Nipun Vats

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy

Graduate Department of Physics University of Toront O

Copyright @ 2001 by Nipun Vats

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Abstract

Xon-Markovian Radiative Phenornena in Photonic Band-Gap Materials

Nipun Vats

Doctor of Phiiosophy

Graduate Department of Physics

University of Toronto

2001

We present theoretical analyses of CO herent and inco herent radiative p henomena fiom

active materiais embedded within photonic crystals (PCs). Fluorescence in PCs is de-

scribed in terms of the local density of electromagnetic modes (LDOS) at the position

of the active elements. We derive expressions for experimentally rneasurable quantities

and test our formalism using various models of the LDOS. The radiative emission of a

classical dipole oscillator in a PC is then described by coupling the systern oscillator to a

large but h i t e set of discrete oscillators describing the resemoir density of modes within

a PC. This classical analysis motivates the study of radiative emission from microwave

PCs.

We next discuss the collective emission of Light Erom N two-level atoms with a reso-

nant fiequency near the edge of a photonic band-gap (PBG). Mean-field theory shows a

macroscopic atomic polarization in the atomic steady state. This suggests the existence

of an associated coherent radiation field IocaIized about the atoms, in the absence of

an external cavity mode. The effects of quantum fluctuations on collective emission dy-

namics are shown to difXer strongly from those in free space, due to the non-Markovian

atom-field interaction near a photonic band-edge. A classical noise ansatz is intzoduced

that simulates the effect of the tempordy corrdated quantum fluctuations of the elec-

tromagnetic reservoir. The laser-& properties of superradiant ernission near a photonic

band-edge Iead us to hypothesize that a band-edge laser system may exhibit a reduced

laser threshold and a laser field that will exhibit phase diffusion significantly different

&om that in a conventional cavity laser.

Finaiiy, we propose as a unit of quantum information (qubit) the single photon oc-

cupation of a localized field mode within an engineered network of defects in a PBG

material. Qubit operations are mediated by opticaüy excited atoms interacting with

these localized States of light as the atoms traverse the connected void network of the

PBG structure. We describe conditions under which this system can have independent

qubits with controiiable interactions and very low decoherence, as required for quantum

information processing.

Dedicat ion

In loving memory of my grandfather, Mr. Suraj Prakash Bates.

Acknowledgement s

Firstly, 1 would like to thank Prof, Sajeev John for his careful reading of this thesis, his

hancial support, and his patience. The great freedom he gave me to follow my o n

research interests is surely rare in a supervisor, and was instrumental in shaping this

thesis.

Throughout my graduate studies, 1 have been most fortunate to have the opportunity

to work with exceptional people who have provided me with much encouragement, guid-

ance, and knowledge, both inside and outside of physics. There are two individu& who

deserve a speciai thanks bordering on eternal gratitude. The first is Dr. Tran Quang,

without whom this thesis could not have begun. His enthusiasm for the field, and his

constant encouragement to "get things done" got me started in the right direction. The

insights he shared taught me a great deal about my field. And his sense of responsibility

towards the students around him (such as myseif) set an example that I d l take with

me weil beyond my physics career. Thank you Quang; your departure from physics was

t d y a great los.

The second is Dr. Kurt Busch, withoiit whose help this thesis would likely have never

been completed. Kurt has aiways been wiilhg to share his considerable physics knowledge

and insight, and as a result, he has been a very generous teacher and collaborator.

On a more personal note, his friendship and counsel have dso helped me though some

particularly rough spots dong the way to this degree. Working with you has b e n and

continues to be a pleasure, Kurt.

1 shouid also single out the "Aussie Contingent" : Prof. Barry Sanders' approachability

and enthusiasm during his visit to Toronto led to many illuminating discussions, The

encouragement he has given me over the past few years has b e n of great help to me.

Behind Dr. Terry Rudolph's swagger and his (formerly) long locks of hair c m be found

a good-natured physicist with a head chock full of good ideas. Thank you Terqy for

thinking enough of our joint ideas to see our coliaborative work through to fruition.

Finaiiy, 1 must thank those near and/or dear to me who have helped me though this

degree. Thanks to my mother Neena and my siçter -4ngel for their unconditional love

and support at aU times, most especially on the phone in the middle of the night. Thanks

to my office mates, particularly Bruce Elnck and later Marian Florescu, for making it

a pleasure to come to work, and to Ovidiu Toader, whose computer wizardry and great

patience enabled me to complete this degree without blowing out (or blowing up) a single

computer. It was a distinct pleasure to learn about the field of photonic crystals jointly

with my hiend and fellow student, M e s h Woldeyohannes. My sincere appreciation to

aii my friends who helped me to keep some semblance of sanity, most especially Anne

Cohen, the inimitable Jenny Krestow, and my dear fnend and colleague, Rob Spekkens.

A final extra special thanks to Lora Ramunno, for her friendship, her Herculean support,

and for sharing her enormous heart and her very special mind - thank you.

1 wish to acknowledge the Physical Review, the Journal of Modern Optics and my

collaborators for their permission to include in this thesis previously published and co-

authored works respectively.

Contents

1 Introduction 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Photoniccrystals 1

. . . . . . . . . . . . . . . . . . . . . . 1.2 Active media in photonic crystals 6

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Outline IO

2 Theory of fluorescence in photonic crystals 13

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction 15

2.2 Atom-field coupling in a photonic crystal . . . . . . . . . . . . . . . . . . 16

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Equations of motion 21

. . . . . . . . . . . . . . 2.4 Evaluation of fluorescence spectra and dynamics 24

. . . . . . . . . . . . . 2.0 Fluorescence for mode1 photon densities of states 30

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Discussion 42

3 Radiating dipoles in photonic crystals 45

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction 45

3.2 Classicd field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3 Projected iocai density ofstates, m a s renormalization and Lamb shift . 51

3.4 Discretkation of the reservoir . . . . . . . . . . . . . . . . . . . . . . . . 54

. . . . . . . . . . . . . . . . . . . . 3.5 Numerical results for a mode1 system 56

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Discussion 59

4 Non-Markovian quantum fluctuations and superradiance near a pho-

tonic band-edge 62

4.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2 Low atoniic excitation: harmonic osciiiator mode1 . . . . . . . . . . . . . 68

4.3 High atomic excitation: mean field solution . . . . . . . . . . . . . . . . . 74

4.4 Band-edge superradiance and quantum fluctuations . . . . . . . . . . . . 83

4.5 Sirnulated quantum noise near a band-edge * . , . . . . . . . . . . . . . 94

4.6 Conclusions . . . . . . - . . , , . . . . . . . . . . . . . . . . . . . . . . . 98

5 Quantum information processing in localized modes of light within a

p hotonic band-gap mat etial 102

6 Conclusions and future directions 114

A Outline of photonic band structure calculations 120

B Classicai field theory for a radiating dipole 124

B.1 Free-field Hamiltonian - . , a . . . . . . . - . . . . . . . . - . . . . . . 124

B.2 Radiating dipole embedded in a Photonic Crystal . . . . . . . . . . . . . 126

C Isotropic dispersion and photon density of states 128

D Caiculation of the memory kernel 130

E Evaluation of [ A A ( ~ ) )*

Bibliography 135

Chapter 1

Introduction

1.1 Photonic crystals

Photonic crystals (PCs) are periodic dielectric structures that strongly moc& the dis-

persion (energy-momenturn) relation of iight [l]. This is achieved through the carefully

engineered interplay between scattering resonances from individual elements of the pe-

riodic array and Bragg scattering hom the dielectric lattice. Fkom this definition, these

materials are seen to be the photonic analogue of semiconductor materials as the latter

relate to electrons. PCs are of great scientific and technological interest for their ability to

drasticaiiy alter the nature of the propagation of light [2,31, as weil as for their associated

ability to modi& the radiative properties of optically active materials embedded within

them [4]. It is this Iatter property that forms the basis for the phenornena described in

this thesis.

The most drastic effect on iight propagation occurs when a PC is designed so that the

propagation of light is prohibiteci in aU three spatial directions over a continuous range

of frequencies. This propagation-hee frequency range is known as a photonic band-gap

(PBG). It has been some 14 years since the initial propasal of the concept of the PBG

[3,4], and 10 years siuce the creation of the ht PBG materiai at microwave frequencies

[5]. Nevertheles, it is oniy in the past few years that great strides have been taken in

the production of PBG materiais at the technologically important optical and near-IR

kequencies. This is a resdt of the rather arduous conditions that must be satisfied for the

fabrication of a material exhibiting a cornplete PBG. To produce a PBG materiai typically

requires a dielectric lattice whose periodicity is comparable to the relevant wavelengths

of light. We mua aiso ensure that we obtain the fiequency overlap of stop bands for

photon propagation that arise when the Bragg condition is met for light traveling in

various directions. This requires 6rst that the stop bands in each direction be sufficiently

large, requiring a large dielectric contrast between the constituents of the dielectric lattice.

.k weli, the iikelihood of attaining overlapping stop bands is maximized by choosing a

crystal structure for which the Bragg condition is as similar as possible for al1 incid.ent

directions. This motivates the construction of a PC with a FCC-like structure, for which

the Brillouin zone is as nearly spherîcal as is allowed for a threkimensionally periodic

structure. Along with these conditions, there are additional constraints on the topology

of the structure (formation of a PBG is favoured by a connected, network topology), as

weli as on the dimensions of the constituents of the dielectric lattice.

One of the strengths of the PBG concept is its applicability at any Iength and he-

quency scaie, as there is no fundamental length scaie contained in Maxwell's equationsL.

Despite this fact, the above conditions are clearly difficdt to achieve on the micron length

scales of optical PCs. However, in the past two years advances in micro-lithography [61

and in the fabricatioc of seif-organizing colloida1 systems [7] have produced materials

wbich prohibit photon propagation over a large range of directions, giving rise to a pho-

ton propagation pseudogap. More recently, materials that suggest the presence of a full

PBG at fiequencies in the near-IR have been produced [a, 91. In particuiar, an inverse

'Our non-reiativistic treatment of the atom-fieid interaction in this thesis requixes the introduction of an ultraviolet cutoff in order to avoid the indusion of Iogarithmic divergences at high frequencies (see Ch. 2). This sets a fundamental length scale at the cutoff wavelength, the Compton wavelength of the eiectron 10-'*m, which is six orders of magnitude d e r than optical wavelengths. -4s a result, we c m dectively neglect the influence of this length scaIe-

Figure 1.1: SEM image of a fabricated inverse opai structure. Source: Ref. [81

opal PBG materiai has been constructed by infiltrathg a dielectric material into the

voids of an FCC colloidai synthetic opal crystd grom by self-assembly, and then etch-

ing away the initiai colloidai template [8]. Tho result is a PBG material that can be made

highly ordered over hundreds of lattice constants and that c m eventually be produced

in a cost-effective manner (Fig. 1.1).

It is because of the very recent advent of such materials that until the past year

technological efforts had been focussed almost excIusively on easier-twfabricate one-

and two-dimensional anaiogues of three-dimensionai PBG materiais. While such lower-

dimensionai materiais have very interesting properties in their own right, their effects on

photon propagation are, in general, not as drastic as in the case of full three-dimensional

inhibition of propagation. Furthemore, as we shaii see, the nature of the photon disper-

sion relation is dimensionally dependent, niaking the three-dimensionai case fundamen-

taliy different from its lower-dimensional counterparts. In what foiiows, we shail focus on

threedimeflsionai PCs exhibithg either a photon propagation pseudogap or, most often,

a photonic band-gap- We shall assume that the dielectnc materiai making up the PC is

Figure 1.2: Bandstructure for a fcc lattice of air spheres in silicon ( E z 11.9). Source: Ref. [12]

linear, frequency-independent, isotropic and essentidy non-absorptive in the fiequency

range of interest; materials that sati* these properties at sufEciently low field intensities

for fiequencies in the near-IR include Si and Gap. In so doing, we however acknowledge

the rich potential for novel non-linear opticai processes in photonic crystai materials [IO].

We note that while much of our work relates to PCs at opticai and near-IR length scales,

in Chapters 3 and 5 we also discuss applications of PBG materials to radiative phenom-

ena at microwave wavelengths. The more advanced state of fabrication techniques in the

microwave regime suggests that the investigation of microwave radiative phenornena may

also prove to be a h i t f u l avenue of research.

The photonic dispersion relation associated with a photonic crystal may be under-

stood by making an analogy with electronic band structure. -4s described in greater detail

in -4ppendix A, the periodic dielectnc plays the role of a periodic potential for Light which

obeys an eigenvalue equation for the magnetic field based on Maxwell's equations [Il].

The solutions to this equation are a discrete set of eigenfrequencies for a given value of

Figure 1.3: Density of states (DOS) for a fcc Iattice of air spheres in silicon (E N 11.9). Source: Ref. [12].

the wavevector k in the 6rst Brilloui. zone, w,,,k, and the associated eigenfunctions for

the electric field, Enlk (obtained from the magnetic field solution), where the band ind~,u

n labels the specific eigensolution. The resulting photonic bandstructure for an inverse

opal photonic crystai is shown in Fig. 1.2. We clearly see that the dispersion relation

deviates strongly from the linear dispersion exhibited by light in Uee space. There is also

a clearty identifiable PBG between the 8th and 9th bands. !hile this procedure may

seem completely analogous to the electronic case, we m u t be careful to note that there

are very strong differences between electronic and photonic bandstructure. For example,

the fact that photons are bosons rather than fermions means that many photons may

occupy the same state, By summing over the states available at a given frequency for al1

wavevectors, we may obtain the density of photon modes, or densiv of states (DOS), at a

given frequency, N(w) , associated with a given photonic bandstructure. -1s an ewnpIe,

the density of states corresponding to the bandstructure of Fig, 1.2 is shown in Fig. 1.3,

We shall explicitly use and expand upon the concept of the density of states in a PC in

Chapters 2 and 3; it also serves as a pedagogical tool for the discussion below.

1.2 Active media in photonic crystals

It has Iong been established that the nature of atomic spontaneous emission is not a~

immutable property of an atom, but is iustead determined by the atom's environment

[131. This fact may be glimpsed by the use of time-dependent perturbation theory in the

form of Fermi's Golden Rule, which gives the transition probability R per unit tirne for

an atomic electron to go from an initial state 12) to a final state 1 f ) as

where X IabeIs the two photon polarization states, V is the interaction Hamihonian

between the atomic electron and the electromagnetic vacuum, and N(W+~) is the density

of states (DOS) at the transition fiequency between the initial and finai states. Such a

description of spontaneous decay processes is a reasonable approximation only when the

set of possible final states forms a broad and featureless continuum. This is clearly not

the case in a PBG material. Nevertheless, it demonstrates that the decay process is

strongly dependent on the density of photon modes available to an atom in the vicinity

of the atomic transition frequency. in general, the smooth w2 dependence of N(w) in

free space may be modified by imposing boundary conditions on the electromagnetic

field on length scales comparable to the wavelength range being considered. Such a

condition is satisfied, for example, by microcavities used in conventional cavity quantum

electrodynamic experirnents [141. As we have shown, photonic crystals may also strongly

modify N ( w ) Erom its free space value. in contrast to microcavities, PCs d o r d scalability

to any Erequency rangeL, as well as providing a much richer photonic mode structure t hat

may be used to realize a host of novel quantum optical phenomena.

When discussing the atom-field interaction in a photonic crystal exhibit ing a PBG,

there are essentiaiiy three regimes of interest, as depicted schematicaiiy in Fig. 1.4: (i)

deep inside a PBG, (ii) near the edge of the gap, and (üi) in the vicinity of a feature in the

continuum, e.g. a van Hove singularity. For an atom with a radiative transition frequency

deep inside the gap (labelled (i)), the absence of electromagnetic modes, or in the language

of quantized fields, of electromagnetic vacuum fluctuations, means that the atom is unable

to spontaneously emit a photon through a single photon process. Furthemore, whïie

two-photon emission is permitted, this process has a very low probability. ..As a remit,

a strongly coupled photon-atom bound state is formed [15], in which the photon is

localized about the atom. In this regime, a group of active atoms placed in a PBG

materid will interact with each other primariiy through a modified resonance dipole-

dipole interaction, which rnay give rise to novel collective states of the active medium

[16]. By fabricating a localized defect of the appropriate dimensions within the crystal

structure, one may also produce a single Iocalized defect mode deep within a PBG 1171-

Frequency

Figure 1.4: Schematic drawing depicting the basic features of the photon DOS in a PBG material. The red curve describes the DOS as a function of frequency. The blue shaded regions identify Çequency ranges of particuiar interest. Region (i): deep inside the PBG. Regim (ii): near a photonic band-edge. Region (5): in the vicinity of a van Hove singularity or another "strong" feature in the continuum of modes.

As we suggest in Chapter 5, such defect modes may possess a very high quality factor,

thus providing an extremely promising environment for cavity QED experirnents.

Most of this thesis will focus on the second regime of interest, in which the relevant

atomic transition fiequency lies near the edge of a PBG ((ii) in Fig. 1.4). In this case,

the atom feels the influence of both the gap and the electromagnetic continuum of modes

outside the gap, giving rise to a completely new type of atom-field interaction. Near

the band-edge, the photon density of states is restricted and rapidly varying, making

it dramatically different from the w2 dependence in free space, More iundamentally, we

may describe the situation in terms of a system-resemoir interaction, ic wtiich we have an

atomic system coupled to an electromagnetic resemoir of photon modes that are subject

to quantum fluctuations [18]. In these terms, the DOS near the band edge is such that

the correlation tirne of electromagnetic vacuum fluctuations is not negiigibly small on the

time scaie of the evolution of the atomic system. In fact, the reservoir exhibits iong-range

temporal correlations, making the temporal distinction between the atomic system and

electromagnetic mervoir difficult to identify. This is in marked contrast to the free space

case, in which a smooth and broad reservoir density of modes impiies that the reservoir

quickly relaxes to its initiai state, and thus exhibits no memory of its previous state on

the tirne scale of the atomic dynamics. This fact permits the application of the Born-

Markov approximation scheme to £ree space quantum optical systems. CIeariy, such

an approximation is invalid near a photonic band-edge, where reservoir memory effects

are the source of both the novei behaviour and the theoretical complexity of band-edge

radiative systems.

