 107 A/(rad2m2V) [9], about an order of magnitude smaller than the thermal brightnesslimit given by Eq. (4.3). The emittance achieved is limited by nonlinear space chargeforces. Recently, it was shown that the detrimental effect of space charge forces can bevirtually eliminated by proper shaping of the radial intensity profile of a femtosecondphoto-excitation laser [15]. This would make it possible to reach the thermal brightnesslimit corresponding to T = 103 − 104 K.

Our proposed pulsed UCP source is similar to an rf photogun in the sense that itproduces (sub)ps bunches with a comparable bunch charge from a comparable source area.The much lower temperature of the source, however, implies a potential increase of thebrightness by up to three orders of magnitude. to realize a UCP source in practice wepropose a four-step procedure, illustrated schematically in Fig. 4.1:(I) A cold (T < 1mK) cloud of atoms is trapped in a Magneto-Optical Trap (MOT) in avolume of a few mm3 with densities up to 1018 m−3 [16].(II) Part of the cold atom cloud is excited to an intermediate state with a quasi-continuous,µs laser pulse.(III) Then, a pulsed laser beam propagating at right angles to the excitation laser, ionizesthe excited atoms only within the volume irradiated by both lasers. Here a UCP is formed[11]. In this way electron bunches of up to 100 pC can be created in a volume of ∼ 1 mm3.By exciting the atoms to just above the ionization limit with a multi-ns laser pulse, theelectrons are created at T ≈ 1 mK. Ponderomotive heating of the free electrons in theoptical field is negligible. Subsequently, ns-timescale heating processes inside the plasmaincrease the temperature to T ≈ 10 K1 [12].

1An initial temperature of 40 K is more realistic [12], but has no significant effect on the final outcome.

53

Chapter 4.

Figure 4.1: Schematic of the four-step procedure to realize a pulsed UCP electron source.

(IV) The bunches are extracted by an electric field at least an order of magnitude strongerthan what is minimally required for pulling the electrons and ions apart. For a 1-mm-sized 100 pC bunch this typically means applying a voltage of 1 MV across a 1 cm gap,which should be switched on extremely rapidly (< 1ns) to prevent space-charge-inducedemittance growth during acceleration. Such rapid switching of high voltages is possible byusing, for example, laser-triggered spark gap technology [17]. For an MOT a loading rateof over 1× 1011 atoms/s is possible [16], so a total charge of up to 10 nC may be extractedper second.

The extracted bunches are automatically compressed in the drift space after accelerationbecause the electrons in the back of the bunch experience a larger acceleration potentialdifference than those in the front. This ”velocity bunching” leads to sub-ps bunch lengths.

The fact that the initial electron density is proportional to the product of the intensitiesof the excitation and the ionization laser beams in the region of overlap, offers an excellentopportunity for control over the initial charge distribution. As we showed recently [15], thedetrimental effects of space charge forces may be virtually eliminated by the combinationof lowering the dimensionality of the bunch and proper shaping. A highly desirable initialcharge distribution, for example, is a pancake bunch (bunch length much smaller thanradius R) with a half-circle radial charge density distribution [15]

ρ(r) ∝√

1− (r/R)2. (4.4)

Such a distribution automatically evolves into a uniform, ellipsoidal bunch, which is charac-terized by perfectly linear space charge fields and thus zero space-charge-induced emittance

54


growth. A second initial distribution is a cigar bunch (radius R much smaller than bunchlength) with a parabolic longitudinal charge density distribution, which will also evolveinto a uniform, ellipsoidal bunch. Using Eq. (4.2) one may show that the normalized rmsemittance of such objects is given by

ε = R√

kT/5mc2. (4.5)

Due to the two-step ionization scheme it is possible to create the UCP in either the’half-circle-profile pancake’ or the ’parabolic-profile cigar’ configuration, each with its ownspecific advantages: as we will show, the pancake bunches are characterized by a highcharge, a small energy spread, and robust, stable behavior, while the cigar bunches offer alow emittance and high compressibility. Note that, in spite of the ideal initial distribution,rapid acceleration is still necessary, because initially the electron bunch is still subjectedto nonlinear forces due to the ion cloud. The time it takes to separate the electrons fromthe ions should be kept as short as possible.

To investigate the feasibility of the proposed source, we performed an extensive set ofsimulations with the General Particle Tracer (gpt) code [18]. The starting point is anMOT containing rubidium atoms in a spherically symmetric gaussian density distributionwith an rms radius of 2 mm and a density in the center of 1× 1018 m−3.

To create a pancake bunch, a fraction of the atoms is excited with a radial distributiongiven by Eq. (4.4), with R=2 mm. The ionization laser beam then cuts out a longitudinalslice of 15 µm thickness. Assuming an overall ionization efficiency of 50%, this results in10 pC charge. To create a cigar bunch, the atoms are excited within a radius of 80 µm fromthe axis. Subsequently, the ionization laser cuts out a parabolic longitudinal density profilewith a total length of 1 mm, resulting in 1 pC of charge. The initial electron temperatureof both bunches is set at 10 K.

For the accelerating stage, a cylindrically symmetric field geometry is assumed in whichboth the cathode and the anode are thin conducting plates with a circular hole of 5 mmradius, separated by a distance d = 20 mm, as is shown in Fig. 4.2(a). The hole inthe cathode enables optical acces for both the trapping and excitation laser beams. Theelectric field is modeled by the product of an electrostatic field due to a voltage V0 = 1 MVacross the diode, calculated with Superfish [19], and a linearly increasing time factort/τr, with τr = 150 ps. The time dependent electric field gives rise to an azimuthalmagnetic field Bφ, which is calculated on the basis of the Superfish field map. Nearthe axis Bφ ≈ −V0r/(2c

2τrd), acting as a positive lens. The validity of this approach hasbeen verified with the 3D time domain code MW-Studio [20]. In the gpt simulationsspace charge forces are calculated with a 3D anisotropic Poisson solver, tailor made forbunches with extreme aspect ratios [21]. The effect of the ions on the electron bunchduring extraction has also been included, but turns out to be negligible.

In Fig. 4.2(a) the acceleration electric field geometry is shown, indicated by equipoten-tial lines, as well as the radial bunch envelope as a function of z. In Fig. 4.2(b) and (c),respectively, the rms normalized emittance ε and the rms bunch length σz (in fs) are plottedas a function of z. The pancake bunch is started upstream at z = −5 mm. As a result the

55

Chapter 4.

(a)

(b)

(c)

Figure 4.2: (a) Field geometry and radial bunch envelope as a function of z; (b) rms normalizedemittance as a function of z; (c) rms bunch length as a function of z. Solid lines: cigar bunch;dashed lines: pancake bunch.

bunch is initially focused by the non-uniform electric field, as can be seen in Fig. 4.2(a),thus partially compensating for the radial space charge forces and the defocusing ”exitkick”of the diode. The final energy of the pancake bunch is 470 keV. The cigar bunch isstarted at z = 0, as its small initial radius does not require any additional focusing. Thefinal energy of the cigar bunch is 270 keV.

As is shown in Fig. 4.2(b), ultra-low normalized emittances are achieved of the order ofε ≈ 0.1 µm for the pancake bunch and even lower for the cigar bunch. The behavior of ε as afunction of z is similar for both configurations. Initially ε ≈ 0.04 µm for the pancake bunchand ε ≈ 0.0015 µm for the cigar bunch, in agreement with Eq. (4.5). After initiation, ε firstrises sharply due to space charge forces and then levels off to slow monotonous growth,only briefly interrupted by a temporary rise while passing through the non-uniform fieldin the hole of the anode.

Figure 4.2(c) shows that in the proposed setup sub-ps bunch lengths can be realizedindeed, without the use of ultra-fast lasers or magnetic compression. Compression is solelydue to velocity bunching, which is particularly efficient for the cigar bunch: at z = 42 mman rms bunch length σz = 20 fs is achieved, resulting in a peak current I > 25 A. Theposition of the bunch length minimum can be conveniently adjusted over a range of severalcm by shifting the initial position by a few mm.

The pancake bunch, on the other hand, almost immediately reaches a respectable bunchlength value of σz = 150 fs, corresponding to I = 25 A, which is maintained over manycm of its trajectory. This steady behavior reflects a balance between space charge forcerepulsion and velocity bunching, which is relatively weak due to the small accelerationpotential difference experienced by pancake bunches.

The combination of ultra-low emittances and ultra-short bunch lengths results in ex-

56


tremely high brightness values: after leaving the diode, the pancake bunch attains a con-stant value B⊥ = 5×1013 A/(rad2m2), Br = 1×108 A/(rad2m2V). This value is an order ofmagnitude higher than state-of-the-art rf photogun performance [9]. The cigar bunch per-forms even better: at the bunch length minimum, z = 42 mm, B⊥ ≥ 5× 1014 A/(rad2m2),Br ≥ 1 × 109 A/(rad2m2V), comparable to CNT performance [8]. The cigar bunch con-figuration clearly offers the highest peak brightness and the shortest bunch lengths, butonly at specific positions and with a relatively large energy spread. The pancake bunchis typically less bright, but exhibits robust, stable behavior with a relatively small energyspread.

We therefore conclude that UCP-based electron sources have enormous potential foradvancing the state-of-the-art in ultra-short electron bunch brightness. This potential gainin brightness is due to the combination of a low initial thermal emittance and a shortbunch length that results from velocity bunching. The brightness values resulting from thesimulations are impressive, but still 1 − 2 orders of magnitude removed from the thermallimit. In principle, therefore, even higher brightness values may be attained, for exampleby optimizing the accelerating diode structure.

Interestingly, UCP sources do not require any conditioning and do not suffer from aging,in contrast with most solid state (photo and field) emitters (cf., e.g., [8]). Essentially, foreach shot a new, fresh source is used, which may be beneficial in terms of reproducibilityand lifetime. Furthermore, a pulsed UCP source is equally suitable for producing ultra-bright ion beams, which may be of interest for Focused Ion Beam (FIB) applications.

Finally, we speculate that even lower electron temperatures may be realized, for examplewhen the atoms are not photo-ionized but excited to a high level. From there they canbe ionized by a high-voltage pulse, which simultaneously extracts the electrons from theplasma. If the excited level is chosen such, that in a field of ≈ 107V/m only its highestStark-shifted sublevel is ionized, then a pancake slice of electrons is extracted from theplasma within a few ps. Since this is much faster than the time scale (100 ps) at whichplasma heating occurs, mK electron temperatures may be retained.

It is intriguing to note that at T=1 mK, the electron thermal De Broglie wavelengthλth = h/

√2πmkT is 2.4 µm, comparable to the interparticle distance n−1/3 at densities

n ≈ 1018 m−3. This would imply the production of fermi degenerate electron bunches, andthus the ultimate, fundamental quantum limit in electron beam brightness.

We thank M.J. van der Wiel, H.W.C. Beijerinck, and M.C.M. van de Sanden for stim-ulating discussions. This work is part of the research programme of the Stichting voorFundamenteel Onderzoek der Materie (FOM, financially supported by the NederlandseOrganisatie voor Wetenschappelijk Onderzoek (NWO)).

Bibliography

[1] B.J. Siwick et al, Science 302, 1382 (2003).

[2] J. Cao et al, Proc. Natl. Acad. Sci. USA 96, 338 (1999).

[3] R.W. Schoenlein et al, Science 274, 236 (1996).

57

Chapter 4.

[4] TESLA, the superconducting electron-positron linear collider with an integrated X-ray laserlaboratory, DESY technical design report (2001),available from http://www-hasylab.desy.de/facility/fel/.

[5] C.W. Siders et al, Science 286,1340 (1999).

[6] F.V. Hartemann et al, Phys. Rev. E 64, 016501 (2001).

[7] V.A. Lobastov et al, Proc. Natl. Acad. Sci. USA 102, 7069 (2005).

[8] N. de Jonge, M. Allioux, J.T. Oostveen, K.B.K. Teo, W.I. Milne, Phys. Rev. Lett. 94,186807 (2005).

[9] For a recent overview, see P. Piot, in The physics and Applications of High BrightnessElectron Beams, edited by J. Rosenzweig, G. Travish, and L. Serafini (World Scientific,Singapore, 2003), p. 127.

[10] C.H. Garcia and C.A. Brau, Nucl. Instr. Meth. Phys. Res. A483, 273 (2002).

[11] T.C. Killian, S. Kulin, S. D. Bergeson, L. A. Orozco, C. Orzel, and S. L. Rolston, Phys.Rev. Lett. 83, 4776 (1999); for a recent review, see T.F. Gallagher et al., J. Opt. Soc. Am.B 20, 1091 (2003).

[12] S.G. Kuzmin and T.M. O’Neil, Phys. Rev. Lett. 88, 065003 (2002).

[13] C.A. Brau, ref. [9], p. 20.

[14] P.W. Hawkes and E. Kasper, Principles of Electron Optics II: Applied Geometrical Optics(Academic Press, London, 1996).

[15] O.J. Luiten, S.B. van der Geer, M.J. de Loos,F.B. Kiewiet, M.J. van der Wiel, Phys. Rev.Lett. 93, 094802 (2004).

[16] W. Ketterle, K.B. Davis, M.A. Joffe, A. Martin, and D.E. Pritchard , Phys. Rev. Lett. 70,2253 (1993).

[17] G.J.H. Brussaard et al., IEEE Trans. Plasma Sci., 32, 5 (2004).

[18] http://www.pulsar.nl/gpt.

[19] J.H. Billen and L.M. Young, POISSON SUPERFISH, Los Alamos Nat. Lab. Rep. LA-UR-96-1834.

[20] CST Microwave Studio Version 5, CST GmbH, Germany (2003).

[21] G. Poplau, U. van Rienen, S.B. van der Geer, and M.J. de Loos, IEEE Trans. Magn. 40,714 (2004).

58

5

First generation pulsed electron source setup

5.1 Introduction

In Chapter 4, we argued that an Ultracold Plasma (UCP) can serve as a high brightnesselectron source. In Chapter 1 we discussed that a UCP is typically created by photoioniza-tion of the atoms from a Magneto-Optical Trap. As a consequence the starting point foran experiment aiming at creating a cold-atom electron source is a Magneto-Optical Trap(MOT) as described in Chapter 1. For this a robust, stable and reproducible setup hasbeen designed and tested to trap large numbers of cold atoms.As described in Section 1.2.2, an MOT consists of three orthogonal pairs of red-detuned,counter-propagating, circularly-polarized laser beams intersecting at the center of a mag-netic quadrupole field. Atoms entering the trapping volume determined by the regionwhere the laser beams overlap with a velocity below the capture velocity are trapped.To build an MOT one therefore needs a laser setup that produces a sufficient amount ofpower at the right frequency, with a stability better than the atomic linewidth.A second requirement is a vacuum setup in which the MOT can be made. The vacuumis essential since elastic and inelastic collisions with hot background atoms lead to traploss. A third step consists out of creating the proper magnetic field gradient G and theproper diagnostics to measure the number of trapped atoms and the volume from whichthe density can be derived.To create a UCP from an MOT, one has to excite the atoms in the MOT with a laser thatgenerates a sufficient amount of power at the right wavelength. Furthermore the properdiagnostics to detect the UCP have to be constructed.

5.2 Loading a Magneto-Optical Trap

Since the first creation of an MOT, several different trapping schemes have been tried [1–3].The essential differences can be found in the configuration of the laser beams and the waythe trap is loaded. In a typical setup the MOT is loaded with atoms from a slow atomic

59

Chapter 5.

CaptureVolume

A

B

C

D

Figure 5.1: Illustration of the principle of a vapor-cell Magneto-Optical Trap, where atomsare trapped from the background vapor. Atoms entering the trapping volume defined by theregion where the MOT beams intersect are cooled and forced towards the center. Only atomswith a velocity below the trapping velocity vc are trapped (traces A, D). Atoms entering thetrapping volume with a velocity above vc are merely deviated (traces B, C).

beam [1]. The velocity of the atoms entering the trap region should be smaller than thecapture velocity vc ≈ 40m/s [4]. An alternative way of loading atoms in an MOT is directlyfrom a background vapor in a so-called vapor-cell MOT [2]. The number of atoms in athermal distribution at room temperature with a velocity below the capture velocity islarge enough to create a substantial MOT. The principle of a vapor-cell MOT is illustratedin Fig. 5.1.In general one can trap more atoms in an MOT loaded from a beam (1010) than from abackground vapor (109), but at the cost of a more complicated beam apparatus (Chapter2). The main reason to use an atomic beam (for alkali atoms) is that the MOT can beoperated at a lower background pressure, resulting in a longer lifetime of atoms in the trap.This is of great importance for experiments aiming at creating a Bose Einstein Condensate(BEC), since the creation of a BEC can take up to a minute during which one wants tolose as few atoms as possible [4].If one is not interested in the specific atom at hand but merely in the number of particles,the two most convenient atomic elements are rubidium and cesium. For these atomicspecies cost effective and practical diode lasers are commonly available. For the cold

60


electron bunch project the element rubidium was chosen, mainly for the reason that therubidium isotope 87Rb is the most practical element to Bose condense [5]. This can berelevant for future experiments aiming at the production of quantum degenerate electronbunches as described in Chapter 7.In our experiment we eventually aim at creating a MOT containing 1010 atoms. Becauseof this we designed both an MOT and a 2D+ MOT [6]. The 2D+ MOT can trap andcool rubidium atoms from a background gas in two directions; in the third and remainingdirection, the atoms can move freely. The result is an intense beam of slow and cold atomsthat can be used to load the MOT. As a first step however, we made a vapor-cell MOTbecause of its simplicity.The rate equation describing the loading behavior of an (vapor-cell) MOT is given by:

dNt

dt= RL −NtΓBG − β

∫n2dV, (5.1)

where Nt is the number of trapped particles, RL the loading rate of the MOT, ΓBG thelinear loss term, n the density in the trap and β the two-body rate constant. The two-bodyloss rate is calculated by integrating βn2 over the the trap volume dV . The capture rateis given by [2, 7]:

RL =nBGV

23 v4

c

2v3mp

, (5.2)

with nBG the density of rubidium atoms in the background vapor, V the capture volumeand vmp the most probable velocity of the background atoms.The linear loss term defined by ΓBG represents linear trap losses. The dominant linear lossprocess is loss of atoms due to collisions with background atoms. In an elastic collision, ahot background atom colliding with a trapped atom can transfer momentum, resulting ina velocity larger than the capture velocity of the MOT. In an inelastic collision, the atommay be transferred to a hyperfine state that is no longer trapped. The two-body losses aredue to collisions of two trapped atoms; the dominant effect comes from collisions of groundstate atoms with excited state atoms and hyperfine state changing collisions. In contrastwith e.g. Ne∗ (see Chapter 2), the two-body loss term can often be neglected for alkaliatoms in an MOT [8].In a first order approximation, neglecting two-body losses, the loading of a MOT is de-scribed by:

Nt =RL

ΓBG

1− exp(−ΓBGt). (5.3)

Practically this means that the MOT fills on a timescale determined by the loss due tobackground atoms ΓBG, to a steady state number of atoms N∞ given by:

N∞ =nBGV

23 v4

c

ΓBG2v3mp

. (5.4)

This implies that in order to trap a large number of atoms it is important to have large MOTbeams, resulting in a large capture volume V and a high capture velocity vc. For a vapor-cell

61

Chapter 5.

MOT, both the loading rate and the loss rate are in a first approximation linearly dependenton the partial rubidium background pressure. This makes the steady state number ofatoms virtually independent on the rubidium background pressure (this approximationbreaks down when the mean free path of the atoms becomes of the same order as the trapdimensions). The typical filling time (1/ΓBG), however, does scale inversely proportionallyto the rubidium background pressure.As mentioned in Chapter 1, the largest MOT in terms of atom number at the highestparticle density reported so far by our knowledge is reported by Ketterle et al. [9] et al.They used sodium, where the MOT was loaded from a Zeeman slowed beam. They wereable to trap about 1010 atoms at a density of almost 1012 atoms cm−3. Schoser et al. wereable to trap over 6×1010 atoms but at a lower density [10], using a 2D-MOT [11]. Vaporcell MOT’s containing over 1010 atoms have also been reported, at the cost however of arelatively high background pressure [12]. A high background pressure is disadvantageousfor creating low emittance electron beams due to the interaction of the electrons withbackground atoms.

5.3 2-D MOT

In order to make bright electron bunches one wants a maximum of particles at a maximumdensity (see Chapter 4).As a consequence, if one wants high particle numbers, a high density, a low backgroundpressure and a high loading rate one needs to load the MOT with a slow atomic beam. Theloading rate is important since it defines the repetition rate of the electron bunch source.Commonly used slow atomic beams are for example Zeeman slowed beams, see Chapter2, or beams using a chirped slower [13]. Zeeman slowed beams are capable of producing abeam flux of over 1012 atoms/s, at the cost of a high beam divergence [1]. The experimentalapparatus however, is rather complex and large (see Chapter 2).Other, more compact sources are based upon extracting atoms from a 3D MOT loadedfrom a background vapor through a differential pumping stage. A well established sourcewas reported by Lu et al. [14], they developed a low velocity intense source (LVIS-source).Here the atoms are extracted from the 3D MOT by having a shadow in one of the MOTbeams. Due to the intensity imbalance a slow beam of atoms is extracted from the MOT,with a flux of 5×109 atoms/s and a mean longitudinal velocity below 20 m/s.A different approach is a two-dimensional configuration where atoms are extracted directlyfrom the background vapor. Two similar approaches are the 2-D MOT [11] and the 2-D+

MOT [6], both capable of producing a high flux (1010 atoms/s) of slow atoms (8 m/s and75 m/s for a 2-D+ and a 2D-MOT respectively).A standard 2-D MOT consists of two pairs of counterpropagating laser beams with oppositecircular polarization. These beams have to intersect at the center of a two dimensionalquadrupole field. Atoms from the background vapor entering the cooling volume definedby the region where the laser beams overlap are cooled and driven towards the axis. Inthe direction perpendicular to the laser beams the atoms feel no force, and as such they