Whiie not a major focus of the present work, we note that interesting radiative effects

are not limited in fiequency to the immediate vicinity of a PBG. It is evident kom F i s .

1.2 and 1.3 that the photonic dispersion relation and the associated density of states

possess a rich structure even within the aiiowed continuum of electromagnetic modes.

For example, the DOS of a PC d l exhibit van Hove singularities, corresponding to

saddie points in the dispersion relation, for which the h t derivative of the DOS is

undefined ((iii) in Fig. 1.4) [19, 201 . There may &O be rapid variations in the DOS over

small frequency ranges. Such features of the DOS can be expected to have a considerable

effect on the emission properties of active media. In Chapter 2 we provide a formalism

for determining the effect of these and other features of the DOS on radiative atornic

emission. More specifically, we argue thz'c one must explicitly take into account the

spatial variation of the electromagnetic field modes in a PC, motivating the introduction

of a local density of States (LDCiS j. .Ah in Chapter 2, we show that, even in the absence

of a complete gap, the presence of a strong pseudogap in the DOS may have a very

significant effect on spontaneous emission.

For the purposes of simplicity and concreteness, we take the active medium to consist

of two- or three-level systems with transitions in the frequency range of interest. In this

thesis, we use the terms "atomsn and "active medium" interchangeably. In practice, the

"atomic" system may take the form of an actual atomic vapour, quantum dots ("artificial

atoms"), or electron-hole pairs in a semiconductor. In Chapter 2, we briefly describe how

our results for atomic systems may be extended to more complicated active media (e-g.,

fluorescing dyes) where appropriate. With the exception of the work presented in Chapter

5, the effects we describe are evident in, and often require, a system of many atoms. -As

a result, these "buik" phenornena should be much more readily realizable experimentally

than those that involve the precise control of single atoms.

Out line

The remainder of this thesis is divided into five chapters. In Chapter 2, we present a

formalism for the accurate description of fluorescence Erom an active medium within a

PC. This anaiysis is also relevant to the description of the atomic decay contribution to

quantum optical processes in PCs. In the process, we derive the form of the atom-field

coupling in a PC. Our formalism naturaiiy takes into account the fact that the density of

states at a given position in a PC (the LDOS) is determined not only by the dispersion

relation, but also by the d u e of the electromagnetic field modes at that point. We then

proceed to develop expressions for experimentaily measurable quantities in fluorescence

experiments, and we test Our approach on severai mode1 densities of states. In Chapter 3,

we obtain the emission dynamics of a classical electric dipole osciiiator in a PC by treating

the dipole's coupling to a large but finite nuniber of discrete EM resewoir oscillators with

a spectral density appropriate to the density of modes in the crystal. This description

of radiative dynamics is appropriate to osciilating dipoles in the microwave regime, and

also provides a description of atomic emission in the optical regime in situations where

saturation effects are negligible.

The fluorescent emission described in Chapter 2 involves the incoherent emission

£rom an initially excited collection of atoms. In contrast, in Chapter 1 we Great the

collective coherent ernission from a dense cokction of identical atoms embedded in a

PBG material, each with an atomic resonance fiequency near a photonic band-edge.

This is the phenornenon of superradiance, or superfiuorescence. Superradiant emission is

characterized by the decay of a collective atomic dipole moment, which may be treated

semiclassically, triggered by quantum fluctuations of the electromagnetic vacuum at early

times. We first provide an analytical treatment of the band-edge supenadiance in the

limit of low initiai atomic excitation. This is foiIowed by a detailed analysis of the mean-

field or semiclassical regirne of superradiance. Next, we explicitly treat the influence

of band-edge quantum fluctuations on superradiant ernission and thus show how the

temporal correlations of the electromagnetic reservoir distinguish the band-edge case

fiom superradiance in free space. Finailx we introduce a classical noise ansatz that

successfully sirnulates the effect of band-edge vacuum fluctuations on the superradiant

system.

We then move fiom the treatment of collective radiative phenomena near a photonic

band-edge in Chapter 4 to an investigation of a single atom phenornenon deep inside

a PBG in Chapter 5. Specilicdy, we investigate the potentiai of tramferring single

photons to locaiized defect modes by means of single atoms traversing the void regions

of a PBG material. We elucidate the advantages of this coafiguration over conventionai

cavity QED systerns, and discuss how the localized photons thus created may be used as

quantum bits (qubits) for quantum information processing applications.

Chapter 6 presents a sumrnary of the thesis, dong with suggestions for promising

theoretical extentions and experirnental applications. The Appendices contain additionai

details about our work.

Chapter 2

Theory of fluorescence in photonic

cryst als

In the next two chapters, we present a description of radiative phenomena in PCs (in-

cluding those that do not exhibit a full band-gap) using a reaiistic description of the

modal density seen by active eIements pIaced withia such a crystai,

Central to this discussion are the photonic bandstructure of a photonic crystal and the

associateci photon density of states. -4s outlined in Chapter Il the photonic bandstructure

gives the photon eigenfrequencies, un*, for each unique mvevector in the first Briiiouin

zone of a photonic crystal (Fig. 1.2). The corresponding total density of photon states

for a given frequency, iV(w), is obtained by counting the modes available for a given

frequency. However, the total density of states does not accurately describe the modal

density seen at a particular point in the unit cell ofa crystal, as the field modes available

for a given frequency vary from point to point within the crystal. This is to be expected

due to the absence of translationai syrnmetry within a unit ceU. Figure 2.1 strikingIy

demonstrates how the local fieId can change throughout a photonic crystai. Here, we

compare the electric field intensity in a siiicon inverse opat (FCC) photonic crystal for

the W-point of the Brillouin zone at the upper and Iower band edges. As expected fiom

CHAPTER 2. THEORY OF FLUORESCENCE IN PHOTONIC CRYSTALS

Figure 2.1: Band-edge electric field intensity in an inverse opai silicon photonic crystai. Outlined circle in centre of the image corresponds to a cross-section of a spherical void region of the crystai. Red denotes high field intensity, whereas blue denotes low intensity. (a) Field intensity at lower band edge (8th band); (b) Field intensity at upper band edge (9th band). Figure generated by O. Toader.

energy considerations [Il], at the lower band edge the field lies almost exclusively in

the dielectric fraction of the crystai, whereas the field is primarily in the void region at

the upper band edge. An accurate description of the atom-field coupling must therefore

incorporate the local density of states, Ni (r, w ) , the evaiuation of which requires explicit

knowledge of both the eigenfrequencies and the eigenfunctions for the fields at a specific

point in a unit cell.

In the present Chapter, we develop a quantum formalism for the description of fiub

rescence from active media embedded in photonic crystals pmessing either a pseudogap

of a fidl photonic band-gap. In the next Chapter, we demonstrate how the radiative

dynamics of an osciiiating eIectric dipoie can be described by a classical treatment that

makes explicit the connection between the photon density of states and the avdable

modes of the electromagnetic reservok These chapters are thus aimed at interfacing

realistic caicuiations of the availabIe photonic modes with calculations of the atom-fieId

interaction in a PC. For the tirne being, we present a formalisrn for the description of

radiative emission in PCs, and demonstrate the vaiidity of this formalisrn through a series

CHAPTER 2. THEORY OF FLUORESCENCE iN PHOTONIC CRYSTALS

of mode1 calculations.

Introduction

As outlined in the introductory Chapter, recent advances in microfabrication have re-

sulted in the creation of photonic crystals which possess photonic dispersion relations

that are strongly modiiied from free space at irequencies in the optical and near-IR. To

date, we have seen the fabrication of materials with possess strong photon propagation

pseudogaps [7, 211, which prohibit photon propagation in certain directions, and materi-

als which appear to possess a full photonic band-gap [8,9], for which photon propagation

is prohibited in ail three spatial directions. Such modiications of the photon dispersion

relation, and of the associated photon density of States, have been predicted to strongly

modify the radiative dynamics of optically active materials placed within a PC. C:n-

til now, most predictions have however been based on idealized models that focus on

specific features of the photonic dispersion.

It is the aim of this work to provide an efficient formalisrn for interfacing realistic

calculations of the photon dispersion relation (and the associated spatial distribution

of the EM modes) in a PC with calculations of the emission properties of active media

embedded in these materiais. In particular, we treat the phenomenon of fluorescence fiom

a dilute distribution of active elements pIaced within the high or low dielectric fraction

of a PC [22]. The study of fluorescence from within a PC is of considerable interest for

a number of reasons. First, it provides an important tool for the characterization of a

PC. -4ctive elements within the crystai may coupie to modes that are inaccessible from

outside the crystai due to the &match in symrnetry between Bloch modes within the

crystd and external phne waves [23,24]. -4.5 a result, fluorescence fiom a PC may prove

to be a more reliable means of determinhg the presence of a full PBG than reflectioa

and transmission experirnents [25, 261. Second, our fonnalism permits an evaluation

CHAPTER 2. THEORY OF FLUORESCENCE IN PHOTONIC CRYSTALS 16

of qualitative treatments of radiative emission from a photonic crystai based on mode1

photon dispersion relations. Furthemore, our method enables a quantitative description

of the interaction between an atom and the electromagnetic modes available in a PC,

which is centrai to the description of quantum opticai phenomena in these materiais.

The outline of this Chapter is as follows. In Section 2.2, we develop a quantum

description of the atom-field interaction in a realistic PC in terms of the natural (Bloch)

modes of a periodic crystal. In Section 2.3, we derive the integroiifferential equation

describing fluorescence fiom active media. in the process, we introduce the concepts of

the projected local density of states, and the orientationdy averaged local density of

states, which describe the local electromagnetic fields seen by radiating atomic dipoles

in this system. Section 2.4 derives the expressions for fluorescence spectra and dynamics

starting from the local photon density of states, including a detailed treatment of the

Lamb shift in a PC. We then test our forrnalism on idealized models of the dispersion

relation in a PC in Section 2.5. Fiaiiy, in Section 2.6, we give a qualitative discussion of

how our formalism may be appiied to interpret actuai fluorescence experiments in PCs.

2.2 Atom-field coupling in a photonic crystal

We airn to describe the fluorescence spectrum and emission dynamics of an active materiai

placed in either the high- or low-index region of a photonic crystal. Physical realizations

oE such a system include dilute solutions of fluorescent organic dyes in the void regions

[27] and luminescent rare earth ions embedded in the dielectric backbone of an air-

dielectnc crystai [28]. The active material is modeied as a coiiection of twdevel atoms

situated at random positions. These atoms are fiuthermore assumed to be present in a

d c i e n t l y low density so as to eliminate the possibility of collective coherent emission.

The ciifferences between realistic active eiements and our somewhat idealized system are

discussed in Section 2.6.

CHAPTER 2. THEORY OF FLUORESCENCE iN PHOTONIC CRYSTALS 17

The Hamiltonian for an ekctron in an electromagnetic fieId rnay in generd be wrïtten

in the form:

p and me are the mornenturn and mass of the atomic transition electron respectively,

and A(r) and @((r) are respectively the electrornagnetic vector and scalar potentials.

Using &Iaxweli's equations and their relations to the associated potential functions, the

equations of motion for the classicai scalar potential and the vector potential A can

be written as [29]

where the dielectric permittivity, given by ~ ( r ) = E ~ ( ~ ) Q ~ is assumed to be iinear and

frequency-independent in the frequency range of interest. The spatially varying dielec-

tric function E,, (r) describes the periodic modulation of the dielectric constant wit hin a

photonic crystd, cp(r) = cP(r + R), where R is a vector of the direct Bravais lattice.

R = Ci Rai, nj E 1, the aj being basis vcctors of the periodic Iattice. To simplify our

expression for A, we choose to work in a gauge in which cP = O. Eq. (2.3) reveals that

this condition can be satisfied provided that:

The consequences of this constra.int are discussed below.

A ciassical theory for the electromagnetic field in a photonic crystai based on the

above equations is developed in detail in Re&. [29] and [30]; the resuits are summarized in

Appendix B. The classical equations may be quantized in the usual manner [29,31]. The

appropciately quantized sohtion of Eq. (2.2) for the vector potential may be expanded

CHAPTER 2. THEORY OP FLUORESCENCE IN PHOTONIC CRYSTALS

in the general form

where âk,.(t) = ô). .(0)e-kt is the annihilation operator for a field mode with wave

vector k and with polarization state o = 1,2, and satisfies the boson commutation

relation âL,&] = d(k - k')JUnd. The mode fuaetions Ak,.(r) may in general be any

complete set of basis functions spanning the region under consideration. In free space,

where there is complete translational symmetry, it is natural to choose as basis functions

simple plane waves, Ak,@(r) = e'kmrek,v, where a , is a unit vector in the direction of the

polarization state a for a given wavevector k. In a photonic crystal, the periodicity of

the dielectric breaks this full translationai symmetry. As a result, the field seen by an

active atom varies from point to point within a unit ce11 of the crystal [32]. One may

express Â(r) at a specific point using a plane wave basis, however such an approach

would not elucidate or take advantage of the syrnrnetry properties of the periodic crystal.

It is therefore highly advantageous to use a basis of Bloch modes, which satisfy the

Bloch-Floquet theorem,

Ak(r f = eibRAk(r), (2-6)

as we may then conveniently restrict our attention to a single Wigner-Seitz ceii of the

lattice. If we then adopt a reduced zone scheme for k [19], we may mite the vector

potentiai in a photonic crystal as

where V is the volume of a unit ceil of the lattice, n is the energy band index in the

6rst Brillouin zone, and the wave vector integration is over each band in this region of

k-space. Mode functions Iabeled by n are henceforth understood to be Bloch modes

of the crystal. Unlike in free space, Merent polarization states for a given wavevector

are not necessarily degenerate in energy. Therefore the band index n also counts the

polarization states for a given wavevector k.

From (2.1), we see that the quantized interaction Hamiltonian of the atom and field

for an atomic electron at position ro is given by

In this expression, we have neglected the term involving A2 in the Hamiitonian (2.1), as it

describes photon-photon interactions, which are negligible at low energies. Yote that in

general the electron momentum and the vector potentiai do not commute: [Â(r), fi] =

ihV -Â(r). However, in a spatialiy homogeneous dielectric, Eq. (2.4) reduces to the

condition V . A = 0, and we recover the weU-known p -Â form of the minimal coupling

Hamiltonian. Clearly, this is not the case in a periodic dielectric. We may however assume

that the electromagnetic field varies littIe over the spatial extent of the electronic wave

function, thus allowing us to keep only the dipole contribution of the electronic charge

distribution. -4s pointed out by Kweon and Lawandy 1291, when such an approximation

is valid, we may then evaluate the vector potentiai at the position of the atomic centre

of m a s . Since the electron m a s is very s m d compared to that of the atomic nucleus,

this is equivalent to edua t ing A at the atomic nucleus, whose motion is independent of

the electronic motion. We may then wcite

where ro is now understood to be the position of the atomic nucleus. Alternatively,

we may simply note that the spatial variation in the dielectric constant occurs over the

Iength scaie of the lattice constant of the crystal, which is orders of magnitude larger

than the spatial extent of the individuai active atoms. As a result, we may treat ~ ( r ) as

CHAPTER 2. THEORY OF FLUORESCENCE IN PHOTONIC CRYSTALS

a constant over the length sale of the active elements, thus validating Eq. (2.9).

At this point, we proceed to rewrite the interaction term in the tom D Ê [18l in the

electric dipole approximation, where D is the usual electric dipole operator, and Ê is the

electric field operator obtained fiom Eq. (2.7). We note that in principle one may derive

the fi *& form of the interaction directly fiom the Hamütonian (2.1), without recouse to

the approximation scheme presented here, thereby avoiding issues relating to the acausal

nature of the vector potentiai (see, e.g, Ref. [33]). Nevertheless, our approach results in

the correct form for the atom-field coupling.

In a rotating wave approximation, the full Hamiltonian for a two-level atom and the

electromagnetic field in a photonic crystal c m now be written as

The index f i labels the energy band and wavevector of a given field mode, p = (n, k), and

the cj ( j = +, -) are the usuai Pauli operators for a two-level atom with a (bare) atomic

resonance frequency W ~ L . We have also dropped the circumfiexes denoting operators,

as in what foilows the distinction between operators and ordinary functions should be

self-evident. The position-dependent atom-field mode coupling constants, g,. are given

by

where and d are respectiveIy the magnitude and the direction unit vector of the

dipoIe matrix eIement for the atomic transition. Whereas the condition V -A = O in free

space implies that the plane wave modes are transverse (k - A = O), condition (2.4) for a

photonic crystal does not necessarily give transverse polarization states for Bloch modes.

CHAPTER 2. THEORY OF FLUORESCENCE IN PHOTONiC CRYSTALS

2.3 Equations of motion

We wish to analyze the atomic emission in a SùuMinger equation fomaiism 134, 331.

Atom-field interactions that involve more than one photon are more easily (and often

necessarily) described by a density matrix or by Heisenberg operator equations, and

much of o u analysis can be carried over to such systems; see Section 2.4.3. In the single

photon sector, the system wavefunction for a two-level atom with dipole moment d21d is

h (d , ro, t) and bl,,(d, ro, t) label the probability amplitudes for the excited atom plus

an electromagnetic vacuum state, and a de-excited atom with a single photon in mode

p respectiveiy at a given position ro of a Wigner-Seitz ce11 in a photonic crystd; A, =

w, -o?l. In a frame that is CO-ratatingwith the bare atomic resonance frequency, w 2 ~ , Eq.