62


can move freely along this axes. The result is two thin and dense atomic beams travelingin opposite directions perpendicular to the laser beams. The atoms moving along the axisexit through a small aperture to a different vacuum chamber where they can be trapped ina 3-D MOT. This small hole acts both as a geometrical filter for the longitudinal velocityof the atoms and as a flow resistance. The contribution of hot atoms moving through thehole is negligible. The interaction time is determined by the ratio of the distance they haveto travel and the longitudinal velocity of the atoms, This also implies that the longer the2-D MOT, the higher the total flux is at the expense of a higher longitudinal velocity. thediameter of the hole acts as a filter for the longitudinal velocity of the atoms, since onlythose atoms that are driven to a radial position smaller than the radius of the hole areallowed to pass.There are two other criteria that need to be fulfilled in order that a particle will exitthrough the aperture. For one, the radial velocity of the atom must be smaller than theradial capture velocity of the 2-D MOT. A second criterion is that the background pressureis not too large so that the main free path of the atoms is larger than the dimensions ofthe 2-D MOT.The 2-D MOT with the largest atom flux to our knowledge used for loading a 3-D MOT hasbeen reported by Schoser [11]. They achieved a maximum beam flux of 6×1010 atoms/swith an average longitudinal velocity of 75 m/s. They were able to load a Rb MOTcontaining 6 × 1010 atoms, to our knowledge the highest particle number reported in anMOT so far [10].A simple extension to a 2-D MOT leading to a large increase in beam flux has been reportedby Dieckman et al. [6], who added an additional pair of laser beams in the longitudinaldirection, one pushing beam copropagating with the atomic beam and one slowing beamcounterpropagating the atomic beam. The slowing beam is sent into the vacuum from theside and is reflected on a 45 degree aluminum mirror. At the center of this mirror theexit hole was drilled, as depicted schematically in Fig. 5.2. The result of this is that thedownward beam has a shadow so that atoms close to the axis are mainly pushed. Theseextra beams lead to an increase in beam flux and a lower longitudinal velocity comparedto a pure 2-D MOT. Dieckmann et al. obtained a beam flux of 9×109 atoms/s with alongitudinal velocity of 8 m/s. These results were obtained with almost a factor of 20 lesslaser power compared to Schoser et al. [11].In our setup we basically copied the construction of Dieckman et al. regarding the 45degree aluminum mirror with a 1 mm aperture. In contrast to Dieckman et al., we do notuse retro-reflecting beams but four separate beams. This was done since the atoms absorba significant fraction of the light resulting in a shadow in the reflected beam, leading toan imbalance. The vapor cell consists out of a standard glass cuvette glued to a metalconstruction with epoxy. For this we copied a design reported by Voigt [15]. This provedto be a low cost, simple and robust way to mount the glass cell to a metal flange.At this moment we have not actually tested our 2D+ MOT yet.

63

Chapter 5.

cooling laser

exit

Slowing beam

Pushing beam

30 mm

Figure 5.2: Illustration of the principle of a 2D+ MOT. Atoms from the background vapor ina standard cuvette, entering the volume where the 2D MOT beams intersect, with a velocitybelow the radial capture velocity are cooled and driven towards the axis. In the direction ofthe axis the atom feels no force and can leave the cuvette trough a hole. The hole is drilled ina 45 degree mirror on which a laser is reflected, resulting in a extra slowing beam. From theback of the cuvette a pushing beam is sent in.

5.4 Laser setup

The main technical challenge in building a stable and reproducible MOT is to build a stableand robust laser system, preferentially with a lot of laser power. In our experience the lasersetup is the key parameter determining the up-time of the setup. For laser cooling andtrapping of rubidium one essentially needs two lasers; one has to be locked to the trappingtransition and one has to be locked to the repumper transition as described in Chapter 1.The trapping laser has to be stable in terms of intensity but especially the linewidth of thelaser has to be lower then the natural linewidth of the used transition, and the laser hasto be locked to a fixed frequency relative to the atomic resonance. For the repumper laserthe conditions on the frequency are less stringent.In our case, the trapping laser is a commercial diode laser (Toptica DLX), generating ap-proximately 950 mW of laserlight with a wavelength of 780 nm. A second commercial diodelaser (Toptica DL 100) is used as a source for the repumping light; this laser generates 150mW at a wavelength of 780 nm. In an MOT one typically needs a saturation parameterof s = 1 − 10 for all six laser beams, while the diameter of an MOT beam is typically 1cm. To achieve this, one needs about 10-100 mW, well below the maximum output powerof both lasers.The setup of both lasers is depicted in Fig. 5.3. Both lasers are mounted on one end of a120 cm × 240 cm optical table and are mounted on shock absorbing rubber to minimizethe effect of acoustical vibrations. The lasers and the optics needed to lock the laser tothe atomic resonance and set the frequencies are mounted in a closed box. This is done to

64


eliminate the effect of air flow around the laser which can cause frequency instabilities andthermally induced motion of the optics. Special care is taken to make sure all electronicsand the optical table are connected to the same electrical ground.The trapping laser is locked to the 5S1/2,F=3 → 5P3/2,F=4 transition of 85Rb using apolarization spectroscopy technique, that is thoroughly described by Wijtvliet [16]. Typ-ically, 4 mW of light is split off and sent through an acousto-optical modulator (AOM)in double pass configuration (see Fig 5.3). The AOM has a center frequency of 80 MHzand can be tuned from 60-100 MHz. The AOM is driven by the amplified output of acomputer-controlled voltage-controlled oscillator (VCO). With this AOM, the frequency ofthe split-off beam can be changed in a frequency window of 80 MHz without changing thealignment of the laser. After the double pass configuration, about 2 mW of light detunedby 160 MHz to the red remains, that is used to lock the laser with the polarization spec-troscopy technique, which generates an error signal corresponding to the deviation of thelaser from the resonance frequency. This error signal is fed back to the piezo controller ofthe laser through a PI controller (Proportional, Integral). The stability of the locked laserwas determined by monitoring the error signal. The peak to peak fluctuations correspondto 2 MHz, smaller than the natural linewidth of 85Rb (Γ =5.98 MHz, it has to be notedthat the bandwidth of the error signal ranges from D.C. to 10 kHz). The ultimate testhowever is the stability of the MOT, discussed in Chapter 6.The main beam exiting the laser is split in two and each beam is sent through a separatedouble pass configuration. As such we have two laser beams, each containing about 200mW that we can detune separately over about 80 MHz from resonance. One beam is usedto create the MOT; the second beam will eventually be used for the 2D MOT+ (see section5.2).The light from the repumper laser is first sent through an optical diode after which about5 mW is split off with a glass plate. This light is used to lock the laser to the 5S1/2,F=2→ 5P3/2,F=3 transition using a saturated absorption technique [16]. The obtained errorsignal is fed back to the piezo control of the laser through a PI controller. The frequencystability of this laser is not measured directly since it is less critical and sufficient based onthe stability of the MOT.

5.5 Vacuum chamber

The central part of our setup is a large vacuum chamber, depicted in Fig. 5.4. The size ofthe trap chamber was chosen to fit a future accelerator structure (see Section 5.11). Thechamber has 12 CF40 side ports, mainly for optical access, 4 CF200 ports to install theaccelerator and the diagnostics, one CF100 port to connect the cuvette of the 2D+ MOTand one CF160 port to connect vacuum pumps. We chose to connect 3 pumps to the trapchamber. This is depicted in Fig. 5.4.(i) A turbo molecular pump (tmp) (250 l/s) that can be isolated from the trap chamber byclosing a manual valve, backed by a turbodrag pump and a diaphragm pump. The tmp isused to pump down the trap chamber to a pressure of about 10−6 mbar. At approximately

65

Chapter 5.

DLX

DL100

!"# $%&#"&'()*+,-* ./-0)123*4+21-*5

678898

:;<<=<>?@A>?@

BCDEEFGCDHI

JKLM NOPQRSTUV

WXYXZY[\

]^_`abcbde fg`hi cj_bkkga

lmnmompqmrqmss

tuvwxyz|

Figure 5.3: Impression of the optics used to stabilize the lasers and divide the laser beams.Both lasers and the optics are placed in a covered box.

this pressure two ion-getter pumps can be switched on. Once the ion-getter pumps areswitched on, the tmp can be switched off after closing the manual valve. This is mainlydone to eliminate mechanical vibrations coming from the pump.(ii) A 150 l/s ion-getter pump that has to reduce the pressure further to the final valueof about 10−9mbar. This pump is connected to the CF160 port of the vacuum chambertrough a CF160 cross to minimize any flow restriction that reduces the effective pumpingspeed.(iii) A 20 l/s ion-getter pump has been added to the system. This has been done so wecan switch off the large ion-getter pump during the experiments, facilitating the build-up

66


!"#$% &"'&

()*+,-./0123 42536

78 9:;:9<=>9

?@ABCDE @EDAF?GH

IJJ KK

LMNOP

Figure 5.4: Illustration of the vacuum setup. A central trap chamber can be pumped down bythree separate vacuum pumps. Two ion getter pumps and one turbo molecular pump that canbe isolated from the trap chamber by a manual valve. A small rubidium reservoir is connectedto the trap chamber, rubidium is let in in the system by opening a valve. Above the setup anoptical platform is placed for the optics to steer the MOT beams.

of a sufficient partial rubidium pressure.The rubidium is let in in the system by opening a small mechanical valve (as depicted inFig. 5.4). On the other side of the valve there is a small tube in which we placed rubidium.The rubidium was released by cracking a glass ampule containing solid rubidium. Thereservoir can be heated to increase the pressure. Note that the reservoir has to be belowthe valve, since by heating the rubidium this can become liquid and enter the trap chamber.Around the flanges of the trap chamber we installed heating elements to facilitate the bake-out of the trap chamber.

5.6 Trapping beams

As a first step we constructed an MOT consisting out of 6 laser beams of which 3 beams areretro-reflected. This makes it easier to align and to make a first MOT. The disadvantageof using a retro-reflection MOT, compared to an MOT consisting out of 6 separate beams.

67

Chapter 5.

Figure 5.5: Illustration of the beam paths of the MOT. The black lines denote beams at ahigh level, the grey lines indicate the low MOT beams. The beam paths of the radial beams arebuild symmetrically, to facilitate the future implementation of a dark spot MOT. The repumplaser is mixed with the horizontal beams.

is that, due to absorption, the MOT casts a shadow in the laser beams that causes animbalance. Since this is not critical for the first experiments we did not yet implement the6 independent beams. However, the optics were placed such that changing to a 6-beamMOT is straightforward.An idea of the MOT beam alignment is given in Fig. 5.5. The alignment of the MOT beamswas chosen such that two opposing MOT beams are derived from the same beam splitter.This makes it easier to adjust the intensity balance per beam pair. The beam paths ofthe opposing beams in the radial beam pairs (see Fig. 1.1) were made symmetrical. Thismakes it easier to implement a dark-spot MOT in future experiments [4]. In a dark-spotMOT, there is a dark spot in the center of the repumper beam, leaving the center of thetrap without repumper light. As a result the atoms at the center in the F = 2 groundstate no longer fluoresce. This reduces the re-absorption of emitted photons that wouldnormally result in a outward radiation force, limiting the achievable density. This becomesimportant at densities of about 1010cm−3. The total power in the MOT beams is about 110mW; we shaped the laser beams using two telescopes resulting in a beam with a diameterof about 2 cm. The saturation parameter s is thus about 70. The total power in therepumper beam is about 50 mW; also here the beam diameter is about 2 cm. The actualMOT beams are clipped by an aperture to prevent straylight from casting reflections onour camera.

68


Figure 5.6: Measured dependence of the electric and the magnetic field during a typicalexperiment. The magnetic field is turned on and off in about 150 µs.

5.7 Magnetic fields

In order to make an MOT one also needs a quadrupole magnetic field which can be madeby placing a pair of coils in a anti-Helmholtz configuration. A typical gradient is about10 G/cm in the longitudinal direction, i.e. the direction along the symmetry axis of theMOT coils (the x-direction in Fig. 5.8). Since our trap chamber is relatively large, placingthe coils outside the vacuum would result in a set of large coils (diameter 500 mm) whichis unpractical. As a consequence, we placed a pair of coils inside the vacuum. The coilswere made out of hollow, water cooled copper tubing, we used 4 turns for each coils. Theradius of the coils is about 100 mm and the distance between the coils is about 100 mm(the Helmholtz condition for a pair of coils states that the radius of the coils equals thedistance between the coils). The current sent through the coils is typically 100 A and cango as high as 200 A, limited by the current supply. We installed a switch that can switchthe magnetic fields on and off within 150 µs (see Fig. 5.6). The gradients in both ther-direction (z- and y-direction in Fig. 5.8) and the x-direction were 0.04 G/(cm A) and0.08 G/(cm A) respectively.At the center of the trap one wants to have a magnetic minimum. This will lead tosub-doppler cooling [13] giving lower MOT temperatures. To null the magnetic field atthe position of the MOT, we installed large, so-called compensation coils around the trapchamber. As an extra feature, the compensation coils can be used to carefully move theMOT.

69

Chapter 5.

MOT Lens LensPinhole Detector

Figure 5.7: Illustration of the fluorescence setup. Two lenses make an image of the atom cloudon a calibrated photodiode. At the focus of the telescope we installed a pinhole to minimizethe effect of stray-light. The entire system is covered by a plastic tube.

5.8 MOT diagnostics

The main diagnostic tool for the MOT is measuring the fluorescence [13] emitted by thetrapped atoms. This is done with both a CCD camera and with a calibrated photodiode.From the CCD images of the MOT we can extract information of the size of the sample.The total power detected by he photodiode or the CCD camera is a measure for the totalnumber of atoms in the MOT. Information of both the size of the sample and the numberof trapped atoms can be used to calculate the density.The fluorescence power P , emitted by N trapped atoms is given by:

P = ~ωΓfN, (5.5)

with ω the frequency of the transition used, Γ the decay rate from the excited state and fthe fraction of atoms in the excited state as defined in Chapter 2:

f =1

2

CS

1 + 4δ2 + CS, (5.6)

where S is the total saturation parameter for the six MOT beams combined and δ the laserdetuning expressed in units of Γ. The quantity C is a phenomenological factor which liesbetween the average of the squared Clebsch-Gordan coefficients of all involved transitionsand 1 For the analysis of the rubidium MOT we use C=0.7 based upon the work ofTownsend et al. [17]. The light emitted by the atoms is then imaged either on the cameraor on the photodiode through a lens as indicated in Fig. 5.7. The current I generated bythe photodiode is then given by:

I = ηPΩ

4π. (5.7)

Here η=0.5 A/W is the power-to-current calibration factor of the photodiode at 780 nm,and Ω=6.6×10−4 sr, the solid angle of the lens. The fluorescence detection system isdepicted in Fig. 5.7. The uncertainty in the number of particles is typically a factor oftwo. Thus, when one knows the detuning δ and the intensity S of the MOT beams onecan measure the number of atoms. In our experiments we have a typical S of over 70and are detuned by -3 × Γ, this results in a calculated excited state fraction of about30%. However, for most analysis, we assume an excited state fraction of 50 %, giving an

70


underestimate of the number of atoms.When one integrates the total pixel content of a CCD camera image this is also a goodmeasure for the number of atoms. We used the number of atoms determined from aphotodiode measurement to calibrate the CCD images in terms of numbers of atoms. Assuch we have all critical information with one CCD image, i.e., number of atoms, volumeand density.

5.9 Excitation

For excitation of the atoms in the MOT to a plasma or a Rydberg state, we use a tunablepulsed dye laser (Quanta-Ray PDL3) (λ ≈ 480 nm, pulse width ≈ 10 ns, bandwidth ≈15 GHz, with about 1 mJ per pulse). The laser is pumped by the third harmonic of aNd:YAG laser with a repetition rate of 10 Hz. The wavelength of the laser is determinedby sending it through a calibrated monochromator and can be adjusted with a computer.The absolute accuracy of this system is about 0.03 nm. The wavelength of the laser iscontinuously measured with a computer, this can be used to compensate for drift in thelaser wavelength. The surface area of the beam going to the atom cloud is about 1 mm2.

Figure 5.8: Illustration of the setup. Indicated in the figure are the MOT coils (C), the MOTbeams (M), the detector (D), the ionization laser (I) and the CCD camera (E).

71

Chapter 5.

5.10 Plasma diagnostics

The Rydberg atoms and the plasma components can be detected by applying an electricfield pulse over a rod structure that surrounds the atom trap as depicted in Fig. 5.8 (copiedfrom the Virginia design, see Section 3.3 [18]). The electric field pulse field ionizes theRydberg atoms down to a principal quantum number n of about 35 and pulls the plasmaapart. The pulse has a rise-time of about 2 µs and can go as high as 2 kV. Dependingon the polarity of the electric field it pushes the ions or the electrons to a MCP detector.We calculated the static electric field of the rod structure with Superfish to convinceourselves that the field was homogeneous over the typical size of the MOT (1 mm), thiswas the case within 10 %.A measurement of the magnetic and electric field during a typical experiment is given inFig. 5.6. Due to the relatively long risetime of the electric field pulse, and the differentfields at which the Rydberg states ionize and the plasma is pulled apart, we can stateselectively detect the Rydberg atoms and discern between Rydberg atoms and plasma.As a detector we use a Multi Channel Plate (MCP) detector with a phosphor screen.The MCP detector amplifies the charge (maximum gain 104) which is accelerated to thephosphor screen where it casts a light image that is detected with a CCD camera. From thisimage we can determine the width of the electron or ion bunch hitting the phosphor screen.The current resulting from the electrons hitting the phosphor screen is also measured. Atthis point we have not yet calibrated our MCP detector, for example with a Faraday cup.Neither have we tested the linearity of the MCP detector yet, typically we have less thanone electron per MCP pixel.

5.11 Second generation accelerator

Figure 5.9: Design of the first 30 kV accelerator structure; indicated are the inner and theouter conductor and the position of the MOT.

72


The work described in this Section has mainly been performed by Taban and van derGeer.A 30 kV (10 ns rise time) accelerator structure as depicted in Fig. 5.9 has been developedas the next step in generating electron bunches. This accelerator is designed to fit in thevacuum chamber described in Section 5.5, and will make use of the same MOT coils. Lowenergy, low emittance electron bunches are for example useful for time-resolved ultra-fastelectron diffraction experiments [19]. Furthermore, this accelerator should allow us to provethe potential of a using an MOT as a low emittance electron or ion source. Simulationsperformed with the General Particle Code depicted in Fig. 5.10 suggest that an emittanceof 0.15 mm mrad can be reached at a distance of one meter from the source with a peakcurrent of about 50 mA. If proven successful, one can pursue higher voltages and shorterrise times.

5.12 Conclusions

We constructed a dedicated setup that can be used to prove the validity of the cold atomelectron source concept. We constructed the necessary vacuum setup to trap cold atomswith enough space to install the first accelerator. We constructed a flexible and stablelaser-setup that can be used to trap cold atoms. We can measure the number of atomsand the density with a standard fluorescence technique.Furthermore we constructed the diagnostics necessary to create and detect Rydberg atomsand ultracold plasmas.

73

Chapter 5.

Figure 5.10: Simulations performed with the General Particle Tracer Code, simulating thefirst version of a 30 kV accelerator. Depicted in the graph are the calculated emittance (fullblack line) and the peak current (dashed grey line).

74


Bibliography

[1] W. DeGraffenreid, J. Ramirez−Serrano, Y.−M. Liu, and J. Weiner, Rev. Sci. Instrum. 71,3668 (2000).

[2] C. Monroe, W. Swann, H. Robinson, and C. Wieman, Phys. Rev. Lett. 65, 1571-1574(1990).

[3] W. Hansel, P. Hommelhoff, T.W. Hansch, and J. Reichel, Nature 413, 498 (2001).

[4] E. van Ooijen, Ph.D. thesis (University of Utrecht 2005).

[5] E. W. Streed, A. P. Chikkatur, T. L. Gustavson, M. Boyd, Y. Torii, D. Schneble, G. K.Campbell, D. E. Pritchard, and W. Ketterle, Rev. Sci. Instrum. 77, 023106 (2006)

[6] K. Dieckmann, R.J.C. Spreeuw, M. Weidemuller, and J.T.M. Walraven, Phys. Rev. A 58,3891 (1998).

[7] P. A. Molenaar, Ph.D. thesis (University of Utrecht 1996).

[8] L. Karssen, Master thesis (University of Utrecht 2002).

[9] W. Ketterle, K. B. Davis, M. A. Joffe, A. Martin, and D. E. Pritchard, Phys. Rev. Lett.70, 2253 (1993).

[10] J. Schoser, Ph.D. thesis (University of Stuttgart 1996).

[11] J. Schoser, A. Batar, R. Low, V. Schweikhard, A. Grabowski, Yu. B. Ovchinnikov, and T.Pfau, Phys. Rev. A 66, 023410 (2002).

[12] G. Labeyrie, F. Michaud, and R. Kaiser, Phys. Rev. Lett. 96, 023003 (2006)

[13] H.J. Metcalf, P. van der Straten, Laser Cooling and Trapping (Springer, Berlin HeidelbergNew York, 1999).

[14] Z.T. Lu, K.L. Corwin, M.J. Renn, M.H. Anderson, E.A. Cornell, and C.E. Wieman, Phys.Rev. Lett. 77, 3331 (1996).

[15] D. Voigt, Ph.D. thesis (University of Amsterdam 2000).

[16] R. Wijtvliet, internal report (2004).

[17] C. G. Townsend, N.H. Edwards, C.J. Cooper, K. P. Zetie, C. J. Foot, A. M. Steane, P.Szriftgiser, H. Perrin, and J. Dalibard, Phys. Rev. A 52, 1423 (1995).


[19] B. J. Siwick, J. R. Dwyer, R. E. Jordan, R. J. D. Miller, Science 302, 1382 (2003).

75

6

Experimental results

6.1 Introduction

In Chapter 5, we discussed the design and construction of a setup to create the first electronbunches from a cloud of cold atoms. In this Chapter4 we present the first results. The firststep in our proposed scheme is to make a stable Magneto-Optical Trap (MOT) containinga large number of atoms. The second step is then to excite the atoms from the MOT to aRydberg gas or an Ultracold Plasma (UCP) and extract the electrons or ions.

6.2 MOT characterization

As discussed in Chapter 5 we use a vapor-cell MOT configuration. Here we report its mostimportant operational characteristics.

6.2.1 Number of trapped atoms

In Fig. 6.1 we show the measured number of atoms as a function of the total laser powergoing to the MOT for both the repumper laser (top) and the trapping laser (bottom). Thelongitudinal magnetic field gradient was set at 8 G/cm and the partial rubidium pressurewas 5×10−8 mbar, as we determined by absorption measurements. The detuning of thetrapping laser was set at -18 MHz. We assumed the excited state fraction to be 50 %.In the bottom part of Fig. 6.1, we show the number of atoms trapped as a function ofthe trapping laser power. The total laser power in the repumper laser was set at 60 mW.There is a clear linear dependence between the number of atoms trapped and the totallaser power, indicating that the loading rate keeps increasing with total laser power. It issufficient to conclude that we can trap over 109 atoms in the vapor-cell MOT at a power ofabout 120 mW, corresponding to a saturation parameter s ≈ 70. This is a good startingpoint for our further experiments.

4Part of the work described in this Chapter, will be submitted for publication.

77

Chapter 6.

Figure 6.1: (Top) Number of atoms trapped (measured with fluorescence detection) as afunction of the power of the repumper laser beam. (Bottom) Number of atoms trapped as afunction of the power in the trapping laser beam, the grey line is a linear fit to the data toguide the eye.