(2.12) dong with the Hamiltonian (2.10) give the equations of motion for the amplitudes,

Fomally integrating (2.14) and substituting the solution into (2.13), we anive at an

equation for the excited-state amplitude,

G(d,ro,t - t') is a t imde lay Green functioa, or memory kernel, which describes the

mean effect of the electromagnetic vacuum on the atomic system [36] at position roi it is

dehed as

G(d, ro, r ) r 8(r) lg,,(d, ro)12 e4afir (2.16)

CHAPTER 2. THEORY OF FLUORESCENCE LN PHOTONIC CRYSTALS 22

Here 9(?) is the Heaviside step function, which ensures that G(d, ro, r) = O for r < O?

as required by causality considerations.

Making explicit the band and wavevector contributions to the wave vector surn,

(2.16) becornes

where a! = ~~ , ( - , / 16h~7? , and the k-space integration is over the k t Brillouin zone.

Here we have added a frequency integration over a Dirac deltz function, which does

not affect the value of G (d, ro, r). The fiequency integral is defined only over positive

frequencies, as there are no negative energy photon modes. Note that Eq. (2.11j does not

contain within it a conventional total density of states (DOS), which counts the number

of modes available at a given frequency,

N(w) E / d3k b(w - w.,~). IBZ

Such a DOS fails to account for either: (i) the relative orientation of the atomic dipole

and a given field mode, or (ii) the local contribution of the mode at the position ro.

Tt is therefore more usefui to consider a projected local density of states, defined as

For a specific atom, or for coherent, collective emission fiom a group of atoms (e.g.,

Iasing or superradiant emission), one must explicitIy consider the relative orientation of

the atomic dipole and the various Bloch modes in (2.19). In the case of fluorescence,

however, we have a collection of independentIy emitting atoms with essentiaiiy random

dipole orientations. As a result, in order to describe the ''mean" emission characteristics

of the system, we average d over ail solid angles, giving a factor of 113. We may further

introduce a distribution functicn, p(r), which describes the density of fluorescing atoms

at a given point in the crystal. We shall assume that the atomic distribution is the

same for each unit celi. Perforrning an average over both dipole orientation and the

atomic distribution within the crystal, we obtain an expression for the fluorescence Green

function,

where, after performing the angular integration, the local density of states (LDOS) is

defined as

( ), and ( )e are used to denote the spatiai and orientational averages respectively, and

in (2.20) we have absorbed dl numerical factors into the prefactor P = ~ & & / 1 2 t i r ~ ~ ~ .

The spatial integration is performed over the density distribution function for the active

atoms in a Wigner-Seitz celi, such tbat J&rp(r) = Ne, the total number of active

atoms within this unit cell. The replacement of G(r, T ) by Gf (r) in Eq. (2.15) gives

the equation of motion for the probabüity amphude of the excited state population in

fluorescent emission; we denote this norrnaiized fluorescence amplitude by bf(t). The

resulting fluorescence equation is then

As discussed in Refis. [12] and [32], it is the local density of states (2.21) that one must

evaluate in order to determine the electromagnetic modes in a given frequency range

available to the active atoms in fluorescence, as the Bloch mode of a periodic dielectric

for a given band n tends to reside preferentiaiiy in either the high or low didectric region

of the crystal. Different modes may therefore have very different spatial distributions,

and accordingly can couple very differently to an active atom at a given position in the

crystal. We note that Eq. (2.21) corresponds to the local radiative DOS of Ref. [32].

However, because we have made a field expansion in terms of the natural Bloch modes

of the crystai, in our case the distinction between a local DOS and a local radiative

DOS does not arise. The relation between the LDOS and the total DOS is given by the

expression

which shows that for a small dielectric modulation in the crystal, which implies a weak

interaction between the dielectric and the electrornagnetic field, the total DOS can pr*

vide a reasonable description of the field at any point in the crystal. Clearly, such a

condition is not satisfied by a crystal exhibiting a strong pseudogap or a hii photonic

band-gap [12].

2.4 Evaluation of fluorescence spectra and dynamics

Below, we describe the method of calculation of experimentally measusable quantities

fkom fluorescence experiments for a given LDOS. For onvenience, we shall presently

consider the case of a single radiating atom in each unit ceil at the position ro, such that

and Ne = 1. The fluorescence Green function (2.20) is then

where it is understood that Nl(w) is evaluated at the position ro. This simplification

is made only to make our subsequent analysis more transparent; spatial averages over

more compiicated atomic density distributions may be introduced in a straightfomard

manner, due to the linear nature of the averaging process, Because of the complexity

of calculating Nl(r, w ) throughout the active fraction of the crystal it is, in fact, more

practical to evaluate G f , ( r ) at a few representative points within a unit cell. We note

that for a crystal comprised of many unit ceiis, we are stiU justified in performing an

average over dipole orientations, as the dipole orientations of the single atoms in each

unit ceil are uncorreiated.

Central to our analysis is the Fourier transform of the probabiiity amplitude b f ( t ) ,

which is given by

The factor of e-"ll' in the integrand accounts for the fact that b f ( t ) has been defined in

a rotating frame in Eq. (2 .22) . Evaluating (2 .25) , we obtain

in which G ( Q ) is the Fourier transform of the memory kernel (2.24);

Changing the order of integration and performing the time integration in (2 .27) yields

p denotes a Cauchy principal value integral. We may thus re-express Eq. (2.26) in the

We see that the last term on the RHS of this expression appears to shift the bare atomic

frequency, and is in fact the source of the atomic Lamb shift, as described beIow.

2.4.1 The Lamb shift

As is well-known from the theory of fiee space spontaneous emission, the dressing of

an atom by virtual photons leads to a shift of its bare atomic resonant frequency [181.

En photonic crystals, the modifieci electromagnetic vacuum near a photonic band gap or

pseudogap may produce an anomalous Lamb shift [15]. In particuIar, calculations for

simple mode1 systems have suggested that, near the edge of a full gap, the strong dressing

of an atomic system by real, Bragg reflected photons may be sufficiently strong so as to

split a formerly degenerate atomic level into a doublet that is repelled Erom the band edge

both into and out of the gap. This effect could then give rise to fractionai localization

effects and vacuum Rabi oscillations in the atomic emission dynamics [34]. The possibiiity

of detecting such effects in redistic photonic crystaIs is discussed in Section 2.5.

The energy eigenvalue equation for the dressed atomic frequency(ies) is given by an

equation for the real part of the poles of &(R) after analytic continuation to a complex

kequency space; the imaginary part is responsible for atomic decay. From Eq. (2.29),

the implicit eigenvalue equation for the dressed atomic frequency, L&, is

where the principal value integration is assumed when the dressed frequency lies in the

aliowed electromagnetic continuum, Nl(w) # O.

Because the density of states for large frequencies should approach the free space

DOS, i -e . ~ y ( w ) oc w2 for large w, we see that the right-hand side of this equation is

formally divergent. A complete treatment of this divergence would require a relativistic

quantum field theoretic approach; instead we appeai to the non-relativistic prescription

of Bethe [37]: The right hand side of (2.30) can be written in the aiternative form

4 ( 4 (w1)2 (W21 - w')

- f l /a dwtNio. (2.31) (w1I2

The last term in this equation is linearly divergent, and is related to the fact that the bare

electronic mass is also dressed by the electromagnetic field. It can thus be removed from

the equation if we include a mass renormalization counterterm in our initiai Hamiltonian.

This leaves only the first term, which is at most only logarithmically divergent. This

latter divergence can be treateà by introducing a cutoff in the frequency integration at

the electron's Compton frequency, ue, as higher energy components would probe the

relativistic structure of the electron and can therefore be neglected in our andysis, The

Lamb shift is thus given by the solution(s) to the equation '

In free space, the atom-field coupling strength, given by 8, is weak (< W ~ I ) , and

Nl(w) is a smoothly-varying hinction. +4s a result, we may assume that the pole of &,(O)

is only slightly shifted from its undressed value, which amounts to setting =

on the nght hand side of Eq. (2.32). This pole approximation, dong with the free

space DOS, N(w) = w2/2 , gives the usual Wigner-Weisskopf resuit for the fiee space

Lamb shift, = -w218 h(we/uZI)/~. Near a photonic band gap, ,8 is unchangeci

'We note that by starting wïth a rotating wave approximation to the dipole coupling Hamiitonian, we have negiected a "co-propagating" contribution to the Lamb shift, which would add an additional term to the RBS of Eq. (2.32) with the substitution (621 -W.) (&1+ w') in the denominator of the integrand; see Ref. [38].

CHAPTER 2. THEORY OF FLUORESCENCE IN PHOTONIC CRYSTALS 28

from itç free space value, however N&) can in principle vary sufliciently strongly as to

modify the W igner-Weisskopf picture. We therefore retain the full expression (2.32) in

our consideration of the Lamb shift in photonic crystals. Finally, we note that because

of the explicit functional dependence of the Lamb shift on the bare atomic fiequency and

the DOS in a photonic crystal, we cannot a priori transform the equations of motion for

the fluorescence dynarnics to a rotating frame at a constant Lamb shifted frequency, as

is cornmonly done in free space. It is for this reason that we have chosen to work in a

rotating hame at the bare atornic fiequency.

2.4.2 Emission spectra

The fluctuation (or emission) spectrum for fluorescent emission as a hnction of frequency,

R, is given by the Wiener-Khintchine relation [181,

Extracthg the real part of (2.29), the emission spectrum (2.33) for an arbitrary DOS is

given by [35]

Here, we have again denoted the Lamb shifted atornic frequency by Wzr, as defined by Eq.

(2.32). We see explicitly that the form of the emission spectrum is compIetely determined

by the LDOS in the crystal, and by the position of the (dressed) atomic transition ire-

quency. The emission spectrum thus defineci corresponds to the total spectmm obtained

by considering the radiation emitted into al1 directions from the active medium.

2.4.3 Emission dynamics and electromagnetic reservoir corre-

lat ions

The dynamics of fluorescent emission are given by the evolution of the excited state

atomic population, [bl(t)12. Because our input parameter is the LDOS of the crystal, we

wish to evaluate b (t) from the inverse Fourier transform of b (R), i.e.,

where we have transformed back to a rotating frarne, and we have made expIicit the

fact that Nf (w) is defined only for positive frequencies by the use of the step function,

8(w). From this expression, we see that for t = 0, the fact that Nf(R) a Q2 for large

frequencies means that the memory kernel wiii be Iogaritbmicaily divergent. However?

we are interested only in the behavior of this function on the time scale of the atomic

dynamics; this is, in general, much longer than the natural t h e scaie in (2.20), which is

set by the atomic resonance Gequency, ~ 2 ~ - We therefore impose a high fiequency cutoff

on (2.35) for t = O without any Ioss of information on the timescaie of atomic emission.

We choose to apply a smooth cutoff of the form eJ/"$, and we choose w, such that

our result is insensitive to perturbations about this choice of cutoff (in practicai tems,

w, EZ 3wzr). This transform is then well-dehed for a given N&), and can be efficiently

caiculated by standard Fourier integral methods. We note that in contrast to the Lamb

shift, which probes the high-frequency behaviour of the LDOS and the associated vittual

photon contribution, the presence of the phase factor ei(*-l)t in Eq. (2.35), coupled

with the fact that the remaining argument of the integrand falls off at large frequencies

implies that the emission dynamics are determined only by the LDOS in the vicinity of

the atomic tiequency.

We rnay also evduate the memory kernel, G(T), which represents the temporai au-

tocorrelation function for the electromagnetic reservoir. As previously mentioned, this

Function plays a central role in the description of the atom-field interaction and therefore

aliows us to characterize the nature of this interaction in a given photonic crystal. From

Eq. (2.20), G(T) may be defined in terms of Nl(w) as

Upon evaluation of Eq. (2.36), the resulting fimction G(T) may be used to evaiuate the

emission dynamics by direct integration of Eq. (2.22) in the tirne domain. This method

is numericaily more straight-forward than the evduation of Eq, (2.35). However it is

considerably more computationally intensive, as it requires that we expiicitly integrate

over all previous values of bl( t l ) in order to obtain b f ( t ) .

2.5 Fluorescence for model photon densities of states

We now apply the methods of Section 2.4 to simple modeis of the photon dispersion

relation and of the associated density of states as a test of our rnethod. We explicitly

consider three cases: free space, a model DOS for an anisotropic photonic band-edge? and

a model DOS for a pseudogap in a photonic crystal. For simpiicity, the DOS in these

models is chosen to be position-independent. Nevertheless, in light of the computational

complexity of calculating a realistic LDOS, such idealized models provide an invaluable

means of developing a qualitative and quantitative understanding of the atom-field in-

teraction in a photonic crystal. While the chosen modeis provide an analytic form of the

DOS, we note that out method does not require that such an andytic form euists, in

contrast to previous attempts to describe the spontaneous emission of an atom in a PC

[34, 391.

CHAPTER 2. THEORY OF FLUOFLESCENCE IN PHOTOMC CRYSTALS

2.5.1 Free space

-4s is well known, the free space photon dispersion relation is linear and isotropic, i.e.,

u k = c Ikl. The corresponding DOS is therefore given by N(w) = 2w2/c3, where the Factor

of 2 has been included to account for the two photon polarizations that are degenerate

in energy. The Lamb shift for this case has been discussed in Sect. 2.4.1, and is given

approximately by JLad = wzLa ln(w, /~~~)/c? = y l n ( ~ , / w ~ ~ ) /27r, where we = mec2/h,

and me is the electron mas. For 7 = 108s-1 and wzl = 1015~-L, we arrive at a value

of dLamb = 2.2 x LO~S-' . Since JLad is essentiaily constant, we incorporate it into Our

definition of the atomic resonant frequency, ~ 2 ~ .

The exact spectrum evaluated fiom Eq. (2.34) is given by

where a = 7 / 2 ~ ~ ~ . For atomic transitions in the opticai and near-Et, we have y/wzl

so that we may approximate (2.37) by the usual fiee space Lorentzian emission

spectrum with a linewidth given by y,

in agreement with the result obtained in the Markovian approximation. -4s eupected,

the corresponding emission dyniimics shows the decay of the upper atomic state to be

highly exponentiai in nature, with a decay rate of y. Both in free space and in the case

of a PC, our results are obtained in the absence of a Markovian approximation [18] for

the form of the memory kernel.

2.5.2 Anisotropic band-edge model

In order to describe the atom-field interaction near a photonic band-edge, we consider an

aniciotropic effective mass model for the photonic dispersion [36, 391. The band-edge of a

threedimensional photonic crystal is associated with a set of n high symmetry points on

the surface of the 6rst Brillouin zone of the crystai, whose positions in reciprocal space

are given by the vectors &, i = 1, n. For exarnple, in an inverse opal PBG material, the

band-edge for the PBG between the 8th and 9th bands occurs at the W-point, which is

highly degenerate.

We expand the photon dispersion relation about the upper band edge, wu, to quadratic

order in k, giving 2 W ~ = W ~ + A I L - ~ I . (2.39)

We note that by choosing to expand the dispersion relation about the upper band edge,

we are describing field modes that reside predorninantly in the void region of the crystal

(the "air" band) [Il], a fact that is borne out by an explicit calculation of the LDOS [LP]

(see Fig. 2.1). Accordingly, this expansion is applicable to the description of emission

from active elements in the void regions at frequencies near the upper band edge. In this

case, we may neglect the influence of the lower band. Similar considerations may be used

to motivate an expansion about the lower band-edge for active elements in the dielectric

fraction of the crystai.

In a PBG materiai, the degree of curvature of the dispersion relation near the band

edge wiii strongly depend on the specific structure and dielectric materiai being consid-

ered, as well as on the direction of the expansion about the band-edge [40]. Therefore, it

is more accuate to express the expansion coefficient A as a tensor quantity, to be deter-

rnined from a microscopic caiculation of the dispersion near a band-edge; this is however

beyond the scope of the present work. For our purposes, we shall assume that A îs a

scalar constant, a condition that is satisfied exactly for crystai geometries in which the

band-edge wavevector poçsesses cubic symmetry within the Brillouin zone [15], and is

otherwise a reasonable approximation for the dispersion relation near a band-edge aftei-

averaging over al1 directions. From ( M g ) , the DOS can be written as

The (w - wU)'l2 dependence of N(w) is characteristic of a three-dimensional phase space

[41], and is in agreement with the band edge LDOS computed for an inverse opal PBG

material [12]. The physical quantities we wish to compute require the evaluation of the

product PN(w), which may be expressed as

Here BA is the characteristic iiequency for band edge dynamics in the anisotropic modeI,

and is given by

From this expression, it is clear that the determination of the frequency and time scales

for band-edge fluorescence wiU depend on an accurate determination of the expansion

parameter A for a specific PBG material. The d u e ,BA may thus be deduced from a

careful calculation of the LDOS in the vicinity of the band edge of a given crystal. In

the present work, we shail instead rescaie the reIevant quantities to the frequency scde

PA; a preliminary estimate in Ref. [421 however suggests that fiA should f d within the

range of -017 < BA < 107. The ambiguity inherent to this simple model demonstrates

the need for a more realistic caiculation of the LDOS in order to obtain a quantitative

evaluation of the atom-field interaction in a PBG material.

The Lamb shift computed fkom Eq. (2.32) for the anisotropic model is shown in Fig.

2.2. We see that the shift is frequency4ependent near the band-edge, showing that the

CHAPTER 2. THEORY OF FLUORESCENCE M PHOTONIC CRYSTALS

Figure 2.2: Plot of the Lamb shift as a function of fiequency near an anisotropic band- edge.

standard Wigner-Weisskopf approach is not applicable. In order to obtain a quantitative

estimate of the Lamb shift at the band-edge, we take the representative values of -1 =

108s-' , BA = -017 and wu = 1 x 10%-', which gives a value of JLamb(wu) z 2 x 109~-L: this

value is an order of magnitude larger than the fiee space Lamb shift. The accuracy of our

calculation is however compromised by the fact that we have neglected the contribution of

the lower band-edge to the frequency integration. Additionally, the density of states for

our model does not accurately take into account the structure of the DOS at frequencies

weii above the band-edge. Nevertheles, our model captures the qualitative behaviour of

the Lamb shift, and should give a rough estimate of its band-edge vaiue in a real PBG

materiai.

The spectrum is derived from Eq. (2.34), and has the form

Here, b = 3&/2. We see from this expression that the functional form of the Lamb shift

CHAPTER 2. THEORY OF FLUORESCENCE IN PHOTONIC CRYSTALS

Figure 2.3: Emission spectnun near an anisotropic band-edge for various values of the detuning of the atomic fiequency from the band-edge fiequency, b = w21 -wu.

contribution ensures that the spectrum is finite for al1 vaiues of the bare atomic hequency,

wzl, inchding the value wzl = wu. This spectrum is plotted in Fig. 2.3. As expected,

there is no emission of radiation in the forbidden band-gap, and the emission goes to

zero at the photonic band-edge due to the absence of electromagnetic modes at wu, We

see that the amount of emitted radiation increases as the atomic resonance Erequency is

moved farther out of the gap, and there is radiation emitted even for wzi inside the gap.

The form of the spectrum is non-lorentzian, implying a non-exponentiai decay of the

excited atomic state population. We however find that for larger detunings of wzl into

the allowed band the spectrum approaches a Lorentzian shape (centered at the atomic

fiequency) that is cut off for frequencies in the gap. We observe a long spectral tail that

extends far into the ailowed electromagnetic continuum for aii detunings of ~ 2 1 near the

band-edge. This is a resuIt of the JW-U, dependence of the DOS, which results in a

slow decay of the spectrum at higher fiequencies when compared with free space. We

expect that this spectral tail would be diminished when using a more accurate mode1 of

the DOS, in which the slowly increasing square-root dependence of the DOS does not

Figure 2.4: Temporai evolution of the exited state population for an initiaily excited two-level atom near an anisotropic band-edge for various values of the detuning of the atomic frequency from the band-edge frequency, 6 = u21- wu.

extend throughout the allowed band.

We now turn Our attention to the dynamics of the population of the upper atomic

state for an initiaily inverteci active medium. The excited state population is plotted in

Fig. 2.4 for various values of the detuning of the atomic transition frequency from the

band edge. We observe a non-zero population in the steady state for W ~ L within the gap.

This is a result of the fractionai localization of the emitted radiation about the atom in

the steady state. For w21 at the band-edge, or within the aliowed band, we b d that

the excited state population decays to zero in the steady state. The population decay

becornes exponential for sufficiently Iarge detunings into the continuum of modes, with a

decay rate proportional to the density of states, as one would expect from a perturbative

solution for atomic decay. We note that the degree of localization of the upper state

population for w2l within the gap is influenced by the DOS in the continuum of modes,

even for atomic transitions well within the gap, as the relevant integrds extend over

al1 frequencies. This accounts for the absence of a completeiy locaiized state (excited

date population of unity) for W ~ L deep in the gap within our model. Our results for the

band-edge dynamics are very similar to those of Yang and Zhu [39], mhich were obtained

by the method of Laplace transfom. However, there are quantitative differences, likely

oming to the fact that their treatment used an approximate form of the memory kernel

(2.36) associated with the DOS for the anisotropic rnodel, Here, we have made no

such approximation. As discussed in Sect. 2.4.3, the fact that the emission dynamics

probe only the DOS near the atomic resonant frequency impiies that the results we have

obtained should not be greatly affected by the inaccurate high frequency limit of the

DOS in our band-edge model.

Finally, it is straightforward to show that G f ( t - t') evaiuated from Eq. (2.36) for the

DOS (2.40) has the form

Where O ( x ) is the error hinetion, Q(2) = (2/Jii) ë t 2 d t . This r e d t is in agreement

with the previously derived result for the anisotropic model 1361 (see Chapter 4). This

may be compared with the free space Markovian result, Gf(t -t') = (: + id~=,,,*) 6(t - t l ) ,

which impiies that the atomic system in free space has no memory of its state at previ-

ous times on the time scaie of atomic emission. We therefore observe that the non-zero

temporal correlations contained Eq. (2.43) are the source of the deviations from the

Markovian behaviour for atomic emission. In general, Gf( t - t'), or where appropriate,

G(d, ro, r) (Eq. (2.17)) fdly characterize the interaction between an active element and

the electromagnetic reservoir. This memory kerneI is therefore of relevance to the d e

scription of quantum optical phenornena within a PC, as it describes the spontaneous

decay contribution to the evoiution of a quantum optical system.

CHAPTER 2. THEORY OF FLUORESCENCE IN PHOTONIC CRYSTALS

2.5.3 Pseudogap model

We now treat the case of a pseudogap, for which the stop band does not extend over al1

propagation directions, thus resulting in a suppression of the DOS rather than the forma-

tion of a full PBG. In contrast to the two cases treated above, it is not a straightforward

matter to develop a model dispersion relation for a pseudogap, as this would require a

more explicit treatment of the directional dependence of the photon dispersion relation.

h t e a d , we propose a model DOS which recaptures the basic qualitative features of a

pseudogap; it is plotted in Fig. 2.5, and has the form

Here, h (which is dimensionless) and I' (in units of wo) are parameters describing the

depth and width of the pseudogap respectively, and wo is the central frequency of the

pseudogap. We see that the pseudogap is açsumed to have a Gaussian profile, and

approaches the fiee space DOS away €rom uo; ie . , N(0) = O and N ( w » w0) = w 2 / 2 .

Furthemore, we obtain the fiee space DOS for h = O, aliowing us to unambiguously

compare resuits obtained for the pseudogap model with the corresponding values iB kee

space.

In Fig. 2.6, we pIot the difference between the Lamb shift computed for the pseudogap

model and the free space Lamb shift, AI,&. We see that in the vicinity of the bare atomic

fiequency, ~ 2 1 - ug, the Lamb shift is fiequency-dependent, as was the case near the

anisotropic band-edge of Section 2.5.2. As we bave p r e s e d the correct high and low-

hequency behaviour of the DOS in the present modeI, we can infer that the fiequency

variation in the Lamb shift in the band-edge case is not an artifact of our band-edge

model which does not possess the correct high and low fiequency behaviour. Therefore,

it is clear that both pseudogap and band-edge emission phenomena cannot simply be

treated by means of a Wigner-Weisskopf approximation, as has been suggested in Refs.

Figure 2.5: Plot of the DOS (Eq. (2.44)) for the pseudogap model. The Nidth of the gap is set by the parameter I' = .O%, which impIies that the pseudogap width is 10% of its central frequency. Various depths of the pseudogap (set by h) are shown.

Figure 2.6: Plot of the ciifference be- the pseudogap Lamb shift and the fiee space Lamb shift, AL&, for system parameters 7 = 108s-L : ~ 2 1 = 1015s-L and ï = . O h . PIots for various values of h are shown.

CHAPTER 2. THEORY OF FLUORESCENCE IN PHOTONIC CRYSTALS 40

[43I - At w21 = WO, we find that the Lamb shift for the pseudogap mode1 is identical to

the free space value, independent of the values of h and II. This is attributable to the

symmetry of N(w) about wo for frequencies within the pseudogap, which negates the con-

tribution of the pseudogap in the eduation of the Cauchy principal-value integral, Eq.

(2.32). The calculated shift may be greater or l e s than the free space value, depending

on whether w21 is greater or l e s than wo, and, as expected, the deviation of the pseudo-

gap Lamb shift from the free space d u e increases as the strength of the pseudogap is

increased by enlarging the d u e of h. It is interesting to note that the maximal positive

and negative d u e s of hLomb for h e d values of i' and h occur at the "edges" of the

pseudogap, which occur at the d u e s w 2 ~ = wo k r. This is clearly due to the fact that

the DOS exhibits the greatest asymmetry about these fiequencies? thereby giving the

Iargest variation when performing the Cauchy principal d u e integration in Eq. (2.32).

This is consistent with the fact that the maximal variation of the Lamb shiEt from the

free space value for a system exhibithg a full PBG occurs at the band-edges. For a

sufiicientiy strong pseudogap, the maximal value of IALadl may be on the order of 15%

of the free space d u e (see Fig. 2.6), a difîerence that should be readily measurable using

conventional measurement techniques.

Spectral and dynamical results for the pseudogap mode1 are presented in Figs. 2.7 (a)

and (b) respectively. The spectrum for this case is given by the expression

where ü = y/2wo. The resulting spectrum is highly Lorentzian in nature, with a linewidth

that depends on the DOS in the vicinity of the atomic transition. -4s a result, we see

that there is a narrowing of the iinewidth and a corresponding increase in the peak of

the ernission spectrum for a hed value of w2l within the pseudogap as the d u e of h is

CHUTER 2. THEORY OF FLUOWSCENCE IN PHOTONIC CRYSTALS 4 1

Figure 2.7: (a) Emission spectrum for a two-level atom with resonant fiequency coin- cident with the centrai frequency of a pseudogap, W ~ L = WO. = . O h 0 . (b) Tempord evolution of the excited state population for an initially excited two-level atom with resonant fiequency coincident with the centrai frequency of a pseudogap, -1 = WQ.

r = .05wo. Plots for various values of h are shown.

increased. This is in contrast to the case of an atomic transition in the vicinity of a PBG,

for which the fiactional localization of light in the vicinity of the emitting "atoms" means

that the integrated emission intensity is not necessarily preserved as the parameters of

the system are changed. As expected, the corresponding curves for the emission dynamics

(Fig. 2.7b) are highly exponential, with a decay rate equal to the spectral linewidth for a

given set of system parameters. We thus see that, in contrast to the case of a PBG, the

spectral and dynamical characteristics of active media with radiative transitions within

a pseudogap may be treated using a perturbative approach, in which we define a decay

rate proportional to the DOS at the atomic resonant fkequency. Such an approach is

valid in the present case because of the smoothness of the DOS in our pseudogap mode1

within the vicinity of the atomic transition fiequency. h more accurate characterization

of the LDOS in a strongly-scattering PC however shows that, even in the absence of a

PBG, there will be a number of sharp features in the DOS and the LDOS, in particular

van Hove singularities [12], whose effect on the radiative properties of an active medium

cannot be described by such a perturbative treatment. Our formalism is therefore useful

in obtaining a complete characterization of the present problem, including the Lamb

CHAPTER 2. THEORY OF FLUORESCENCE IN PHOTONIC CRYSTALS 42

shift, and in a broder sense dows us to describe the effect of virtually any feature of

the DOS within a PC on radiative emission within the same, straightforward h e w o r k .

Discussion

The formalism developed and applied in the preceding sections applies exactly to the case

of a system of two-level atoms in a defect-free photonic crystal. Clearly, real systems

will in general dSer significantly fiom this idealized configuration. h y largescale PC

microfabricated at optical wavelengths will likely posses a significant number of defects,

which may take the form of point defects, dislocations, and grain boundaries within the

bulk of the crystal. The explicit incorporation of these effects into our formalism, though

possible in ptinciple, would be extremely computationally intensive. Qualitatively, we

expect that there may be emission in directions for which photon propagation is prohib-

ited in a perfect crystal. This is a result of the scattering of radiation into the direction

of a PC stop band by defects that are close enough to the crystal surface so that the

scattered light passes through only a small number of crystal layers before reaching the

crystal boundary, and therefore does not feel a significant Bragg scattering e£Fect. As

discussed by Megens et al. [251, this "defect-assisted" emission would be eiiminated for

an atomic transition fiequency deep inside a PBG, as the active elements would not be

able to emit into any direction within the bulk of the crystal. Therefore, the absence of

emitted radiation at frequencies within the band-gap is a strong signature of the exis-

tence of a full PBG, even in the presence of defects. It is also interesting to note that the

presence of a small number of defects may actually aid in the characterization of a PC

via fluorescence experiments, as the presence of defects breaks the exact mode symmetry

of a given Bloch mode. This may permit us to observe emission fiom a Bloch mode of

the crystai that may otherwise be uncoupled to externally propagating modes.

We have aIso made certain ideaiizations with respect to our description of the active

CHAPTER 2. THEORY OF FLUORESCENCE IN PHOTONIC CRYSTALS 43

medium. First, we have negiected the effects of various broadening mechaniam. The

effect of a srna11 amount of homogeneous broadening wil i not monifv the qualitative be-

haviour of the system, and may be minimized by considering an active medium at low

temperatures. lnhomogeneous broadening &ects may be introduced into our formalism

by convolving our results with a probability distribution, F(w) , over the transition fie-

quencies of the constituents of the active medium being considered. It has been pointed

out that certain active media, such as organic dyes, possess both a smdl degree of homo-

geneous broadening, dong with substantial inhomogeneous broadening [26]. The narrow

linewidth of the individual moiecufes in such a dye allows one to probe the LDOS over

srnaIl frequency ranges, whereas the broad distribution of emission fiequencies permits

one to scan the full range of frequencies for which a modification of the emission properties

may be expected (for a full PBG, this may correspond to 5-20% of the midgap frequency,

depending on the structure being considered.). Su& dyes are therefore ided candidates

for the characterization of PCs via fluorescence experiments, and their emission may be

weil described using our formalism.

Findy, we note that for active elements located near dielectric surfaces, and wit hin

the buIk of the dielectric, the atom-field coupling rnay be modified by so-called local

field effects [33, 441. These effects are a result of the microscopie interaction between

individuai active elements and the constituent atoms of the dielectric materid, which

resdts in a radiation reaction on the active elements. Local field effects may then serve

to modify the time scaies for the emission dynamics, as well as the d u e of the Lamb

shift. Therefore, our description wiiI apply most accurately to active eIements located

within the void region of a PC, away from dielectric surfaces. It is clear that each of the

effects outlined above shodd be considered when interpreting the resdts of fluorescence

experiments. However such considerations do not detract significantly h m the usefulness

of o u formalism in the characterization of fluorescence fiom active media in PCs.

In summary, we have developed a general formalism for the description of fluorescence

CHAPTER 2. THEORY OF FLUORESCENCE IN PHOTONiC CRYSTALS 44

fiom active media in photonic crystals. We have used a Bloch mode expansion of the

electrornagnetic field modes in order to express the fluorescence properties of the system

in terms of the local density of modes avaiiable to the active elements. In the process,

we have derived general expressions for the Lamb shift, emission spectnun, and emission

dynamics in PCs that are readily amenable to numerical calculation in the absence of

an anaiytic form for the local density of states. Our formalkm was then applied to

model densities of states in order to demonstrate the validity of the approach. Most

notably, we treated the case of an anisotropic effective m a s model of a photonic band-

edge. Vie showed that while this simple model provides a reasonable characterization of

band-edge emission behaviour, the limitations of the model motivate a more accurate

determination of the band-edge density of states in order to provide a quantitatively

accurate description. Finally, we have discussed how our idealized description may be

modified by various effects inherent to experirnental systems. The formalism presented

here may 6nd application to the characterization of photonic crystals via fluorescence

experiments, as well as to the description of the interaction between an atom and the

electromagnetic reservoir, which is of relevance to virtually any radiative phenornenon

within a photonic crystal.

Chapter 3

Radiating dipoles in photonic

cryst als

3.1 Introduction

As discussed in the Introduction, theoretical studies of atomic transitions coupled to the

electromagnetic modes of a PC within an optical PBG predict a number of novel quantum

phenomena. However, despite sigdicant advances in microfabrication techniques, high

quality photonic crystals at opticai wavelengths currently remain difficult to produce. By

contrast, high-quality PBG materiais at microwave frequencies have been available for

some time [451. Sizeable band-gaps with center frequencies ranging fiom a few GHz up to

2 THz have been reported; these crystals have thus proven the soundness of the concept of

the PBG. Microwave PBG materials may be relatively easiiy manufactured using micro-

machining techniques, and are currently of interest for applications such as the shielding

of human tissue fiom microwave radiation, and for improving the radiation characteristics

of microwave antennae . Aithough PBG materiais at microwave kequencies have been

extensively studied, the behavior of radiating dipoiar antennae embedded in microwave

PCs has not received the same degree of attention. In the microwave domain, a dipole

CHAPTER 3. RADIATING DPOLES IN PBOTONIC CRYSTALS 46

antenna could, for example, take the form of an electricdy excited metaüic pin with a

high Q (quality) factor.

The radiative dynamics of the above system can be described by a charged, one-

dimensionai simple harmonic osciilator (SHOJ. Such an electric dipole osciilator can ais0

provide an excellent description of the radiation of two-level atoms in the opticai domain

when the total excitation energy of the atorns is mil below saturation [18]. Moreover,

the radiation reservoir can itseif be modeled as a bath of many independent SHOs:

Radiative damping arises from a linear coupling between the systern SHO and the large

number of reservoir oscilIator modes. The sirnilarities between the microwave and optical

systems, coupled with the mature state of microwave technoiogy, suggest that many of the

predicted efects for atomic dipoles in the opticai domain could be redized and studied

first in the microwave domain.