The top part of Fig. 6.1 shows the number of atoms trapped as a function of power in therepumper laser. In contrast with the previous measurement, the number of atoms trappedsaturates at a total power of 30 mW corresponding to a saturation parameter s ≈ 10. Thetotal laser power in the trapping laser was set at 60 mW.

6.2.2 Lifetime and loading rate

In the lower part of Fig. 6.2, we show the filling of the trap for different detunings of thetrapping laser, together with fits using Eq. 6.1, where RL, the loading rate of the MOTand ΓBG, the linear loss rate, were set as free parameters:

Nt =RL

ΓBG

1− exp(−ΓBGt). (6.1)

In these experiments we suddenly switched on the trapping lasers and measured the numberof atoms trapped with fluorescence detection.The extracted lifetimes are shown in the top part of Fig. 6.2. The lifetime of the atomsincreases with detuning up to a value of about 0.75 s at a detuning of -18 MHz the power inthe trapping laser was 80 mW, the power in the repumper laser about 30 mW. This lifetimeis long compared to the typical timescales for plasma extraction (<100 µs). The maximumnumber of atoms we trap here is about 6×108 atoms, this results in a loading rate of aboutRL ≈ 8×108 atoms/s. This would allow us to extract electron bunches containing 10 pCat a repetition rate of almost 10 Hz. For the first experiments this is sufficient; adding the

78


Figure 6.2: (Bottom) Measured loading curves for different detunings of the trapping laser,the dashed lines are fits to the date using Eq. 6.1. Indicated is the detuning in MHz. (Top)The fitted trap lifetime as a function of the detuning of the trapping laser. The power in thetrapping laser was set at 80 mW and the power in the repumper laser was set at 30 mW.

2D+ MOT described in Chapter 5 can potentially increase the charge or the repetition rateby two orders of magnitude.

Figure 6.3: Result of a release and recapture experiment, aiming at measuring the temperatureof the trapped atoms. Depicted is the fraction of atoms recaptured after a time ∆t of ballisticexpansion. The dashed line is a fit to the data using Eq. 6.2. The power in the trapping laserwas set at 80 mW, the power in the repumper laser was set at 30 mW and the detuning of thetrapping laser was set at -18 MHz.

79

Chapter 6.

6.2.3 Temperature and density

In Fig. 6.3, we present the result of a simple release and recapture measurement aimed atmeasuring the temperature of the atoms in the MOT. We switch off the MOT laser lightwith the AOM (< 1 µs). After a variable delay time we switch the light on again (< 1 µs).During the time where there is no light (∆t), the atom cloud expands ballistically. As aresult, the atoms leaving the region defined by the MOT laser beams (radius Rc) no longercontribute to the signal when the MOT laser beams are turned on again. By measuringthe fraction of atoms remaining after a variable delay time ∆t, the temperature can bedetermined. In Fig. 6.3 we plot the fraction fr of atoms recaptured as a function of delaytime and fitted the signal with a function derived by Molenaar [1]:

fr = erf

(vc

vT

)− 2√

π

(vc

vT

)exp

(−

(vc

vT

)). (6.2)

Here vT =√

2kBT/m is the thermal velocity of the atoms in the MOT at a temperatureT and vc = Rc/∆t. The fitted temperature is 200 ±100 µK. This is close to the Dopplertemperature, i.e., 142 µK. The uncertainty comes mainly from the uncertainty in Rc. Thepower in the repumper laser was 30 mW, the power in the trapping laser was 80 mW, thedetuning of the trapping laser was set at -18 MHz.The density has been determined by fitting the fluorescence profile of the MOT with aGaussian distribution (exp[−(x/2σ)2]) and using for the density n0:

no =N

(2π)3/2σxσyσz

. (6.3)

From our measurement we get the dimensions (σx,y,z) in only two directions, i.e. the x-direction (0.18 mm) and the z-direction (0.20 mm) (see Fig. 5.8), at the maximum particlenumber (1.2×109 atoms). We assumed the width in the y-direction to be equal to thatof the x-direction. The density we obtained was 1010 cm−3. The rms fluctuations in theMOT signal, derived from the fluorescence signal, are less then 2 % (measured over 20 s).The main result of this Section is therefore that we are able to make a stable vapor-cellMOT containing over 109 atoms at a density of 1010 cm−3. This is an excellent startingpoint to perform the first experiments aiming at creating the first electron and ion bunches.

6.3 Electron and ion beam generation

In this Section we present the results of the first experiments aiming at creating electronand ion bunches. The motivation of these experiments was to show the ability of the setupto create a UCP and extract the ions or the electrons as well as to test the diagnostics.The results of the experiments will be discussed qualitatively, a quantitative analysis of theresults based upon simulations using the General Particle Tracer code is being pursued,the results will be reported in [2].

80


Electric Field pulse

Time

delay t

Excitation

A.lasersB-field

B.

B-field

C. D.lasers

Figure 6.4: The typical experimental procedure to create a UCP from a gas of cold Rydbergatoms. (A) Both the MOT laser-and magnetic fields are on. (B) The laser-field is switched off.(C) The magnetic field is switched off . (D) the laser-field is switched on again. After this theatoms are excited to the 44 d state, after a variable delay t, the field ionization pulse is applied.

6.3.1 Experimental routine

The typical experimental routine is depicted in Fig. 6.4. The starting point is an MOTas described in the previous Section. In the second (Fig. 6.4B) step we quickly switch off(<1 µs) the MOT laser beams. The atoms no longer feel a trapping nor a cooling force.In the third step (Fig. 6.4C), we quickly switch off the magnetic field (<150 µs) as describedin Chapter 5. This is done since the MOT magnetic field affects the motion of chargedparticles. Immediately after the magnetic field is switched off, we switch the MOT laserbeams on again to repopulate the 5P3/2,F=4 state. About 20 µs later, we excite the atomsin the 5P3/2,F=4 state to the 44d Rydberg state with the pulsed laser (λ ≈ 480 nm, 1 mJin 1 mm2). We then wait for a delay time ∆t ranging from 0 to 80 µs and apply a fieldionization pulse.This field ionization pulse is depicted in Fig. 6.5. Approximately 1 µs after the triggerpulse, the field ionization pulse rises in 2 µs to a maximum value of about 200 V/cm(we calibrated the electric field on the ionization field of the 44d state, 97 V/cm). Theionization pulse ionizes all Rydberg atoms with a principal quantum number n > 37, whilethe Rydberg atoms with n < 37 are not ionized. The field pulse strips the electrons fromthe Rydberg atoms and pulls the plasma apart. Depending on the polarity of the pulse itpushes either the electrons or the ions to the MCP detector. The MCP detector amplifiesthe charge (maximum gain 104) which is accelerated to the phosphor screen where it castsa light image that is detected with a CCD camera. From this image we can determine thewidth of the electron or ion bunch hitting the phosphor screen. The current resulting fromthe electrons hitting the phosphor screen is also measured.Immediately after this routine we turn the MOT magnetic fields on again and about 100ms later (our YAG laser works with a 10 Hz repetition rate) we repeat the experiment.

81

Chapter 6.

Figure 6.5: Measured electron traces for different delays between laser excitation and fieldionization, together with the field ionization pulse, the traces were normalized to the totalsurface area of the signal. The maximum stripping field ionizes atoms in the n=37 state. Theenergy in the pulsed laser was 1 mJ (10 ns) in a surface area of about 1 mm2.

6.3.2 Electron signal

Typical electron traces measured with the MCP detector are depicted in Fig. 6.5. Here weplot detected charge as a function of time for different delay times between the excitationpulse and the field ionization pulse. To be able to compare the graphs, we set the peakcorresponding to the 44d state at time t = 0 (the trigger pulse of the field ionization pulseis set at -2 µs). The flight time of the electrons to the MCP detector is typically 200 ns.All graphs are normalized to the total charge, i.e. the surface area of the signal.The central peak at t = 0 represents the electrons stripped from the atoms in the 44d state.On the left of the 44d peak one can see the electrons stripped from all Rydberg states withn > 44 and the electrons from the plasma. On the right side one can see the electronsstripped from the Rydberg states with n < 44 down to a value of about n = 37. For a lowdelay of about 0 µs, there is a sharp peak with a broader wing on the right. The sharppeak represents the original 44d state and the 46p state which are hardly resolved. Thep-peak appears because of dipole-dipole interactions between 44d Rydberg atoms [3]. Thebroader wing on the right is a signature of l-mixing collisions: collisions of electrons withRydberg atoms that result in a distribution over different l states (at the same n) whichfield ionize at higher fields [5].After a delay of about 8 µs there appears a signal at a small electric field of only a fewV/cm, signifying very weakly bound electrons. This indicates the formation of a weaklybound plasma. Also the fraction of atoms in states lower than the original state increasesas is expected. After a delay of 22 µs, a very clear plasma peak appears at very lowelectric fields. After this long delay, the plasma cloud has expanded, so that the electronsare weakly bound (Eq. 3.4), resulting in an electron signal at lower electric fields. The

82


Figure 6.6: Measured ion traces for two different initial Rydberg atom densities. At a lowdensity (grey line) there is only a signal from the original Rydberg state. The curve measuredat higher density shows a clear plasma peak, that is shifted to the left relative to the original44 d state (black line). The Rydberg atoms were created at t = 0, after a delay of 7 µs theywere field ionized and the ions were detected after an extra flight time of 10 µs resulting in apeak at 17 µs.

electrons created in ionization processes such as blackbody photoionization and ionizingcollisions between Rydberg atoms initially escape the cloud, the positive ions which areslower, remain. Once enough positive charge has built up to trap the electrons, a UCPis created. The dynamics presented in Fig. 6.5, i.e. l-mixing and plasma formation isqualitatively consistent with other experiments studying the spontaneous evolution of agas of cold Rydberg atoms into a UCP [4, 5].

6.3.3 Ion signal

If we reverse the polarity of the field ionization pulse, the ions are detected instead of theelectrons. A significant difference between ion and electron detection is the time-of-flight:For ions the typical flight time is 10 µs, compared to 200 ns for the electrons. In Fig. 6.6,we plotted two ion traces, one at a small initial Rydberg density, mainly showing the ionsresulting from the original Rydberg state (44d), and one at a large initial Rydberg density,where most of the Rydberg atoms are converted to plasma. The delay time for both traceswas set at approximately 7 µs; these 7 µs combined with the 10 µs flight time, explain whythe ions resulting from stripping the Rydberg atoms in the 44 d state arrive at 17 µs inFig. 6.6.The density was changed by reducing the power in the repumper laser, which reduced thenumber of particles trapped in the MOT.

83

Chapter 6.

Figure 6.7: Measured fraction of atoms in the plasma and in Rydberg states higher than theoriginal 44d state, as a function of delay time. The open circles represent the result derivedfrom the ion traces, the black circles represent the results derived from the electron traces.