Analytical techniques exist for treating simple forms of coupling between the dipoie

and reservoir for certain modal distributions of the reservoir. However, PCs present

complicated coupling distributions and spectral properties which are not easily amenable

to malytical methods. This is due to the presence of a strongly fiequency dependent and

rapidly-varying reservoir mode distribut ion, which invalidates the usuai Born-Markov

type approximation schemes for the system-reservoir interaction. To obtain accurate

results, we solve the system numericdy for a large, but h i t e , nurnber of oscillators in

the reservoir by discretizing the modes of the reservoir foilowing the approach of Ullersma

[46]. In dealing with our system, there are crucial issues concerning obtaining the correct

couphg strength between the osciilator and the reservoir modes, as w d as in employing

the proper renormalization and mode sampling in numerical simulations. When these

criteria are satidied, our numerical algorithm provides a powerfd approach to treating

radiative dynamics in complicated non-Markovian reservoirs.

Here, we devdop a quantitative tteatment of the radiative dynamics of an electric

dipole osciliator coupied to the electromagnetic reservoir within a mode1 PC. In the

CHAPTER 3. RADIATING DIPOLES IN PHOTONIC CRYSTALS 47

process, we provide a sound theoretical basis for this and other approaches [471 to non-

Markovian radiative dynamics which involve the discretization of a model electromagnetic

resemoir. Additionally, we show how our method can be applied to realistic PC's with

complicated dispersion relations and EM mode structures. This Chapter is organized

as follows. In Section 3.2, we devebp a classical field theory for electromagnetic field

modes in PCs, and we derive the coupling constants between a radiating dipole and these

Bloch modes. This leads to the Hamiltonian of the coupled system and the associated

equations of motion. Renormalization issues arising from the non-relativistic nature of

our theory are discussed in Section 3.3, whereas Section 3.4 describes the discretization of

the resemoir and the numerical solution of the equations of motion. In Section 3.5, these

techniques are applied to a highly computationaiiy challenging model, that of a three-

dimensional, isotropic dispersion reIation for a complete PBG. We recapture fractional

localization and related phenornena within the SHO model. In Section 3.6 we surnmarize

the results and emphasize the posibilities for test h g these predictions experimentaily in

the microwave domain.

Classical field t heory

In this Section, we derive the equations governing the dynamics of a radiating dipole

oscillator located inside a PC. TypicaIly the equation of motion for a damped oscillator,

with time-dependent coordinate q( t ) , is written as the second-order differential equation

Here, we have introduced a damping constant y, the natural fiequency uo and the

driving field F(t ) for the amplitude q of the iinear osciiiator. For instance, for a £ceely

osciliating RLC circuit with ohmic resistance R, capacitance C and inductance L, me

have y = R/L, wa = l/LC, F(t) = O, and q(t) is the electric charge. Eq. (3.1) is,

however, not the most general way of incorporating damping into the equations of motion

for a harmonic oscillator. This description can break d o m if, for example, there is a

suppression of modes in the reservoir to which the dipole osciiiator can couple. Such a

suppression of modes is a feature of the EM reservoir present in a PC. A more general

description of damping forces acting on the harmonic osciiiator therefore requires a precise

knowledge of the mode structure of its environment, and the correspondhg coupling of

the systern oscillator to these modes. In the case of a radiating dipole located in a PC,

it is then appropriate to mode1 its emission dynamics with a SHO coupled to a reservoir

of SHOs. The essential ciifference between the vacuum and a PC is then contained in the

spectral distribution, or density of states (DOS), of the reservoir oscillators, and in the

coupling constants between the reservoir modes and the system oscillator.

The characterization of the reservoir is carned out in Appendix B; here we only

sumniarize the salient results. Given a radiating dipole with a naturai Gequency wo, we

obtain the classicai Hamiltonian

The first tenu on the rïght-hand side of the Hamiltonian is the energy of the dipole

oscillator itseif,

Hdip = < w0 l~rl*. (3.3)

The natural Gequency of the isolated osciilator is wo, and is a constant with the di-

mension of energy x time. This permits us to write the energy of a SHO in units of

its naturai frequency w, i e . , E(w) = <W. The system osciliator's complex amplitude is

given by the dimensiodess, time-dependent complex variable a, defined with respect to

the coordinate q(t) of Eq. (3.1) as

The next term in the H d t o n i a n (3.2) corresponds to the fiee evolution of the

radiation reservoir, wwhich is modeled as a bath of independent SHOs,

The natural electromagnetic modes of the PC are Bloch modes (see Appendi~ A), labeied

with the index p (nk), where n stands for the band index and 6 is a reciprocal lattice

vector that lies in the k t Briiiouin zone (BZ). Their dispersion relation, w,, is different

from the vacuum case, and may have complete gaps and/or the corresponding density

of states may exhibit appreciable pseudogap structure, the manifestation of multiple

(Bragg) scattering effects in periodic media.

As we are working within the fiamework of a non-relativistic field theory, we have

introduced a m a s renormaiization counter t e m Hm = -f41a(2 that cancels unphysical

TN-divergent tems 138, 371. The quantity 4 is specified in Section 3.3.

The interaction between the osciilator and the reservoir is given by a linear cou-

pling tem. As the oscillator frequency is quite Iarge, and the effective linewidth of the

oscillation is relatively smdl, it is possible to simplify the interaction by applying the

rotating-wave approximation. In this approximation, couplings in the Hamihonian of

the iorm and its complex conjugate are neglected, as these terms osciiiate very

rapidly compared to the terms of the type cf& and its conjugate. Hence the interaction

Hamihonian can be expressed as

In the case of a point dipole, i e . , when its spatiai extent a is much smder than the

mveIength corresponding to its natural hquency, Xo = 27rwO/c, the coupling constants

g, can be derived from (i) the magnitude of the dipole moment, d ( t ) = aq(t), tocated at

CHAPTER 3. RADLATING DIPOLES IN PHOTONIC CRYSTALS

io, and (ii) the dipole orientation, d, relative to that of the Bloch modes, Ëy(io):

This dependence of the coupling constant on the dipole's precise Iocation within the PC

is the second essential ciifference fiom the k p a c e case. As shown in Refs. [12, 321,

this position dependence may be quite strong, thus making its incorporation a sine qua

non for any quantitative theory of of radiating antennae or fluorescence phenomena in

realistic PCs.

The emission dynamics can be evaluated from the Poisson brackets of the oscillator

amplitudes and their initiai d u e s , &(O) = 1 and &(O) = O (Vp). Our choice of a(0) and

PJO) corresponds to the initial condition of an excited dipole antenna and a completely

de-excited bath. The only nonzero Poisson brackets are

Eqs. (3.2), (3.7) and (3.8), together with the initial values for the osciliator amplitudes,

completely determine the emission dynamics of a radiating dipole embedded in a PC.

In the following sections, we solve the corresponding equations of motion. This task

is complicated by the nature of the reservoir's excitation spectrum: as discussed, the

non-smooth density of states prohibits the use of a Markovian approximation and its

appealing simplifying features [15, 34, 361. Instead, we have to revert to a solution of

the full non-Markovian problem. This is accomplished by fkst rearranging the reservoir

modes in a manner more suitable to both analyticai as well as numericd solutions, and

subsequently solving the equations of motion. In what foilows, we bridge the gap between

previous studies of simplifieci mode1 dispersion relations 115, 34, 361 and band structure

computations [12, 481.

Although we have formally developed our theory for an LC circuit in a rnicrowave

PC, we emphasize that the formalism also applies to a semiclassicd Lorentz osciliator

mode1 of an excited two-level atom, ie., an electron with charge e and mas m which

is bound to a stationary nucleus, for which the energy of excitation is weli below that

required for saturation effects to become relevant. The oscillator coordinate q( t ) m l

then be identified wit h the deviation of t he electron's position from its equilibrium value,

y is the inverse Lie tirne of the excited state, and wo denotes the frequency for transitions

between e~cited and ground state of the two-level atom, This corresponds to making the

substitutions:

L + m, (LQ) + P, t+ fs, (3-9)

where h = 2 d i is Planck's constant.

3.3 Projected local density of states, mass renormal-

ization and Lamb shift

From the Hamiltonian (3.2) we derive the equations of motion for the ampiitudes

for which we seek a solution with initia1 conditions a(0) = 1 and &(O) = O (Vp) . Our

formalism requires however that we bt determine the m a s renormalizationcounter term

A. This is most conveniently done in a rotating fiame with slowly varying amplitudes

a(t) and b(t), defined as a(t) = ca(t)eYOt and fl(t) = 6(19e'~~' respectively:

CHAPTER 3. RADIATING DIPOLES IN PHOTONIC CRYSTALS 5 2

Conversely, Eqs. (3.12) and (3.13) comprise a stiff set of diflerential equations [49]. In

other words, there are two very different frequency scaies in the problem (wo - u, and

4): making it f i d t to obtaiu a numericd solution. Numerical solution of the problem

is more easily performed in the non-rotating fiame, to which we return in Sect. 3.4.

Eq. (3.13) may be fowaiiy integrated,

and inserted into Eq. (3.12) to yield

4 s in the quantum treatrnent of Chapter 2, the Green function G(T) contains aü the

information about the reservoir and is the subject of our studies for the remainder of this

section. It is again defined as

Here, 8(r ) denotes the Heaviside step function, which ensures the causality of G(r) . We

now proceed to evaiuate G(T) for the fonn of the coupling constants g, given in Eq. (3.7).

To this end, we again introduce the projected local DOS (PLDOS) N(6, d, w ) through

where we have repiaced the symbolic sum over C( by its proper representation as a surn

over bands plus a wave vector integrai over the 32. With these changes, we may rewrite

Here, we have abbreviated fl = (7ra22)/ (Lwo). Eq. (3.18) makes more explicit what we

have argued before: The spontaneous emission dynamics of active media in Photonic

Crystais is completely deterrnined by the PLDOS, M(6, & w ) . -4s the PLDOS may be

drastically different fiom location to location within the Wigner-Seitz cell of the PC

[12, 321, it is imperative to have detailed knowledge about where in the PC the dipole is

situated in order to understand and predict the outcome of corresponding experirnents.

k o m Eq. (3.17) we obtaia the Fourier transform of the Green function, G(Q - wo),

centered around the atorn's bare transition fiequency wo:

where p stands for the principal value.

For large w , we have N ( 6 , d, w ) a w2. The imaginary part of G(Q - wo) apparently

contains a linear divergence in the W. This divergence is to be expected for a non-

relativistic theory, anaiogous to the problem of spontaneous emission in vacuum [381,

and is removed fiom the theory by using the m a s renormaiization counterterm, A, as

fk t pointed out by Bethe [37]. ConsequentIy, we decompose the imaginary part oE

G ( R - wo) into

'3 (Gin - WO)) E - [A + ~ ( w o ) ] , (3.19)

CHAPTER 3. RADMTING DIPOLES IN PHOTONIC CRYSTALS

where we have used the notation:

With the foregoing andysis, we have thus determined the mass renormalization counter

term A. The second quantity in Eq. (3.19), b(wo), is the classical version of the Lamb

shift derived in the previous Chapter. In what follows, we are interested only in evaluating

the oscillator dynamics in the time domain, which does not require an e-xplicit evaluation

of the latter contribution.

3.4 Discretization of the reservoir

To solve the equation of motion for the amplitude of the system oscillator, we rewrite

Eq. (3.15) in a more explicit form:

where g2(w) = B/w, and the mass renormalization counter term A is given by

We rernind the reader that a(0) = 1.

We are now in a position to comment on the origin of the linear damping term 7q(t)

that appears in Eq. (3.1): Lf we consider the long time iimit, i e . , t 3 l/wo, and assume

that N ( 6 , d, W) is a mooth function for frequencies around wo, we can approximate the

CHAPTER 3. RADMTING DIPOLES IN PHOTONIC CRYSTALS

kequency integral in Eq. (3.21) by [27rpN(6,d, wD)/u0] 6(t - f) , which leads to

where the decay constant is dehed as

This approximation is is valid only for long times relative to l/wo, and for a sufficiently

smooth density of states. However, in the case of a PC, the PLDOS rnay have sharp

discontinuities and gaps, thuç requiring that the full equations of motion be considered

instead.

To solve the integro-differential equation (3.21) in a PC, we appeal to the literai

meaning of the PLDOS as a density of states: N(fo,dlw) may be interpreted as an

unnormalized probability density of finding a reservoir oscillator with frequency ;J at

position 6 and orientation d. Consequently, we trançform Eq. (3.21) back to a systern

of coupled differential equations by employing a Monte CarIo integration scherne for an

arbitrary function f ( w ) according to

where the normalization constant

depends on the cutoff frequency, R,. There are M » 1 bath osciliators, containeci

within a set of kequencies {wil 1 5 i 5 M), the fiequencies of which are obtained by

CHAPTER 3. RADIATMG DWOLES IN PHOTONIC CRYSTALS 56

randomly sampling the i n t e d [O, ilc] according to the normaiized probability density

p(Fo, d, w ) = N(4, d ,w) /No . Note that the q may be degenerate, as prescribed by

~ 6 0 , d , 4. Applying this Monte Car10 scheme to Eq. (3.21) and transforming back to a non-

rotating frame in order to avoid having to solve a numericaliy stiff problem, we obtain

where gi = g(w,), 1 < i 5 kf, and the mass renonnalization counter term is evaluated

up to the cutoff frequency O,, i.e., A = a ch N(Fo, d , w) /w2 .

When comparing Eqs- (3.27) and (3.28) to our initiai equations of motion? Eqs. (3.10)

and (3.11), we observe that the considerations in the previous section have allowed us

to rearrange the three-dimensionai wave vector sum over the modes p E (n i ) into a

simple one-dimensional sum over a set of frequencies {wi) with a probability distribution

p(Fo, d, w ) that is eady determined through standard photonic band structure compu-

tation [12]. In the following section, we give the solutions of (3.27) and (3.28) for a

model system which has previously been treated by other methods. In particular, we

demonstrate that known results for the radiative dynamics can be recaptured using our

aigofithm. The numerical results do not depend on the the value of the cutoff fiequency

Rc and the number of reservoir oscillators, once these quantities are large enough such

that ail the relevant features of N ( 6 , d, w ) are adequately represented.

3.5 Numerical resuits for a model system

In order to establish the validity of our approach, we now solve Eqs. (3.27) and (3.28)

for a generic model of a PBG, the three-dimensional isotropie, one-sided PBG [34]. In

CHAPTER 3. RADIATING DIPOLES IN PHOTONIC CRYSTALS

Figure 3.1: The DOS for a three-dimensional, isotropic one-sided bandgap mode1 of a PC. The parameters (see Appendix C) are q = 0.8 and wca/2nc = 0.5.

Appendix C, we outline the construction of the modei's dispersion relation and how

to obtain the corresponding model DOS, N,(w). We note that we do not apped to an

effective m a s approximation in the dispersion relation [36], as is done in most treatments

of band-edge dynamics. This allows us to recover the correct f om of the large frequency

behavior of the photon density of States.

In Fig. 3.1, we show the behavior of N,(w) as a function of frequency for values

of the relevant parameters, the gap sue parameter q = 0.8 and the normalized center

frequency w , a / S ~ c = 0.5 ( s e Appendix C). The DOS exhibits a square-root singularity

at the band edge wUa/2irc = 0.6, as weii as a UV divergence, N,,,(w) a w2, as w -+ oo;

these are the characteristic features of t h model, Due to the simdtaneous presence of

both divergences, this model clearly represents a severe numericd test of our approach.

In order to test the method, we thus replace the PLDOS entering Eqs. (3.27) and (3.28)

by iVm (w).

Figure 3.2: The radiation dynamics resuiting fiom the t hree-dimensional, isotropic one- sided bandgap mode1 DOS as show in Fig. 3.1 for various values of the bare dipoIe osciiiator frequency wo relative to the upper photonic bandedge uu. The photonic band- edge is situated at wua/2?rc = 0.6 and the bare dipole osciliator Çequencies are (a) woa/2nc = 0.58, (b) woa/2m = 0.595, (c) woa/2?rc = 0.599, (d) woa/2m = 0.6, (e) woa/2nc = 0.601, (f) woa/2?rc = 0.605, and (g) woa/2?rc = 0.62. Clearly visible are normal-mode oscillations, or vacuum Rabi oscillations, and the fiactional localization of radiation near the photonic bandedge. The coupiiig strength has been chosen such that g(wo) = IO4. For frequencies deep in the photonic bandgap (woa/2?rc = 0.58) and deep in the photonic conduction band (woa/2m = 0.62) we observe negligible and exponentid decay, respectively.

In Fig. 3.2? we present the results of our numerical solution for the radiation dpamics

of a dipole oscillator with frequency wo that is coupled to the modes of a PC, as described

by Eqs. (3.27) and 13-28), for various values of the bare osciilator fiequency, woa/2m,

relative to the bandedge at wua/2m = 0.6. The coupling strength has been chosen such

that g(wo) = IO-" corresponding tu B = 10-8 x w i . Clearly visible are normal mode oscillations, also referred to as vacuum Rabi oscil-

lations, and the fractional localization of the oscillator's energy at Iong times near the

photonic band-edge [34]. As expected, for frequencies deep in the photonic band-gap

(waa/21ic = 0.38), where the system oscillator is effectiveIy decoupled from the bath

oscilIators, we find no noticeable decay of the osciliator amplitude. Deep in the pho-

tonic conduction band (woa/2ac = 0.62), the systern osciiiator is coupled to a bath

with a smooth and slowly-varyllig mode density, as in free space. We therefore observe

exponentiai decay of the oscillator amplitude, though with a time scde that differs signif-

icantly from that in Free space. Due to the Iarge value of the DOS close to the photonic

band edge in this model, the initial decay is faster for bare oscillator frequencies close

to this edge thm for frequencies deep inside the aliowed photonic band. These results

were obtained for a smooth exponential cutoif for the DOS around RCa/2.rrc = 3.0 and

hl = 2.5 x 105 oûcilIators representing the modes of the PC. We also performed numerical

simulations between d l combiations of R, and LW with values Q,a/2nc = 3.0,6.0,9.0

and LW = 2.5 x 105,3 x 105: 106 and found that the numerical values differ by at most

0.2% of the d u e s shown in Fig 3.1. This demonstrates that, despite the presence of the

singularities in the DOS, our approach stiii provides accurate and convergent resdts.