6.3.4 Plasma fraction

In Fig. 6.7 we plotted the fraction of signal that is detected at an electric field below the ion-ization field of the original 44d state (97 V/cm, this corresponds to the signal at t < −0.5 µsin Fig. 6.5). For the ions we used a similar argument, e.g. in Fig. 6.6 we determined thesignal at t < 16.5µs. This signal mainly represents ions from the plasma and to lesserextent the ions resulting from field ionizing the Rydberg atoms in higher states. Initiallythis fraction increases with delay time and the ion and the electron signal are consistent.After a delay of about 8 µs however, the fraction saturates at about 55 % for the electrons,while for the ions it continues to increase to a maximum value of about 90 % at a delayof about 25 µs. After the maximum, the ion fraction decreases again and catches up withthe electron signal at a delay of about 45 µs. During this time the obtained electron signalstays almost constant at 50 %. Finally both signals decay at different timescales.Our explanation for the experimental result that the electron fraction levels off at 50 %,while the ion fraction continues to increase with delay time, is that after a time of about10 µs, the plasma becomes less deeply bound, resulting in a lower detection efficiency forthe electrons. The effect of magnetic (and electric) stray fields is more important for lightand slow particles. As a result, the electron signals from highly excited Rydberg atoms andweakly bound electrons, having a lower velocity, are detected with less efficiency. The ionson the other hand, due to their large mass, suffer a lot less from this effect. We could alsoobserve this in practice; the exact setting of the current through the compensation coilshad a large effect on the electron signal; for the ion signals, there was hardly an effect.We have to note here that because we can not detect deeply bound Rydberg atoms, the

84


actual fractions are lower. When the plasma expands, the size increases, resulting in ashallow potential well for the electrons, leading to loss of plasma electrons. This explainsthe difference in decay times between the ion- and the electron fraction.A full treatment of the problem based upon simulations was performed by Pohl et al. [6].In their simulations they started from a cloud of cold Rydberg atoms in the n=70 state,and let the Rydberg gas evolve. They calculated the plasma fraction as a function of delaytime and saw qualitatively the same behavior. Initially the plasma fraction increases ona timescale of a few µs to a fraction of about 75 %. From this point on, the plasma frac-tion decreases slowly due to the interplay between ionization, recombination and excitingand deexciting collisions. They also calculated the distribution over the different Rydbergstates as a function of delay time and found a redistribution to lower n states. At a delaytime of about 4 µs, they found a clear maximum at n=25.Although their simulations agree qualitatively with our measurements, they did not in-clude the effect of dipole-dipole interactions. Furthermore we can only detect Rydbergstates down to n=37, which makes us overestimate the plasma fraction. In the acceleratorstructure described in Section 5.11, we will be able to detect Rydberg states down to verylow n, allowing us to measure a more complete redistribution to lower Rydberg states.

Figure 6.8: Total charge in the signal above the initial Rydberg state (black dots) and belowthe initial Rydberg state (grey squares), as derived from the electron traces.

6.3.5 Total charge

In Fig. 6.8 and Fig. 6.9, we plotted the total charge (the are under the measured traces) inthe plasma (t<-0.5 µs) and the total charge from the low Rydberg states (t>-0.5 µs) forthe electrons and the ions respectively. The reported charge is only a minimum value since

85

Chapter 6.

Figure 6.9: Total charge in the signal above the initial Rydberg state (black dots) and belowthe initial Rydberg state (grey squares), as derived from the ion traces.

we assumed the detection efficiency to be 100 % at a maximum gain (104). As a result thetotal charge for the electrons and the ions cannot be compared, since they have differentdetection efficiencies and different gains. For both the ions and the electrons one can seethe Rydberg signal decreasing rapidly and the plasma signal going up.For the ions one can see that the total charge in the plasma goes up to a value of almost1 pC; for the electron traces the plasma signal also goes up but far less relative to theRydberg signal than is the case for the ion signals. We explain this by the lower detectionefficiency of plasma electrons as discussed in the previous Section. The decay of the Ry-dberg atom signal is attributed to two effects. For one, the Rydberg atoms are convertedto plasma and redistributed over low n states that can no longer be detected by collisionswith Rydberg atoms and electrons. Second, the Rydberg atoms decay due to spontaneousemission to states that can no longer be detected (n<37).In Fig. 6.10 we plotted the total charge for the ions and the electrons as a function of delaytime. Oliveira et al. [7] found that Rydberg states decay exponentially to non detectablestates due to spontaneous emission and determined the decay time up to n=42. From theirdata we extrapolated the lifetime for a 44d state (τ ≈ 90 µs) and plotted the expected ex-ponential decay in Fig. 6.10. This line indicates that the loss due to spontaneous emissionis of the same order as the observed decay at long expansion times where the density islow. One has to note though that the Rydberg atoms are redistributed over lower n states,which decay faster. For the electrons one can see the total charge decreasing continuouslywith increasing delay time. As explained, this is mainly due to the loss of Rydberg atomsbecause of spontaneous decay to non-detectable states, redistribution to low n states anddue the low detection efficiency of weakly bound plasma electrons. The ion signal is moreintriguing. Initially the total charge also decreases rapidly as for the electrons. But after

86


a delay of about 10 µs the detected charge increases again, reaches a maximum at a delaytime of about 20 µs, and then decreases again with increasing delay time. This structureis correlated with the formation of a plasma. We interpret the appearance of this struc-ture as the excitation (mainly resulting from collisions with electrons) during the plasmaformation of atoms decayed to states with n<37, that were no longer detectable.

Figure 6.10: Measured total charge in the ions an electron traces as a function of delay time,the dashed line represent a exponential decay with a lifetime of 90 µs, corresponding to a n of44.

6.3.6 Spatial distributions

In our setup we have a phosphor screen with which we can image the ions or the electronsas they arrive at the MCP detector (See Chapter 5). In Fig. 6.11 we plotted a few typicalccd images of the phosphor screen for both ion and electron detection, with and withouta plasma. The images without plasma are at a low initial trap density, the images withplasma were taken at a high density. For low densities, the size of the electron bunch issmaller than the dimensions of the phosphor screen (40 mm), for large densities however,it becomes of the same order.In Fig. 6.12 we plotted the width (1/

√e value) of the measured spatial distributions of

the ions in the directions parallel and perpendicular to the rod structure as a function ofdelay time. These widths were obtained from fitting the images with a two dimensionalGaussian distribution, which provides a simple characterization of the data. Also depictedis the total charge, and the charge in the plasma signal as a function of delay time.It seems that the widths in both directions have the same trend as the total charge. Firstthe widths decrease and after a delay time of about 10 µs the width increases again toreach a maximum at about 20 µs after which it decreases again. Furthermore, the widthin the direction perpendicular to the rod structure has an extra offset.

87

Chapter 6.

Figure 6.11: Contour plots of images obtained from the phosphor screen. (A) The electronsignal at a low density, i.e. without plasma. (B) The electron signal at a high density, i.e. withplasma. (C) The ion signal at a low density, i.e. without plasma. (D) The ion signal at a highdensity, i.e. with plasma.

The results for the electrons are depicted in Fig. 6.13. Also here the width seems to followthe charge. Here, however, there is no structure appearing and there is less of a differencebetween the widths in the direction parallel and perpendicular to the rod structure. Thewidth of the signal has at least three contributions (the initial width of the cloud is ≈ 0.2mm).

I The temperature T : without any other forces present the temperature would resultin a ballistic expansion.

II Coulomb expansion due to space-charge forces. The plasma in itself is quasi-neutral.The field ionization pulse however, strips the Rydberg atoms and pulls the plasmaapart, pushing either the electrons or the ions towards the detector. During thisflight time the space-charge forces in the cloud lead to an expansion determined bythe charge density in the cloud.

III As a third effect, the geometry of the rod structure leads to a diverging electric fieldin the direction perpendicular to the rods, leading to an additional spread in thatdirection. In the direction parallel to the rods, however, the electric field is more orless homogeneous.

88


We speculate that the divergence of the electric field in the direction perpendicular to therods is the main reason why in Fig. 6.12 (ions) there is a offset between the widths inthe directions parallel and perpendicular to the rods. Furthermore, as the width seems tofollow the charge we conclude that space-charge forces are important.For the electrons, we could observe that stray magnetic and electric fields, e.g. the MOTcompensation fields have a significant effect on the shape of the electron cloud distribution.As such it is not clear how to interpret this data as yet.

Figure 6.12: Widths obtained from fitting the phosphor images with a 2 dimensional Gaussianfit as a function of delay time. The lower traces show the total charge (open circles) and thecharge in the plasma signal (full circles).

6.3.7 Temperature, emittance and brightness

To get an upper value for the ion and the electron temperature, we plotted the width ofthe electrons and the ions as a function of the total charge in Fig. 6.14 and Fig. 6.15. Fromthese data sets we extrapolate the width to zero charge, which should eliminate the effect ofspace-charge forces. If we attribute this width completely to the effect of ballistic expansion(the measured time-of-flight for the electrons is 200 ns and for the ions, 10 µs) we get anupper value for the electron and the ion temperature. Both for the electrons and the ionswe get a temperature of 40 K. The obtained electron temperature is in general agreementwith other work, [8]. The ion temperature, however, is a factor of ten higher than thevalues obtained by Killian et al. [9], indicating that other effects besides the temperatureare determining the width of the distributions. These values can be used to get a value forthe initial emittance of the bunch using Eq. 4.5:

ε = R√

kT/5mc2. (6.4)

For the electrons we get a value of ε = 0.01 mm mrad, using R=0.2 mm (radius of the MOT)and T=40 K. Using the values obtained at the detector we can determine the maximum

89

Chapter 6.

Figure 6.13: Widths obtained from fitting the phosphor images with a two dimensional Gaus-sian fit as a function of delay time. The lower traces show the total charge (open circles).

emittance at the maximum charge, this gives us an upper value for the emittance (R= 8mm, T= 105 K) of ε= 0.5 mm mrad. This value for the emittance is an upper value sincewe ignored any correlation between x and px (see Section 1.4). At the detector, the peakcurrent at the maximum charge is ≈ 1 µA, resulting a normalized transverse brightnessB⊥ = 1×105 A/(rad2 m2), this value, which is a minimum value since we overestimate theemittance and underestimate the current, is quite low and is mainly limited by the lowpeak current due to the low electric fields.

6.4 Conclusions

In conclusion we created a very stable MOT containing over 109 Rb atoms at a maximumdensity of 1010cm−3. From this MOT we created an Ultracold Plasma by exciting the coldatoms to a 44d Rydberg state. We observed that this state then spontaneously evolves intoa UCP. We measured the evolution of the fraction of plasma as a function of delay timeand found a general agreement with previously published simulation results.We are one of the first to present the spatial images of electrons or ions pushed out of acloud of cold Rydberg atoms and plasma to a phosphor screen. We measured the width ofthese clouds as a function of delay time and found that the dynamics is mainly determinedby space-charge forces.From these measurements we can derive an upper value of 40 K for the ion and the electrontemperature of the cloud. This temperature can be translated to an initial emittance ofε=0.01 mm mrad. The minimum measured value of the transverse normalized brightnessat the position of the MCP detector is B⊥=1×105 A/(rad2 m2). The low initial emittance,validates the potential of using the electrons excited from an MOT as a low emittance

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Figure 6.14: Widths obtained from fitting the phosphor images of the electrons with a twodimensional Gaussian fit as a function of the total charge. The black circles represent themeasured widths in the direction parallel to the rods. The open squares represent the measuredwidths perpendicular to the rods.

Figure 6.15: Widths obtained from fitting the phosphor images of the ions with a 2 dimen-sional Gaussian fit as a function of the total charge. The black circles represent the measuredwidths in the direction parallel to the rods. The open squares represent the measured widthsperpendicular to the rods.

source. At this moment, however, the brightness is also quite low, which is due to the

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Chapter 6.

accelerator structure and the applied voltages. The brightness can be greatly increasedby going to higher voltages, shorter risetimes and higher charges in the second generationaccelerator.

92


Bibliography

[1] P. A. Molenaar, Ph.D. thesis (University of Utrecht 1996).

[2] M. Reijnders, internal report to be published (2006).