3.6 Discussion

In summary, we have developed a numerical algorithm for quantitativdy descrîbing the

radiative emission of an osciilating eIectric dipole located in a PC. The theory is based on

CHAPTER 3. ~ D I A T I N G DIPOLES IN PHOTONiC CRYSTALS 60

the natural modes of the PC, the Bloch waves, and ailows the direct incorporation of re-

alistic band structure calculations in order to obtain quantitative results for the radiation

dynarnics of the dipole antenna. We have shown how the theory must be renormalized

in order to account for unphysical divergences and have identified the classical analogue

of the Lamb shift of the dipole's natural radiation frequency. Our numerical scheme is

based on a probability interpretation of the PLDOS that solves the equations of motion

for the dipole oscillator coupled to the electromagnetic mode reservoir of the PC.

The viability of this approach was demonstrated for an isotropie model DOS for which

we have derived well-known results for radiating atomic systems [34] in the context of a

radiating classical dipole. The model considered contains two divergences, one square-

root-divergence at the photonic band edge and a quadratic UV-divergence, and therefore

clearly comprises the most serious test of our approach. More realistic models of a

three dimensionai photonic band-edge cake into account the anisotropy of the BZ, and

therefore do not suffer from a band-edge singularity [36]. -4s a result, Our formalism is

clearly more than capable of treating more realistic descriptions of the electromagnetic

reservoir within a PC and can be used for quantitative cornparison with experiment.

Though we have developed our theory for an LC circuit in a microwave PC, we have

pointed out in Section 3.2 that the fomalism aiso applies to a semiclassical Lorentz

oscillator model of an excited twdevel atom. Therefore, our approach is applicable

to both microwave antennae and to optical atomic transitions. However, technological

constraints suggest that microwave experiments wili likely be easier to perform than

optical experiments invoIving single atoms. -4s discussed, an appropriate rnicrowave

antema could: for exarnple, take the fom of a high-Q metallic pin placed in or near

a PC. The pin can then be excited by a focused ultrashort laser pulse that generates

fiee carriers at one end; these carriers then undergo several oscillations across the pin

before reestablishing charge equilibriurn. The resulting signal could be detected and

compared with the emission from such an antenna positioned in kee space, or within a

CHAPTER 3. RADIATING DiPOLES IN PHOTONIC CRYSTALS 61

homogeneous sample of the dielectric materiai that makes up the backbone of the PC

under consideration.

In its own right, such a rnicrowave system could have considerable applications in radio

science and microwave technoiogy- For example, the PBG can be used as a iiequency

filter, and can be used to âne tune the bandwidth of a dipoie ernitter with a resonant

frequency near the edge of the gap. It may also be possible to actively rnodify the

photonic band structure, effectively changing the radiation pattern of a dipole emitter.

A feasible scheme for active band structure modification has recently been proposed in

the context of opticai PCs [50], in which the PC is infiltrated with a liquid crystalline

material whose nematic director is aligned using appiied electric fields, By rotating the

director, it was found that the band structure could be significantly modified, and that

PBGs may be opened and closed altogether. SimiIar methods may be applied to the case

of microwave PCs.

Mthough we have concentrated specificaliy on the linear model, the method of cou-

pled osciilators might be extended to treat a nonlinear osciilator. We expect that such

osciiiator models will reproduce some of the effects studied for a single two-level atom

coupled to the modes of a PC without the need for quantizing the field. However, a clas-

sical treatment would need to be abandoneci if mdtiphoton excitations are non-uegligible

[47]. Given that multiphoton effects are difficult to observe in the microwave domain [511

and are even more challenging in the optical domain [52], it is reasonable to expect that

a classical model of radiative dynamics in a PC should be sufEcient for many foreseeable

experiments,

Chapter 4

Non-Markovian quantum

fluctuations and superradiance Near

a photonic Band-Edge

In this Chapter we consider the Dicke mode1 [53,54] for the collective emission of light,

or superradiance, from N identical two-Ievel atoms with a transition fiequency near a

photonic band-edge. The study of superradiant emission is of interest not only in its

own right, but aiso because it provides a valuable paradigm for understanding the self-

organization and emission properties of a band-edge laser. Of late, there has been a

resurgence of interest in superradiance in the context of superradiant lasing action [%il,

and due to the experimental reaiization of a true Dicke superradiant system using laser-

cooled atoms [561. A low threshold microlaser operating near a photonic band-edge may

exhibit unusual dynarnical, spectral and statistical properties. We wili show that such

effects are aiready evident in band-eàge superradiance. -4 preliminary study of band-

edge superradiance for atoms resonant with the band-edge [571 has shom that for an

atomic system prepared initiaiiy with a smaU coUective atomic polarization, a fraction

of the superradiant emission remains in the vicinity of the atoms, and a macroscopic

CHAPTER 4. NON-MARKOVIAN QUANTUM FLUCTUATIONS AND.. . 63

polarization emerges in the collective atomic steady state. In the absence of an initial

atomic polarization, the early stages of superradiance are governed by fluctuations in the

electromagnetic vacuum near the band-edge. These fluctuations affect the dynamics of

collective decay and d l , in part, determine the quantum limit of the linewidth of a laser

operating near a photonic band-eàge.

The organization of the Chapter is as follows. In Section 4.1, we present the quantum

Langevin equations for collective atomic dynamics in band-edge superradiance. In Sec-

tion 4.2, we calculate an approxirnate, anaiytic solution for the equations that describe

the N-atom system with low initid inversion of the atomic population. We show that the

atoms can exhibit novel emission spectra and a suppression of population fluctuations

near a band edge. Sections 4.3 and 4.4 treat the case of high initiai inversion- In Section

4.3, the mean field results of Ref. [57] are extended to the case of atoms with resonant

frequencies displaced from the band-edge. It is shown that the phase and amplitude

of the collective atomic polarization can be controlled by an externai field that Stark

shifts the atomic transition relative to the band-edge. The dissipative effect of dipole

dephasing is also included in the framework of our non-Markovian system. Section 4.4

describes superradiant emission under the influence of vacuum fluctuations by exploiting

the temporai division of superradiance into quantum and semi-classicai regimes. We find

that the system exhibits a macroscopic steady-state polarization amplitude with a phase

precession triggered by band-edge quantum fluctuations. In Section 43, we describe a

method for generating a classicai stochastic function that simulates the effect of band-

edge vacuum fluctuations. We show that, for a sufficiently large number of atoms, this

classicai noise ansatz agrees w d with the more exact simulations of Section 4.4, and may

thus be useful in the anaiysis of band-edge atom-fieId dynamics. In Appendix D , we

give the details of the calcuiation of the electromagnetic reservoir's temporai autocorre-

Iation function for different models of the photonic band-edge. This correlation function

is centrai to determining the nature of atomic decay.

4.1 Equations of motion

We consider a model consisting of N two-level atoms with a transition frequency near the

band-edge coupled to the multi-mode radiation field in a PBG materiai. For simpiicity,

we assume a point interaction; that is, the spatial extent of the active region of the PBG

materiai is Iess than the wavelength of the emitted radiation. This is often referred to as

the small sample limit of superradiance [54]- We neglect the spatially random resonance

dipole-dipole interaction (MIDI) near the band edge, which may have a more important

impact on atomic dynamics when the atomic transition lies deep within the PBG [57,581.

Nevertheless, Our simpUed model should provide a good qualitative picture of band-edge

collective emission. For an excited atomic state 12) and ground state Il), the interaction

Harniltonian For our system can be written as

where a* and a: are the radiation field annihilation and creation operators respectively;

4, = wx - u21 is the detuning of the radiation mode frequency from the atomic

transition frequency uzl. g~ = ( ~ ~ ~ d ~ ~ / f i ) ( h / 2 ~ ~ ~ ~ V ) ~ / ~ e ~ ud is the atom-field coupling

constant, where d2iud is the atoniic dipole moment vector, V is the sample volume7 and

ex = ek,, o = 1,2 are the two transverse polarization vectors. The J, are collective

atomic operators, defined by the relation Jij ~ f = , li)Pk (,jI ; i, j = 1,2, where li),

denotes the ith level of the kth atom. Usiug the Harniltonian (4.1), we may mite the

Heisenberg equations of motion For the operators of the field modes, ax(t) , the atornic

inversion, J3(t) = J22(t) - JLl(t), and the atomic system's collective polarization, Jrz(t):

AND... 65

(4.4)

We may adiabatically eliminate the field operators by formdy integrating equation

(4.2) and substituting the result into equations (4.3) and (4.4). The equations of motion

for the coilective atomic operators are then

Here, q( t ) = Ex gAaA(0)e-iA~t is a quantum noise operator which contains the influence

of vacuum fluctuations. G ( t - t ' ) is the tirne delay Green function, or memory kernel,

describing the electromagnetic resemoir's average effect on the time evolution of the

system operators. The Green Function is given by the temporal autoconelation of the

resemoir noise operator,

We have made use of the fact that (a1(0)aA(0)) r O, as we are deaiing with atomic

transition fiequencies in the optical domain [181. In essence- G(t - t') is a measure of the

reservoir's memory of its previous state on the time scale for the evolution of the atomic

system. In Eiee space, the density of field modes as a function of fiequency is broad and

slowly varying, resulting in a Green function that exhibits Markovian behavior, ~ ( t - t ' ) =

($ + isLad) b(t -t'), nhere 7 is the usual decay rate for spontaneous emission and hma

is the vacuum Lamb shift 1181. Near a photonic band-edge, the density of electromagnetic

modes varies rapidly with fiequency in a manner determined by the photon dispersion

relation, uk. We show that this results in long range temporal correlations in the reservoir

which affect the nature of the atom-fieId interaction.

CHAPTER 4. NON-MARKOVLAN QUANTUM FLUCTUATIONS AND... 66

In order to evaluate G(t - i) near a band-edge, we first make the continuum approx-

imation for the field mode sum in equation (4.7):

In this Chapter, we use an effective mas approximation ta the full dispersion relation

for a photonic crystal. Within thii approximation, we consider two models for the near

band-edge dispersion. The details of the caicuiation of G(t - t') for each mode1 and a

discussion of its applicability is given in Appendix D. As discussed in Chapter 2, in an

anisotropic dispersion model, appropnate to fabricated PBG materials, we associate the

band-edge with a specific point in k-space, k = ko. By preserving the vector character

of the dispersion expanded about b, we account for the fact that, as k moves away from

b, both the direction and magnitude of the band-edge wavevector are modified. This

gives a dispersion relation of the form:

Here, the value of A is determined by the curvature of the dispersion relation about the

band-edge. The positive (negative) sign indicates that wk is e-upanded about the upper

(lower) edge of the PBG, and w, is the frequency of the corresponding band-edge. This

form of dispersion is valid for a gap width w,,, » c Ik - ko[, rneaning that the effective

m a s relation is most directly appiicable to Iarge photonic gaps and for wavevectors near

the band-edge. Furthemore, for a large gap and a collection of atoms which are nearly

resonant with the upper band-edge, it is a very good approximation to completely negIect

the effects of the lower photon bands. The band-edge density of states corresponding

to equation (4.9) takes the form N(w) - (w - u=) ' /~, w > w,, characteristic of a three-

CHAPTER 4. NON-M ARKOVIAN QUANTUM FLUCTUATIONS AND.. .

dimensional phase space. The resulting Green hinction for wc(t - t') )> 1 is

In addition to the anisotropic photon dispersion model, it is instructive to consider a

simpler isotropic model. In this model, we extrapolate the dispersion relation for a one-

dimensional gap to ail three spatial dimensions. We thus assume that the Bragg condition

is satisfied for the same wavevector magnitude for ail directions in k-space. This yields

an effective mass dispersion of the form ~k =: W, + A(lk1 - lko1)2, which associates the

band-edge wavevector with a sphere in k-space, Ikl = ko. Strictly speaking, an isotropic

PBG at finite wavevector [hl does not occur in artificially created, face centred cubic

photonic crystais. However, a nearly isotropic gap near ko = O occurs in certain polar

crystals with polaritonic excitations [59]- -4 simple example of such a crystal is table

salt (NaCl), which has a polariton gap in the ida red frequency regime. The band-edge

density of States in the isotropic model has the form N(w) - (w - w,)-'/~, u > uc, the

square root singularity being ch~acteristic of a one-dimensional phase space. For the

Green function we obtain (see Appendix D),

In both (4.10) and (4.11), 1 5 ~ = wz1 - wc is the detuning of the atornic resonance

frequency from the band-edge, and Pa is a constant that depends on the dimension of the

band-edge singularity. In particular, for the isotropic rnodel, /?;112 = W ~ { ~ ~ & / I ~ ~ E O T ~ ~ * C ? +

while in the anisotropic modelL, /?;" = W&&/SAE~W~ ( ~ 4 ~ ' ~ .

'Note that 83 is equivaent to BA of Chapter 2.

CHAPTER 4. NON-MARKOVTAN QUANTUM FLUCTUATIONS AND... 68

4.2 Low atomic excitation: harmonic oscillat or rnodel

In order to understand the effects of band-edge vacuum fluctuations, we begin by pre-

senting a simplified model that permits an anaiytic solution, and is applicable to a systern

in which only a smail fraction of the two-level atoms are initiaiiy in their excited state.

This discussion demonstrates how Iight emission near a photonic band-edge c m give rise

to novel atomic dynamics, emission spectra, and atomic population statistics. We write

the atomic operators in the Schwinger boson representation [60]:

subject to the constraint on the total number of atoms, bf ( t )b l ( t ) + b i ( t ) b ( t ) = Ri. The

operators b!(t) and bi(t) then describe transitions of the system between the e~cited

state (i = 2) and the ground state (i = 1). In the lirnit of low atomic excitation, the

state Il) has a large population at al1 times, meaning that we can replace the inversion

operator by the classical vahe J3( t ) x -N, and that bl( t ) can be approximated by

b l ( t ) GZ m. In this case, the initially excited two-level atoms behave like a simpIe

harmonic oscillator coupled to the non-Markovian electromagnetic reservoir. -4 fom

of non-Markovian coupling similar to that of bosons to the electromagnetic field occurs

in the context of the output coupling of a cold atom Bose condensate from a trapping

potentiai to the propagating modes of an atom laser [61J. This mathematical analogy

may lead to deeper insight into both the atom laser problem and photonic band-edge

dynarnics. In our model, the Heisenberg equations of motion (4.5) and (4.6), reduce to

CHAPTER 4. NON-MARKOVIAN QUANTUM FLUCTUATIONS AND...

Using the method of Laplace transforms, we can solve for b ( t ) and find

where

and

Ax (t) = CL IS &, ~ ( 4 } . C-' denotes the inverse Laplace transformation, and ~ ( s ) is the Laplace transform of the

general memory kernel, G(t-t'). In this section, we consider the case of an isotropic band-

edge in the effective m a s approximation (equation (4.10)), for which G ( s ) is written as

For this isotropic Green function, we denote the inverse Laplace transform of equation

(4.16) by Br (t)- Br(t) was computed in Ref. [34] in the context of single atom sponta-

mous emission, and a detailed mathematical derivation may be found therein. Here, it

describes the mean or drift evolution of our Heisenberg operator b2 (t). The solution has

the Eorm

where

Figure 4.1: Normalized population of the excited atomic state near an isotropic photonic band-edge for low initial atomic excitation. Various values of the detuning, 6, = w21 - wc, of the atomic resonant frequency w21 from a band-edge at frequency w, are shown. Dashed l i e , 6, = -3; solid line, cfc = O; dotted ihe, 6, = .a. & is measured in units of LW&.

xz = ((A, e- ir /6 - -4 - @6 le-ir/4 , (4.22)

Q (x) is the error hinctioo, B (x) = 5 ~;e-'dt.

The probability of finding the atoms in the excited-state is given by (bi(t)b(t)) =

IB[ (~) 12, and is plotted in Fig. 4.1. We 6nd that the excited state population exhibits d e

cay and osciilatory behavior before reaching a non-zero steady-state d u e due to photon

Iocalization. These effects are due to the strong dressing of the atoms by the radia-

tion field near a photonic band-edge, resulting in dressed atomic states that straddle

the band-edge. Light emission from the dressed state outside the gap results in highly

non-Markovian decay of the atomic population, while the dressed state shifted into the

gap is responsible for the Çactional steady-state population of the excited state. The

consequences of this strong atom-field interaction are discussed in detail for single atom

spontaneous emission in Ref. [34], and for superradiant emission in Sections 4.3 and

4.4 of this Chapter. We note that the degree of steady-state localization is a sensitive

function of the detuning, 6,, of the atornic resonance fiom the band-edge. The decay rate

scales as ~v1~/3~t for the isotropic model. However, there is no evidence for the build-up

of inter-atomic coherence, as very few of the atoms are initially excited.

Equation (4.14) also allows us to calculate the system's emission spectrum into the

modes w for an atom with resonant frequency w.rl using the relation

where ~ ( s ) is dehed in equation (4.17). The spectrum for the isotropic model is then

This spectrum, shown in Fig. 4.2, ciiffers significantly from the Lorentzian spectrum

for light emission in Eree space. In fact, the emission spectrum is not centered about

the atomic resonant frequency, which is what one would expect for an atom decaying to

an unrestricted vacuum mode density. We see that for an arbitrary detuning, &, of wzl

fiom the band-edge, the emission spectrum vanishes for fiequencies at the band-edge and

within the gap, w 5 w,. This is consistent with the locaiization of light near the atoms for

electromagnetic modes within the PBG. -4s u 2 1 is detuned farther into the gap, spectral

results confirm that a greater fraction of the light is localized in the gap dressed state,

as the total emission intensity out of the decaying dresseci state is reduced. Conversely,

as ~ 2 1 is moved out of the gap, the emission proûie becomes closer to a Lorentzian in

CHAPTER 4. NON-MARKOVIAN QUANTUM FLUCTUATIONS AND.. .

Figure 4.2: Collective atomic emission spectrum S(w) (arbitrary units) near an isotropic band edge for low initiai atomic excitation. Various values of the detuning, 6, = u2t -ucY of the atomic resonant frequency wzl from an isotropic photonic band-edge at frequency wc are shown. Dotted line, 6, = -1; dashed line, 6, = 0; solid iine, 6, = 1. 6, is measured in units of LV~/~ ,B~.

form and the total emitted intensity increases. The spectral hewidth ratio between the

isotropic band-edge and free space is of the order of pl/(yN1/3), whiie for an anisotropic

band-edge it is N&/y. This corresponds to the fact that collective emission is much

more rapid near an anisotropic band-edge than in free space, whereas it is slower than

in free space for the isotropic model.

It is also instructive to evaluate the quantum fluctuations in the atomic inversion in

the context of the harmonic osciiiator model. Viriances in the atomic population can be

written in tenus of the Mandel Q-parameter [62],

where n(t) E %(t)b2(t) is the m b e r operator for the occupation of the excited state.

Since both the free space and PBG solutions in our model can be written in the form of

CHAPTER 4. NON-MARKOVIAN QUANTUM FLUCTUATIONS AND ...

Figure 4.3: Fluctuations in the excited state atomic population as measured by the Mandel parameter, Q(t) = ((n2(t)) - (n(t))2)/ (n(t)), for low initial excitation for an atomic resonant Gequency tuned to an isotropic photonic band-edge, 6, = O. Dashed line, Q(0) = 2; soiid line, Q(0) = O. Long-short dashed line denotes fluctuations for Poissonian population variance, Q(0) = 1.

equation (4.15)) we can write the Q-parameter in the general form

Again, 1l3(t)l2 is the normalized probability of finding the initiaily excited Gaction of

the atoms stili in the excited state at tirne t. For an isotropic band-edge, B(t) = Bl(t)

(equatioo (QO)), whereas in free space, B(t) - e-'V*/2, representing the e.xponentia1

decay of the excited state population. Using the identity N G IAA(t) l 2 = 1 - IB (t)12. as

derived in Appendix E, we c m mite the population fluctuations as

For arbitrary initial statistics, atoms in free space decay to the vacuum state with Q (t) =

1; since the atoms decay fdiy, there are no meaningfd atomic statistics in the long t h e

CHAPTER 4. NON-MARKOVIAN QUANTUM FLUCTUATIONS AND. .. 74

Liniit. Q (t) is plotted in Fig. 4.3 for the isotropie band-edge (6, = 0) for the cases Q(0) =

0, 1, and 2. Near the band-edge, photon localization prevents the atomic system from

decaying to the ground state. We h d instead that the steady-state statistics are sensitive

to the statistics of the initial state and to the value of 8,. -4 system initialiy prepared

with super-Poissonian statistics (Q(0) > 1) experiences a suppression of popuiation

fluctuations in the steady-state. In a system that is initiaily sub-Poissanian (Q(0) < 21,

the fluctuations increase, but are held below the Poissonian level by photon localization.

In both cases, the steady-state value of the atomic popuiation fluctuations is controlled

by 6,. Our harmonic oscillator mode1 thus suggests that a PBG system may e-xhibit novel

quantum statistics in the absence of a cavity or extemal fields.

It is important to extend the analysis of collective emission under the d u e n c e of vac-

uum fluctuations to the high excitation (superradiant) regime. In this case, the two-level

nature of the atomic operators will become important and will rnodik the fluctuation

properties from that of the harmonic oscillator picture. This generalization is considered

in the next two sections.

4.3 High atomic excitation: mean field solution

When the atomic system is initialiy M y or neady M y inverted, we expect inter-atomic

coherences, transmitted via the atomic polarizations, to have a strong influence on emis-

sion dynamics. For such high initiai atomic excitation, the quantum Langevin equations

(4.5) and (4.6), paired with the non-Markovian memory kernels (4.10) or (4,ll), do not

possess an obvious analytic solution. Moreover, conventional perturbation theory ap-

plied to these equations fails to capture the influence of the photon-atom bound state

[15], which pIays a crucial role in band edge radiation dynamics However, when the

superradiant system is prepared with an idhitessimal initiai polarkation (JI2(0) # O),

the average dipole moment dominates the incoherent &ect of the vacuum fluctuations

and the absequent evolution is weildescribed by a semi-ciassical approximation 1541.

in this case, it is possible to factorize the atomic operator equations:

The brackets (O) denote the quantum mechanicd average of the Heisenberg operator

O over the Heisenberg picture atum-field state vector, 19) = Iuac) @ [$), where luac)

represents the electromagnetic vacuum state, and Itb) represents the initial state of the

atomic system. Clearly in t h mean field approach, the quantum noise contribution is

neglected, as (q(t)) = O. Recentiy, Bay, Lambropodos and Molmer [631 found that, for

a simpler Fano profile gap model, the dynamics of superradiant emission are affected by

the choice of factorization apptied to the Ml quantum equations, However, the cornplete

factorization used here retains the qualitative features and evolution time scales of more

elaborate factorization schemes. Equations (4.31) and (4.32) were soived numericdy in

reference [57] for an atomic resonance ftequency coincident with the band-edge (6, = 0)

and a small initiai collective poiarization. The initiai coilective state was assumed ta be

of the form

with r <( 1, so that initiaily the atorns are almost W y inverteci. Tn this Section, we

extend the previous analysis to atomic frequencies detuned from the band-edge- Despite

its neglect of vacuum fluctuations, mean field theory üiuminates many of the interesting

features of the system. The relationshïp between mean field theory and a more complete

description including quantum fluctuations is discussed in Section 4.4.

For clarity, we discuss separately the atomic dynamics in our isotropie and anisotropic

dispersion models. Figures 4.4 and 4.5 show the inversion per atom and the average

Figure 4.4: Mean field solution for the atomic inversion, (J3( t ) ) IN, near an isotropic photonic band-edge, starting with an inhniteshal initial polarization, r = IO-^. Various values of the detuning, 6, = W ~ I - w,, of the atomic resonant frequency ~ 2 1 h m a band- edge at frequency w, are shom. (a) 6, = 1; (b) 6, = -5; (c ) 6, = 0; (d) 6, = -5: (e) 6, = -1. 6, is measured in units of P f 3 B t .

Figure 4.5: Mean field solution for the atomic polarization amplitude, 1 (Jt2(t)) [ IN, near an isotropic photonic band-edge, starting with an in f i n i t ba l initial polarization, r = 10-~. Various values of the detuning, dc zs ~ 2 1 - a,, of the atomic resonant frequency W Z ~ hom a band-edge at frequency wc are shown- (a) 6, = 1; (b) 6, = -5; (c) 6, = 0; (d ) 6, = -3; (e) 6, = -1. 6, is measured in units of N2/3@I.

CHAPTER 4. NON-MARKOW QUANTUM FLUCTUATIONS AND ... 77

polarization amplitude per atom respectively for various values of 6, near an isotropic

band-edge. We see from Fig. 4.4 that a kaction of the superradiant emission remains

Iocalized in the vicinity of the atoms in the steady-state, due to the Bragg reflection

of collective radiative emission back to the atoms. This locaiized light exhibits a non-

zero expectation value for the field operator, which in turn leads to the emergence of a

rnacroscopic polarization amplitude in the steady-state. We further note that the decay

rate for the upper atomic state is proportional to N2f3. Accordingly, the peak radiation

intensity is proportional to N5i3. This is to be compared with the values N and 1V' for

the free space decay rate and peak radiation intensity respectively.

As in single atom spontaneous emission near an isotropic band-edge [34], the dressing

of the atoms by their own radiation field causes a splitting of the band of collective atomic

states such that the coiiective spectral density vanishes at the band-edge frequency.

The strongly-dressai atomic states are repetled €rom the band-edge, with some levels

being pulled into the gap and the remaining levels being pushed into the electromagnetic

continuum outside the PBG. In the long time (steady state) limit, the enera contained in

the dressed states outside the bandgap decays whereas the energy in the states inside the

gap rernains in the vicinity of the emitting atoms, It is the Iocalized light associated wit h

the gap dressed states which sustains the fiactionalized steady-state inversion and non-

zero atomic pohrization. For the isotropic model, this splitting and fractional localization

persist even when W ~ L lies just outside the gap (6, > O), and the fraction of locaiized Iight

in the steadystate increases as ~ 2 1 moves towards and enters the gap. In the dressed

state picture, the seIf-induced oscillations in both the inversion and the poIaRzation

which occur d u h g radiative emissïoa can be interpreted as being due to interference

between the dressed states. The oscillation fiequency is proportional to the Erequency

splitting between the upper and lower coiiective dressai states. This is the analogue

of the collective Rabi osciilations of N Rydberg atoms in a resonant high-Q cavity [641.

Rom Fig. 4.4, we see that a dmsed state outside the band gap decays more slowly for

Figure 4.6: Mean field solution for the phase angle (in radians) of the atomic polariza- tion, B(t), near an isotropic photonic band-edge, starting with an infinitesimai initial polarization, r = Various values of the detuning, 6, r u21 - wc, of the atomic resonant frequency u21 from a band-edge at frequency wc are shown. (a) & = 3; (b) dc = O; (c) & = -.75; (d) & = -1. 6, is measured in units of iV?4/31.

atomic resonant fiequencies deeper inside the gap, causing the collective oscillations to

persist over longer periods of time. Clearly, this decay is non-exponentid and highly

non-Markovian in nature. Fig. 4.5 confirms that, as required, the polarization amplitude

for large negative values of dc is constrained by the condition, (JL2(f)) /N 5 1/2.

In Fig. 4.6, we plot the phase angle of the collective atomic polarization in the isotropic

model, B(t) = tan-' {Im (Jlz(t)) /Re (J12(t))). Prior to atomic emission, this phase

angle rotates at a constant rate, and in the vicinity of the decay process B(t) exhibits the

effects of coiiective Rabi oscillations. When the emission is complete, the rate of change

of phase angle, e(t), attains a new steady-state value, e(ts), that depends sensitively

on the detuning frequency 6,. e(tS) is a measure of the energy ciifference between the

bare atomic state and the localized dressed state, f i ( ~ 2 ~ - wloc). Such a poIarization

phase rotation implies that the coiiective atomic Bloch vector of the system exhibits

precessional dynamics in the steady-state, Unlike the conventiond precession 1381 of

CHAPTER 4. NoN-M ARKOVIAN QUANTUM FLUCTUATIQNS AND..,

Figure 4.7: Mean field solution for the atomic inversion, (J3( t ) ) /IV, near an anisotropic photonic band-edge, starting with an infinitesimai initial polarization, r = W6. Various d u e s of the detuning, & = W ~ I - w,, of the atomic resonant frequency 021 from a band edge at frequency wc are shown. Dashed line, 6, = .l; soiid line, 6, = -.1; dotted Iine, 6, = -.3. 6, is measured in units of iV2p3.

Figure 4.8: Mean field sdution for the atomic polarization amplitude, 1 (J12(t)) [ /LV, near an anisotropic photonic band-edge, starting with an infinitesimal initial polarizat ion, T = 10-% .Various d u e s of the detuning, & = ~ 2 1 -uC, of the atomic resonant fiequency

from a band edge at frequency w, are shown. Dashed Line, 6, = -1; solid line, 6, = -.I; dotted h e , cfc = -.3. 6, is measured in units of ~Vj3~.

atomic dipoles in an ordinary vacuum driven by an actemal laser field, Bloch vector

precession in a PBG occurs in the absence of an external driving field. Instead, the

precession is driven by the self-organized state of light generated by superradiance, which

remains localized near the emitting atoms. We see in Fig. 4.6 that for values of 6, such

that ~ 2 1 - WI, < O, &ts) is negative, while for ~ 2 1 - w(, > O, e(ts) is positive, i.e. the

phase is rotating in the opposite direction. At a detuning conesponding to a constant

phase in the steady-state (6(ts) = O), the dressed and bare states are of the same energy:

this occurs for a detuning value of 4 = -0.644Jff381. At this value of 6,, we also find

that (J3(ts)) = O, irnplying that there is no net absorption of light by the atomic system.

This is, in essence, a collective transparent state [38].

Collective emission dynamics near an anisotropic band-edge are pictured in Figs. 4.7

and 4.8. For ~ 2 1 slightly within the gap (6, < O), we again find a fractional atomic

inversion in the steady state (Fig, 4.7). Rabi oscillations in the atomic population are

much less pronounced than in the isotropic model, even for w 2 l detuned into the gap.

This demonstrates that the dressed atomic states outside a physical photonic band-

edge decay much more rapidly than the isotropic model would suggest. Furthemore, in

contrast with the isotropic model, we see that photon Iocalization is lost for even a mal1

detuning of ~ 2 1 into the continuum of field modes outside the band-edge. Therefore,

while we find a macroscopic steady-state polarization and precessional dynamics of the

Bloch vector for 6, < O (Fig. 4.8), for JC 2 O the polarization dies away after collective

emission has taken place. Photon 1ocaIization from an atomic level lying just outside the

gap in a three-dimensional PBG matecial may, however, be realized through quantum

interference effects if there is a third atomic level Lying siightly inside the gap [65I. These

results point to the greater sensitivity of the atomic dynamics to the more realistic

anisotropic band-edge. Because the isotropic model overestimates the momentum space

for photons sati+g the Bragg condition, photon localization effects and vacuum Rabi

splitting are exaggerated in the isotropic model relative to an artificial photonic crystal.

CHAPTER 4. NON-MARKOVIAN QUANTUM FLUCTUATIONS AND.. . 81

In the anisotropic model, the phase space available for propagation vanishes as the optical

frequency approaches the band-edge. -4s a result, vacuum Rabi splitting pushes the

collective atomic dressed state into a region with a larger density of electromagnetic

modes. Consequently, the decay rate of the atomic inversion is proportional to LV near

the anisotropic band-edge, and the corresponding peak radiation intensity is proportional

to N3. Clearly, superradiance near an anisotropic PBG can proceed more quickly and

can be more intense than in free space. As a result, PBG superradiance may enable the

design of rnirrorless, low-threshold microlasers exhibiting uItrafast modulation speeds.

From polarization phase and amplitude results, we conclude that: (i) u'nlike in free

space, the atoms near a photonic band-edge attain a fractiondy inverted state with

constant polarization amplitude and rate of change of phase angle. This corresponds to

a macroscopic atomic coherence in the steady-state anaiogous to that experienced in a

laser. In our case however, "lasing' occurs in the band-edge continuum rather than into

a conventionai cavity mode. (ii) By varying the value of 6,, one can control the direction

and rate of change of the steadystate polarization phase angIe. This rnay be realized by

applying a small extemal d.c. field to the sample which Stark shifts the atomic transition

frequency of the atoms. This type of control over the collective atomic Bloch vector may

be of importance in the area of information storage and optical memory devices [66, 671.

The above andysis rnakes it clear that collective spontaneous emission dynamics in

a PBG are s imcan t ly different from those in fiee space. In a red PBG material, the

dephasing of atomic dipoles due to interatomic collisions or phonon-atom interactions

may also have a significant effect on the evolution of our system over a large range of

temperatures. In the free space Markov approach, dipok dephasing is described by a

phenomenologicai polarization decay constant [68]. Since the Markov approximation

does not apply near a band-edge, one cannot account for dephasing by simply adding a

phenomenological decay term to equation (4.32). However, we expect that the atomic

resonant frequency wiil experience random Stark shifts due to atom-atom or atom-

Figure 4.9: Mean field solution for the atomic inversion (solid line) and polarization amplitude (dashed line) under the influence of coilision broadening for an atomic resonant frequency at an isotropie photonic band-edge, 6, = O. The system is given an infinitesimal initiai polarization, T = IO-=. The simulated stark shift is a Gaussian random distribution with zero mean and standard deviation .5rV2/3B1.

phonon interactions. This efiect can be included in the description of our system by

adding a variation A to the detuning frequency 6, at each time step in a computational

simulation of equations (4.31) and (4.32). A is chosen to be a Gaussian random number

with zero mean. The width of the Gaussian distribution is determined by the magnitude

of the random Stark effect. Such a shulation in hee space would include a random A only

in the equation for the atornic polarization. This is because the slowly varying photon

density of states seen by the a tom at the frequency r~21 +A does not change significantly

with typical homogeneous h e broadening effects. In contrast, we have seen that, near a

photonic band edge, slight variations in 6, may drasticdy change the atomic inversion-

Therefore we Ïnclude A in both system equations, In Fig. 4.9, we plot the evolution of the

collective inversion and polarization under the simulatecl collision broadening described

above. The random Stark shifts lead to the Ioss of macroscopic polarization and the

loss of atomic inversion in the long tirne limit. The latter effect cm be understood by

noting that the random frequency shifts are symmetricaiiy distributed about the mean

resonant frequency. F'requency shifts into the gap promote photon localization, while

those away Gom the gap cause further decay of the atomic inversion. Over tirne, the

net result is that the frequency shifts away Gom the gap encourage the decay of the

atomic population. This is true even in atomic systems for which the mean resonant

frequency lies within the gap. From the above considerations, it is clear that dephasing

is a significant perturbation on photon localization near a photonic band-edge. -4s in

the case of a conventionai laser, the effects of dephasing may be partially compensated

for by external pumping.