[4] W. Li, M. W. Noel, M. P. Robinson, P. J. Tanner, T. F. Gallagher, D. Comparat, B.Laburthe Tolra, N. Vanhaecke, T. Vogt, N. Zahzam, P. Pillet, and D. A. Tate, Phys. Rev.A 70, 042713 (2004).

[5] A. Walz−Flannigan, J. R. Guest, J.−H. Choi, and G. Raithel, Phys. Rev. A 69, 063405(2004).

[6] T. Pohl, T. Pattard, and J. M. Rost Phys. Rev. A 70, 033416 (2004).

[7] A. L. de Oliveira, M. W. Mancini, V. S. Bagnato, and L. G. Marcassa, phys. Rev. A 65,031401 (2002).

[8] J. L. Roberts, C. D. Fertig, M. J. Lim, and S. L. Rolston, Phys. Rev. Lett. 92, 253003(2004).

[9] C. E. Simien, Y. C. Chen, P. Gupta, S. Laha, Y. N. Martinez, P. G. Mickelson, S. B. Nagel,and T. C. Killian, Phys. Rev. Lett. 92, 143001 (2004).

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7

Prospects

In chapter four of this thesis we claimed that an Ultracold Plasma (UCP) can be used asa high brightness electron source, possibly increasing the state of the art brightness with atantalizing two orders of magnitude, this is mainly due to three reasons.For one, the brightness of an electron bunch scales inversely with the electron temperature;since the electron temperatures in an UCP are up to three orders of magnitude lower thanthe electron temperatures in conventional sources, assuming all other parameters are thesame, the potential increase in brightness is about three orders of magnitude.Second, because the initial temperatures are so low one can increase the initial radial sizeof the electron cloud by a factor of ten and still have an initial brightness that is a morethan ten times higher than that of standard sources. In addition, non-linear space-chargeforces, the main detrimental effect in contemporary electron sources are also reduced by afactor of one hundred. It is also possible to shape the space-charge distribution accordingto a waterbag distribution, this gives an even further reduction of non-linear space-chargeforces [1]. And third, the creation of electron bunches, followed by extraction by a fastrising electric field leads to velocity bunching, since the back of the bunch has a largerkinetic energy than the front. This results in a short bunch length, that can be below 100fs in length.To validate this claim experimentally, we built a setup in which the first cold atoms electronbunches were created. We created an UCP from a stable, reproducible Magneto-OpticalTrap (MOT) containing large particle numbers of more than 109 atoms. We are one of thefirst to image the electrons and ions from an UCP on a phosphor screen, a and powerfuldiagnostic tool that can be used to measure the temperature of the plasma and the emit-tance of electron bunches. A 2D+ MOT has been constructed that will be implemented inthe near future, this 2D+ MOT will serve as an atomic beam to load the MOT, hopefullygiving us even more particles in the MOT.The next step in the bunches project will be to implement the 30 kV, 10 ns acceleratordiscussed in Section 5.11. With this accelerator installed we can validate the promise ofan UCP as a bright electron source. Eventually the goal will be even higher voltages andshorter rise-times.

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Chapter 7.

So far we only discussed the potential of using an UCP as a bright source of electrons.However, during the last year our insight in using a cold atom trap as a source for chargedparticle applications grew. This resulted in speculations on which we will comment here.Probably the most imaginative horizon of using cold atoms as a electron source is the cre-ation of quantum degenerate electron bunches, being the fundamental limit in achievablebrightness. Assuming a density of n ≈ 1012 cm−3, which can be obtained in a standardmagnetic or optical trap [2], one would need an electron temperature of about 1 mK inorder to have a quantum degenerate electron bunch (phase-space density of about one). Innowadays experiments this is far from possible. Immediately after creation, the electronsin an UCP can be as cold as a few mK, but the intrinsic lack of correlation in an MOTtranslates itself to rapid heating of the electrons. As a result, the electrons heat up toa temperature of tens of Kelvin. This means that the creation of a quantum degenerateelectron bunch depends on the reduction of this heating effect. Different groups are pur-suing techniques to circumvent the effect of correlation heating. This can be achieved byalready adding correlation to the neutral atoms, for example by exciting the atoms from aoptical lattice [3] or a degenerate Fermi gas [4]. Another possible technique under recenttheoretical investigation is to use the dipole blockade to prevent the ionization of closepairs, close pairs being the main cause of correlation heating. A possible technique thatis being pursued experimentally is to use laser cooling to cool the ions [5]. The ions ofthe atomic species strontium and calcium have practical transitions that can be used forlaser cooling. Cooling the ions can also lead to a reduced electron temperature. As a lastspeculation we could discuss the creation of an electron bunch from a cold Rydberg gasinstead of an UCP. When a Rydberg gas is field ionized, a plasma is only created at theionization field. This gives the plasma far less time to heat up, and might thus lead to theconditions necessary to have quantum degenerate electron bunches.So far we mainly focussed on the electrons due to historical reasons. Actions are takenhowever, to pursue the development of a pulsed and a continuous cold ion source. Sincein the process of photoionization, vital for our scheme, hardly any energy goes to the ionsthe initial ion temperatures are still on the order of microKelvins. And due to their largemass the correlation heating is less dramatic, on a longer timescale, possibly allowing usto extract the ions before significant correlation heating has occurred. Furthermore theions might directly be laser cooled, minimizing the effect of correlation heating and leadingto extraordinarily low values for the emittance [6]. This would be advantageous for bothcontinuous and pulsed ion beam sources [7].The most straightforward and very promising application from a practical view, is theconstruction of a continuous ion beam from a cold atom beam. The main advantage ofusing an ion beam from cold atoms is again the low ion temperature. Furthermore in anion beam the atomic density is low enough to neglect the effect of correlation heating. Thepractical implementation of a continuous ion beam should be relatively simple compared topulsed ion or electron beams. No pulsed high voltage source is necessary and it is sufficientto have only a bright atomic beam, like a 2D+ MOT [7]. A continuous cold ion beam canbe the source for a new generation focussed ion beam (FIB) apparatus [8]. To summarize,the main conclusion of this work is that cold atoms have an enormous potential as a source

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Prospects

for charged particle applications requiring a high brightness.A major technical challenge, however, is to go from an MOT in a large vacuum chamber onan optical table that needs daily tuning, to a product that is turn-key. Very promising hereare the technical advances in the field of cold atom physics. Only ten years ago, when thefirst BEC was created, a setup filling an entire lab was needed. The setups used nowadaysto create a BEC are significantly more compact, e.g. recently a BEC has been created ona chip, in a portable apparatus, the size of a briefcase [9].

Bibliography

[1] O.J. Luiten, S.B. van der Geer, M.J. de Loos,F.B. Kiewiet, M.J. van der Wiel, Phys. Rev.Lett. 93, 094802 (2004).

[2] C.J. Foot, Atomic Physics (Oxford University Press, 2005).

[3] T. Pohl, T. Pattard, J.M. Rost, J. Phys. B: At. Mol. Opt. Phys. 37 L183 (2004).

[4] M. S. Murillo Phys. Rev. Lett. 87, 115003 (2001).

[5] T. C. Killian, Y. C. Chen, P. Gupta, S. Laha, Y. N. Martinez, P. G. Mickelson, S. B. Nagel,A. D. Saenz, and C. E. Simien cond-mat 0407138 (2004).

[6] Yosuke Yuri and Hiromi Okamoto, Phys. Rev. ST Accel. Beams 8, 114201 (2005).

[7] STW Grant Proposal.

[8] J. Orloff, L. Swanson, M. Utlaut, High Resolution Focused Ion Beams (Kluwer AcademicPublishers, 2002).

[9] S. Du, M. B. Squires, Y. Imai, L. Czaia, R. A. Saravanan, V. Bright, J. Reichel, T. W.Hansch, and D. Z. Anderson, Phys. Rev. A 70, 053606 (2004) .

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Summary

In a Magneto-Optical Trap (MOT), realized for the first time in 1987, one can trap andcool neutral atoms to temperatures below a mK. The invention of this device caused arevolution in atomic physics. With an MOT collision and spectroscopy experiments couldbe performed with unprecedented accuracy. The main breakthrough came about 10 yearslater when the atom from an MOT were used to create a Bose Einstein Condensate (BEC),a source of coherent matter. All this work has been rewarded with no less than two No-bel prizes: in 1997, Chu, Cohen-Tannoudji and Phillips received the Nobel prize for thepractical and theoretical development of laser cooling, and in 2001 Cornell, Ketterle andWieman received the Nobel prize for the realization of a Bose Einstein Condensate in adilute gas. In 1999 Killian et al. took trapped atoms in an entirely new direction whenthey succeeded in creating the very first Ultracold Plasma (UCP) from the atoms in anMOT. The temperature of such a plasma is more than 1000 times lower than the temper-atures in conventional cold plasmas. As a result, this type of plasmas received a lot ofattention, both theoretically and experimentally. So far studies of UCPs have mainly beenof a fundamental kind.In this thesis we study both the fundamental aspects of atoms excited from a magneto-optical as well as the feasibility of using the electrons and ions from a UCP as a source forcharged particle accelerators. The work in this thesis therefore pioneers the line betweenatom physics, plasma physics and accelerator physics. The first part of this thesis dealswith an experimental study of the effect of an ionizing laser on cold neon atoms. Herewe study the loss of atoms from a Magneto-Optical Trap caused by photionization. Fromthese measurements we deduce a new value for the near threshold photoionization crosssection 2.05± 0.25× 10−18 cm2 at λ = 351 nm and 2.15± 0.25× 10−18 cm2 at λ = 364 nm,which is a factor of four more accurate than previous measurements. The measured valuesagree with earlier measurements but differ significantly from theoretical values. The valueof this cross section has a practical relevance for the realization of a continuous ion source.The second part of this thesis has been conducted at the University of Virginia. We studythe role of dipole-dipole interactions in the formation of a UCP from a gas of cold Ry-dberg atoms, and the effect of dipole-dipole interactions on the broadening of microwavetransitions. We find that the broadening scales linearly with the density en scales with thefourth power of n, the principal quantum number. These results show the great impor-tance of dipole-dipole interactions in gas of cold Rydberg atoms and are also relevant for

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the possible realization of a quantum computer from Rydberg atoms.In a third part we studied the potential of using a UCP as a source for electron accelera-tors. This was done based upon elaborate simulations performed with the General ParticleTracer Code. The main result of these simulations is that, based upon realistic param-eters, the brightness, i.e., the figure of merit for charged particle sources defined as thecurrent divided by the emittance squared, of such a source can be up to two orders ofmagnitude larger than the brightness of conventional sources, which is a major advance.The emittance is a measure for the the focusability of the bunch, and scales as

√T , with

T the electron temperature. The increase in brightness is mainly attributed to the lowelectron temperatures of approximately 10 K in an UCP, resulting in an initial emittanceof 0.01 mm mrad, compared to 1 mm mrad in a conventional source. With experimentallyproven techniques it is possible to extract 10 nC/s from the source, which is a competitiverepetition rate.In the next part we describe the design, construction and testing of the first setup withwhich we want to build an electron accelerator based upon cold atoms. For this we firstbuilt a vapor-cell magneto-optical trap, in which we can trap over 109 rubidium atoms witha temperature of 200 µK, straight from a background vapor, we also designed an atomicbeam that will be implemented in the near future and can provide an increased repetitionrate. In the last chapter we show how Rydberg atoms excited from an MOT, evolve into anUCP spontaneously. We extracted a minimum of 1 pC from the plasma. As a main result,we were one of the first to show how the ions and electrons from a UCP can be imaged ona phosphor screen. From these images we can deduct an upper value for the electron andthe ion temperature, we extract for both the electrons and the ions an upper temperatureof 40 K, when we translate this temperature to an initial emittance, we obtain a value of0.01 mm mrad, proving the potential of the electrons extracted from atoms excited out ofan MOT as a low emittance source. Our measured current of 1 µA is not yet competitiveand results from the small electric field, but is expected to increase dramatically in thenear future. We will implement a 30 kV 10 ns risetime accelerator with which it shouldbe possible to get a current of about 50 mA. Furthermore we will implement the properdiagnostics to measure the proper charge, current and emittance.In the very last chapter we speculate charged particle sources from cold atoms might ad-vance the state of the art in achievable brightness for both electron and ion beams.

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Samenvatting

In een magneto-optische val, voor het eerst gerealiseerd in 1987, kunnen neutrale atomenopgesloten en gekoeld worden tot temperaturen onder de mK. Dit zorgde voor een warerevolutie binnen de atoomfysica. Zo konden botsing- en spectroscopie-experimenten metongekende preciesie gedaan worden. De grote doorbraak kwam er echter ongeveer tien jaarlater. In 1995 werden voor de eerste keer atomen uit een magneto-optische val gebruikt omeen Bose Einstein Condensaat (BEC) te maken, een bron van coherente materie. Al ditwerk werd dan ook beloond met niet minder dan twee Nobelprijzen, in 1997, kregen Chu,Cohen-Tannoudji en Phillips de Nobelprijs voor de praktische en theoretische ontwikkelingvan laserkoelen. In 2001 kregen Cornell, Ketterle en Wieman de Nobelprijs voor het makenvan een BEC in een ijl gas. Heden ten dage is een MOT uitgegroeid tot het werkpaard vande koude atoomfysica gemeenschap. In 1999 sloegen Killian et al. een nieuwe weg in metkoude atomen door het maken van een ultra koud plasma van atomen uit een magneto-optische val. De temperatuur van een dergelijk plasma is tot duizend maal kouder dan detemperatuur van een conventioneel koud plasma. Sindsdien krijgen dergelijke plasma’s danook veel experimentele en theoretische aandacht. Tot dusver is het onderzoek aan ultrakoude plasma’s echter veeleer van fundamentele aard geweest.In dit proefschrift daarentegen bestuderen we enerzijds fundamentele eigenschappen vanatomen geexciteerd uit een magneto-optische val. Anderzijds bestuderen we de haal-baarheid om de elektronen en ionen van een ultra koud plasma gemaakt uit een magneto-optische val te gebruiken als bron voor een elektronen- of ionenversneller. In dit werkbegeven we ons dan ook op het grensgebied van atoomfysica, plasmafysica en versnellerfys-ica.Zo zijn we eerst de wisselwerking van een ionisatielaser met koude neon atomen gaanbestuderen. Dit hebben we gedaan door te kijken naar het verlies van deeltjes uit eenmagneto-optische val tengevolge van de laser. Uit deze metingen hebben we een waardekunnen halen voor de foto-ionisatie doorsnede die een factor vier nauwkeuriger is danvorige metingen 2.05 ± 0.25 × 10−18 cm2 bij λ = 351 nm en 2.15 ± 0.25 × 10−18 cm2 bijλ = 364 nm. Deze meer nauwkeurige waarden komen overeen met eerdere metingen maarwijken significant af van eerder gedane berekeningen. De waarde van deze doorsnede is vanbelang voor realisatie van een continue ionenbron, een natuurlijke uitbreiding van het elek-tronenproject. Het tweede deel van dit werk is uitgevoerd aan de Universiteit van Virginia.Hier zijn we gaan kijken naar de rol van dipool-dipool interacties in de vorming van een

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ultra koud plasma uit een gas van koude Rydberg atomen. Tevens hebben we het effectvan dipool-dipool interacties op microgolfovergangen bestudeerd. Zo hebben we verbredingvan overgangen gemeten waarbij de verbreding lineair verloopt met de dichtheid en schaaltmet de vierde macht van n, het hoofdquantumgetal. Deze resultaten kunnen relevant zijnvoor een eventuele praktische realisatie van een quantum computer.In een derde deel zijn we gaan kijken naar het potentieel van een ultra koud plasma alsbron voor elektronenversnellers, dit hebben we gedaan op basis van uitvoerige simulatiesgedaan met de General Particle Tracer code. Het belangrijkste resultaat van deze simu-laties gebaseerd op realistische parameters is dat de helderheid van een dergelijke bron, hetvoornaamste getal waarmee geladen deeltjesbronnen vergeleken worden en gedefineerd isals de stroom gedeeld door de emittantie in het kwadraat, tot twee ordegroottes groter kanzijn dan die van bestaande bronnen, hetgeen een enorme stap voorwaarts is. De emittantieis een maat voor de mate waarop een bunch gefocusseerd kan worden en schaalt als

√T ,

met T de elektronen temperatuur. De toename in helderheid is vooral te wijten aan de lageelektronen temperatuur van 10 K in een Ultra Koud plasma, dit leidt tot een emittantievan 0.01 mm mrad, in vergelijking met 1 mm mrad voor conventionele bronnen. Met reedsbewezen technieken moet het mogelijk zijn om 10 nC/s uit de bron te halen, hetgeen eencompetitieve flux is.In het laatste deel van dit werk beschrijven we uitvoerig het ontwerp, de bouw en deeerste tests van de eerste opstelling met dewelke we een elektronenversneller willen bouwengebaseerd op koude atomen. Hiertoe is er vooreerst een magneto-optische val gebouwd, indewelke we meer dan 109 rubidium atomen hebben kunnen opsluiten bij een temperatuurvan 200 µK. De atomen werden gevangen uit een achtergrondgas. In een tweede deel vandit hoofdstuk hebben we laten zien hoe Rydberg atomen geexciteerd uit een magneto-optische val, spontaan overgaan in een ultra koud plasma. Zo konden we minstens 1 pClading vrijmaken. Als hoofdresultaat hebben we als een van de eersten laten zien hoe deelektronen en ionen van een dergelijk plasma met behulp van een fosforscherm afgebeeldkunnen worden. Uit deze afbeeldingen kunnen we een bovengrens halen voor de elektronenen de ionen temperatuur, voor zowel de elektronen als de ionen hebben we een temperatuurvan 40 K gemeten. Wanneer we dit vertalen in een emittantie, krijgen we een waarde van0.01 mm mrad, hetgeen bewijst dat de geexciteerde atomen van een magneto-optische valals lage emittantie bron kunnen dienen. De gemeten stroom is slechts 1 µA, hetgeen teverwachten is gezien de lage elektrische velden. In de nabije toekomst zullen we een 30 kV,10 ns stijgtijd versneller implementeren waarmee we verwachten een stroom van 50 mA tehalen. Verder zullen we de nodige diagnostiek moeten bouwen om een goede meting vande lading en de emittantie te doen.In een allerlaatste hoofdstuk speculeren we dat geladen-deeltjes bronnen gebaseerd opkoude atomen de state of the art in helderheid kunnen vergroten voor zowel elektronen-als ionenbronnen.

102

Dankwoord

Als laatste wil ik graag eindigen met het luchtige gedeelte dat steevast eerst wordt gelezen,het dankwoord. Vooreerst wil ik mijn dagelijkse begeleider Edgar bedanken voor zijn on-voorwaardelijk en bewonderenswaardig geduld tijdens mijn promotie. Zijn steun als fysi-cus en als mens en legendarische interactie met de opstelling hebben me enorm geholpen.Edgar, merci en veel succes! Ook Herman zijn relativeringsvermogen wordt ten zeerstegewaardeerd. Ik zou verder graag mijn commissie willen bedanken voor de snelle en accu-rate commentaar op mijn proefschrift. In het lab heb ik veel plezier gehad met lotgenotenColin (bunny suit), Veronique, Jonathan (dancing queen) en Gabriel, samen hebben we onsvaak met Sisyphus kunnen identificeren. Verder heb ik veel lol gehad in de groep AQT metcollega-aio’s Kenian, Bout, Edwin, Alquin, Paul, Eric, Maarten, die ik veel sterkte wens enBart. Ik heb erg veel geluk gehad met de studenten die ik heb mogen begeleiden, zo waren erde afstudeerders Ralf en Mereijn en stagiairs Raf, Matthias, Ruud en David. Ton zou tocheens een boek (de hitchhikers guide of Nuenen) moeten schrijven over alle straffe dingendie hij heeft meegemaakt, ontploffende wc-rollen en klei-incidenten met muizen, niets is tezot voor Ton. Servaas (sommige theoreten komen ook soms buiten de Hilbert ruimte) benik dankbaar voor de vermeende pogingen me aan het rekenen te zetten. Voor Rina boomwas geen brief noch stempel teveel, merci daarvoor. Na een jaar promoveren kwam er eensamenwerking met de groep fysica en toepassingen van versnellers zonder dewelke ik nooitzou mogen hebben genieten van het enthousiasme en de heldere kijk van Jom, de sceptischeen terechte opmerkingen van Bas en de ideeen en opmerkingen van Marnix. I had a nicetime at the university of Charlottesville in Virginia with Paul Tanner, Ofir Garcia and TomGallagher. Het werk dat we de afgelopen jaren hebben gedaan was nooit mogelijk geweestzonder de hulp van een groep ondergewaardeerde, verschrikkelijk goede technici, Jolanda,Ad, Harry, Wim en uber-vernufteling, vulkaan in wording, Louis. Verder wou ik Bertus enHerman bedanken voor hun hulp voor de kleine negerkindjes, maar vooral Rein voor zijntomeloze inzet en volharding. Ook buiten de TU ben ik mensen dank verschuldigd, zo zijner de Chevaliers die jaarlijks in juli en september voor een chaotisch intermezzo wisten tezorgen. Geert en stijn wou ik bedanken voor het gepalaver onder invloed van het goudengerstenat. Ik ben ook dank verschuldigd aan de families Claessens, Driessen, Lenaers enChristiaens. De zaterdagmiddag in Olympia was steeds een oase van relativering, reflectieen rust door de erudiete en serene conversaties met de para’s. Dichter bij huis wil ik graagde families Schreurs-Van Hove, Vanderstappen-Brouwers, en de familie grispen-Lemmens

103

uit Berg bedanken. Zo ook wil ik graag mijn zus Leen en broer Gijs bedanken die tochniet enkel aangename momenten met hun broer hebben mogen doormaken. Ik wil graagmijn ouders, op wie ik erg trots ben, bedanken voor hun jarenlange steun en liefde. Entenslotte mijn liefste Joke van wie ik zielsveel hou, een stuk van mezelf die er altijd voorme is en waarmee ik hoop nog lang, gezond en gelukkig samen te kunnen zijn, Joke merci,ik hou van u.

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Curriculum Vitae

18 oktober 1978 Geboren te Maaseik,opgegroeid te Neeroeteren

1990–1996 Algemeen Secundair Onderwijs,Heilig Kruiscollege te Maaseik, Belgie

1996–1998 Studie Natuurkunde,Limburgs Universitair Centrum te Diepenbeek, Belgie

1998–2002 Studie Technische Natuurkunde (cum laude),Technische Universiteit Eindhoven

2002–2006 Promotieonderzoek,groep Atoomfysica en Quantum Technologie,Faculteit Technische Natuurkunde,Technische Universiteit Eindhoven

maart–juli 2005 Werkbezoek University of Virginia

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dynamics and applications of excited cold atoms · cover requiem chevalier vampire...

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