Aithough a superradiant system can be prepared in a coherent initiai state of the

type described by equation (4.33) [38], collective emission is typicdy initiated by spon-

taneous emission, a random, incoherent process. Over time, spontaneous emission leads

to the build-up of macroscopic coherence in the sarnple. The effect of vacuum fluctuations

is then of considerable importance in the full description of superradiance, both Gom a

fundamentai point of view, and for potentiai device applications, such as the recently pro-

posed superradiant laser [%l. In the next section, we present a more detailed description

of PBG superradiance that takes into account the role of quantum fluctuations.

4.4 Band-edge superradiance and quantum fluctua-

tions

In order to describe the evolution of the superradiant system's collective Bloch vector

under the influence of quantum fluctuations, we consider atomic operator correlation

functions of the form [691

P = UL~)~(J~L)~) - (4.34)

CHAPTER 4. NON-MARKOVTAN QUANTUM FLUCTUATIONS AND.. . 84

Here the operators are e d u a t e d at equal times. As in fiee space, we expect vacuum

fluctuations to drive the system fiom its unstable initial state with all atoms inverted

to a new stable equilibrium state. Such fluctuations are particularly relevant prior to

the build-up of macroscopic atomic polarization. Indeed, they provide the trigger for

superradiant ernission. In the early-the, inverted regime, we may set J3(t) = J3(0) in

equations (4.0) and (4.6), giving

The resulting equation remains non-linear, and involves products of atomic and reservoir

operators. We may simplib expressions containhg operators in this inverted regime

by considering operator averages over only the atomic Hilbert space. For an arbitrary

Heisenberg operator O(t), we denote the atomic expectation value for an initiai fully

inverted state II) by (O), = (II O II)- We denote by the set ( 1 A)) a complete set

of 2N normalized basis vectors for the atomic Hilbert space including II), such that

(XII) = 6,4~, where 6a,B is the Kronecker delta function. Clearly, (1 1 J3(0) 1 A) = N J l l x .

Since &(O) acts as a source term for &(t) in equation (4.30), we also have the property

( I 1 Jrz(t) 1 A) = O for X # 1 in the inverted regime. This c m be shown by considering

the equation of motion for (II JL2(t) IX):

where p labels a complete set of atomic States. This integro-differential equation satisfies

the initial condition (II J12(0) IX) = O, Since J12(0) acts as a raising operator on the fuily

inverted bra vector (II. For X # 1, the source term in (4.36) is also absent, Ieading to the

solution (II J12(t) IX) = O- Using this property, we may replace the atomic average over

CHAPTER 4. NON-MARKOVIAN QUANTUM FLUCTUATIONS AND.. . 85

products of atomic operators with products of atomic averages, provided that J3(t) =

J3(0). For example,

Here, (O), = (uacl O Ivac) denotes an expectation value over the reservoù Hilbert space.

For an arbitrary moment gpq, we have

We note that such a factorization is valid only for an antinormal ordering of the polar-

ization operators, since (Il JZ1(t) IX) does not vanish in general,

Taking the atomic expectation value of equation (4.35), we obtain

This is a linear equation that has lost its operator character over the atomic variables

but not over the electromagnetic resemoir, as evidenced by the presence of the quantum

noise operator, q(t ) . Equation (4.39) can be solved by the method of Laplace transforrns.

The solution has the form,

where,

CHAPTER 4. NON-MARKOVIAN QUANTUM FLUCTUATIONS

and

AND ... 86

(4.43)

Again, CL denotes the inverse Laplace transfom. The Laplace transformation of the

rnemory k e m l for an isotropic band-edge, Gr(s), is given in equathn (4.19). Despite

the fact that (312(0))A = 0, we retain the first term in equation (4.40) for later notational

convenience,

The early-the quantum fluctuations in a superradiant system prevent us from pre-

dicting a priori the evolution of any single experimental reaiization of the atoms. Instead,

we can determine oniy the probability of a particular trajectory of the coiiective atomic

Bioch vector. In order to obtain the statistics of a band-edge superradiant pulse, we 6rst

determine the statistics of the collective Bloch vector for a set of identically-prepared

systems dter each has passed through the early time regirne governed by vacuum fluc-

tuations. The relevant time scale wiU be referred to as the quantum to semi-classical

evoiution crossouer time, t = to. Our approach is to calculate the phase and amplitude

distributions of the polarization at the crossover time quantum mechanicaüy. The s u b

sequent (t > to) evoiution of the ensemble is then obtained by solving the semi-classicai

equations (4.31) and (4.32) using the polarization distribution function at to. In other

words, the distribution of vaiues of (Ji2(to)) obtained from the early time quantum fluc-

tuations provide the initiai conditions for subsequent, semi-classicai evoiution. In order

to implement this approach, we must kt iden te ta for our system [54, 701. One expects

such a transition to occur in the high atomic inversion regime, (J3( t ) ) = N- It is naturai

to dehe b such that for t > to the expectation d u e of the commutator of the system

operators J21(t) and J12(t) becornes very smail compared to the expectation value of their

product [54]. This gives the condition,

CHAPTER 4. NON-MARKOVLAN QUANTUM FLUCTUATIONS AND.. . 87

Evaluating the above commutator, we have ([J2,(t), JI2(t)]) = (J3(t)), which is equal to

N for full atomic inversion. Rom (4.38) and (4.40), we Cind that

The last equaiity is obtaîned by use of the identity N G ICX (t) l2 = 1 D (t) l2 - 1. as derived

in Appendix E. In lree space, 1D(t)l2 = e f i t , giving the crossover tirne, t p = l/Ny.

One can solve for the crossover t h e near a band-edge, t,PBG, computationaily. in the

isotropic model, for 6, = O we find that gBG z 1.24/pPbL. The crossover time main-

tains this 1/fl/3f11 dependence for ~ 2 1 displaced from the band-edge. The corresponding

time scale for the anisotropic gap is l/1V2b3. The build-up of a macroscopic polariza-

tion then occurs more slowly near an isotropic and more quickly near an anisotropic

band-edge than in free space.

Using a semi-classicai approach, we rnay write the value of the polarization a t any

time t > to in terrns of an amplitude n and a phase 4, ( ~ ~ ~ ( t ) ) ~ ' I J(K, 4, t ) . The

superscript Cl refers to the fact that the expectation value ( is taken in the semi-

classical regime t 2 to. We define P( IG)~K as the probability of finding the amplitude

between rt and rt+ ds, and Q($)d# as the probabüity of ûnding the phase between 4 and

$ + dq5. We may then write the moments of the macroscopic polarization distribution as

For t = to, we assume that the poIarization has the form J(K, 4, t) = rleid, giving for the

moments

The quantum analogue, ( )O, of (4.47) can be wrïtten in the form of equation (4.38)

CHAPTER 4. NON-MARKOVIAN QUANTUM FLUCTUATIONS AND.. .

evaluated at t = to. Substituthg (4.40) and its adjoint into (4.38) yields:

As the reservoir expectation value is taken over the the operators a~ which sath@ a

Gaussian probability distribution, Wick's theorern [18] is applied in order to reduce the

operator averages of products of field operators to averages over products of pairs of field

operators. We then have

This expression has corrections of order Np-L, meaning that it is asymptotically valid

for large N. Equating (4.47) and (4.49), we solve for the distributions P ( K ) and Q(4)

to obt ai. the desired initial polarization distribution for the semi-classical superradiance

equations. The early time distributions for fiee space and the band-edge ciiffer only in

the form of the function D(t) , as the above analysis makes no other distinction between

the two cases. Thus in the band-edge system, as in fiee space, the entire effect of the

early time atomic evolution can be recaptured using the distribution of initial conditions

given at t = to. The phase of the polarization is given by the relation

This shows that Q(4) is un i fody distributed between O and 27r. The initial polarization

amplitude distribution is found from the relation

Figure 4.10: Atomic inversion for superradiance driven by vacuum fluctuations in free space and for an atomic resonant frequency tuned to an isotropic photonic band-edge (6, = O). Solid Lines: result for initiai poIarization distribution at t = O for each system; dashed lines: result for initiai polarization distribution at t = to for each system.

The result is a Gaussian distribution of width NID (ta)12 centered at zero,

Et has been shown via density matLu methods [54] that in &ee space one may choose

the crossover time anywhere in the inverted regime, the simplest choice being to = 0.

This is due to the absence of temporal correlations of the reservoir for t # t'. Figure

4.10 shows the ensembteaveraged collective emission in free space and at an isotropic

band-edge (6, = 0) for N=100 atoms. Both the hee space and band-edge systems are

shown for two choices of initial polarization distribution. The solid lines correspond to

the choice of to = O in the amplitude distribution (4.52) for both free space and the

band-edge. The dashed lines correspond to the choice to = t p and ta = trBC for

the hee space and band-edge systems respectively. ,As per equation (4.50), the initiai

phase of the polarization in aii cases is chosen from a uniform random distribution. -4s

Figure 4.11: Ensembleaveraged atomic inversion, (J3 (t)),, IN, and atomic polarization amplitude, I(J12(t))en,l /N (dot-dashed line), for a system of N = 100 atoms near an isotropic photonic band-edge. The ensemble average is taken over 2000 initial polariza- tion values. Inversion: long dashed curve, 6, = -3; solid h e , 6, = 0; short dashed line, 6, = .S. 6, in units of ,Vf3fl1.

expected, Fig. 4.10 demonstrates that the choice of ta is unimportant in fiee space, so

long as it is chosen in the inverted regime. Near a photonic band-edge, we see that the

choice of ta affects the later evolution of the system. In particular, it affects the onset

time for collective emission. It is clear from these simulations that the details of the non-

Markovian evolution in the quantum regime play a crucial role in the subsequent semi-

classicai evolution of the band-edge superradiance. The long-range temporal correlations

of the reservoir require that we treat the vacuum fluctuations explicitly throughout the

quantum evolution of the system. A similar picture holds in the case of an anisotropic

PBG material. In our anisotropic model, memory of the initial date is expresseci through

the Green function (4.11). In this case, superradiance is aIso highly sensitive to early

stage quantum fluctuations.

Since ensemble average of atomic observables are experimentally measurable quanti-

ties, we consider these in some detail. We use the notation ( ),, to denote an ensemble

CHAPTER 4. NON-MARKOVIAN QUANTUM FLUCTUATIONS AND.. .

Figure 4.12: Distribution of delay times for a system of 100 atoms at an isotropic band- edge (6, = 0) for 4000 reaiizations of the superradiant systern.

averaged quantum expectation value. For illustration, we focus on the & = O and zero

dephasing case for a system of 100 atoms in the isotropic effective m a s model. The

extension to nonzero detuning and b i t e dipole dephasing follows from the discussion

of Section 4.3. From Fig. 4.11, it is evident that the ensemble exhibits a fractional p o p

ulation inversion in the steady-state. The steadystate value of (J3(t))m, for a given

atomic detuning is unchanged from the mean field remit, (J3(t,)) . Since the steady-

state is deterniined by the atom-field coupling strength, and not by the dynamics of the

system, it is insensitive to initial conditions. Fluctuations in the excited state atomic

population may be expressed in terms of the delay time for the onset of superradiant

emission, defined as the time at which the system is exactIy hdf-excited, i.e. (J3) = 0-

Vacuum fluctuations result in a distribution of delay times for the ensemble, asymrnetri-

c d y centered about a peak value, as pictured in Fig. 4.12. The delay tirne distribution is

qualitatively similar to that obtained in free space [70]. However, the width of the distri-

bution scales with the relevant time scale for the isotropic and anisotropic band-edges,

showing that , near a photonic band-edge, atomic population fluctuations during light

CHAPTER 4. NON-MARKOVIAN QUANTUM FLUCTUATIONS AND.. . 92

ernission can be reduced from their ûee space value. Because of the variation in initial

conditions, the Rabi oscillations in (J3( t ) ) for the isotropic gap are much less pronounced

than in mean field simulations. The ciifferences in emission times due to fluctuations

cause the ensemble average inversion to smear out these oscillations. Therefore, one can

no longer directly relate the amplitude and period of the oscillations to the energies of

the collective dressed States.

More striking is the nature of the ensemble's collective polarization under the inAu-

ence of vacuum fluctuations. Figures 4.13 a - d show the evolution of the polarization

distribution from the initial distribution given by equations (4.50) and (4.52) to the

steady-state distribution. Initially, the distribution is sharply peaked about zero. In the

decay region, the polarization amplitude is broadly distributed and has a random phase.

This behavior is reminiscent of the fluctuations of the order parameter in the vicinity of

a phase transition. In the steady state, the polarization amplitude collapses to a very

well-dehed non-zero value. This amplitude is again accompanied by a random phase

that is uniformiy distributed between O and 2w. We may interpret our steady-state result

in the following manner: X fraction of the photons emitted near the photonic band-edge

remain localized in the vicinity of the atoms, causing both the atomic dipoles and the

electromagnetic field to self-organize into a cooperative steady-state. However, vacuum

fluctuations cause this cooperative quantum state to have a random phase, resulting in

a zero ensemble average polarization amplitude, I(Jtz(t)),l = O, as shown in Fig. 4.11.

Measurements of the degree of fi.& and second order coherence of the electromagnetic

field in a band-edge superradiance experiment would provide a probe of the nature of this

self-organized state of photons and atoms near a band-edge. We h h e r note that this

date - well-defined in amplitude but with random phase - is similar to the steady-state

of a conventional laser [71] with a wd-d&ed electric fieId and random phase diffusion.

CF~APTER 4. NON-MARKOVIAN QUANTUM FLUCTUATIONS AND.. .

Figure 4.13: Atomic polarization distribution for a system of 100 atoms at an isotropie band-edge (6, = O ) , subject to quantum fluctuations at early times. 5000 realizations of the superradiant system. (a) t = toPBC; (b) t = 5; (c) t = 11; (d) steady-state- t in units of l/N2f3f11.

4.5 Simulated quantum noise near a band-edge

We have shown that the statisticai properties of a band-edge superradiant system can

be determined because the collective behavior of the constituent atoms leads to a semi-

classical system evolution, triggered by early-time quantum fluctuations. However, a

seamless quantum description of band edge quantum optical systems is extremely di&

cult to obtain, due to the non-Markovian nature of the atom-field interaction. As a first

step, we introduce a method by which to simulate their evolution computationally and in-

clude the effects of quantum fluctuations. Uniike the semi-classical simulations of Section

4.3 which neglected the effect of the quantum noise operator, as (q(t)) = ($ ( t ) ) = O, we

propose to replace (q(t)) in our semi-classical equations by a complex classical stochastic

function with the same mean and two-tirne correlation function a . its quantum counter-

part. This noise function then simulates the quantum noise in our system thraughout

the entire system evolution. We may test the vaiîdity of Our simulated noise ansatz

for band-edge superradiance by comparing the results obtained to those calculated in

Section 4.4.

The classical noise function required to simulate quantum noise near a photonic band-

edge involves a real stochastic function ((t) possessing the underlying temporal auto-

correlation of our non-Markovian quantum noise operator, q(t). In the effective m a s

approximation, this means that (see equations (4.10) and (4.11)),

where again cr = 1 and 3 for isotropie and anisotropic band-edges respectively. P r o b

lems in band-edge atom-field dynamics, such as the present superradiant problem, often

involve non-iinear equations under the d u e n c e of colored quantum noise. It is also

interesthg to note that nonlinear problems involving classical colored noise are of con-

siderable interest in classicai statistical physics [72I. In what foliows, we use the method

CHAPTER 4. NON-MARKOVIAN QUANTUM FLUCTUATIONS AND... 95

kt introduced by Rice [73] and elaborated on by Biilah and Shinozuka [74] in order

to generate colored noise s a t i w g equation (4.53). For noise with a power spectrum

P ( w ) , dehed as the Fourier transform of the autocorrelation function (<(t){(t')), their

algorithm gives,

with equality obtained for N + oo. Here, un = nAw, 4 w = w,,/N, and w,, is a cutoff

frequency above which the power spectrum can be neglected. Each 9, is a random phase

u n i f o d y distributed in the range [O, -?il . By use of a particular set of random phases

{a,) to generate the noise values at each tirne step, we obtain a single "experimentain

realization of the quantum noise in our system. Since me cannot predict a priori the

specific form of the quantum Eiuctuations in a particular experiment, we again average

over many reaiizations of the superradiant system, each governed by a different {(t), in

order to obtain distributions and ensemble averages of relevant quantities. We note that

equation (4.54) clearly gives (((t)),, = O, as desired, since the random a, cause the

ensemble to average to zero. To show that (4.54) also gives the correct autocorrelation

function, we write:

In (4.53), only the k = 1 components, in which the random phases (Jik and QI cancel

each other, survive the ensemble average. -4ii other terms in (4.55) vanish in the ensern-

CHAPTER 4. NON-MARKOVIAN QUANTUM FLUCTUATIONS AND ...

Figure 4.14: Soiid line: ensemble averaged autocorreIation function, (E(T)((T'))~,, of the classical colored noise function c(r) corresponding to vacuum fluctuations near an isotropic band-edge. The dashed line is a plot of the exact autocorrelation function in the effective m a s approximation, ( r - r')-'/*.

ble average. As N + 00, (4.56) becomes the Fourier transform of P(u) , which equals

(<(t)<(t'))tns . Studies have shown that for values of N as smaIl as 1000, the desired autb

correlation may be obtained with as little as 5% error [74], making this a computationally

feasible technique. F'urthermore, unlike other methods for the generation of stochastic

functions (see Refs. [72]), the present method computes the desired function, { ( t ) , using

only a uniform random distribution of phases Qk as input, rather than requiring the

computation of a Gaussian stochastic function as an intermediate step. This decreases

the likelihood of spurious correlations between our random numbers. Figure 4.14 shows

the two-time correlation of c(t) for a = 1, for an ensemble of 2000 realizations of the

noise function generated by the algorithm of equation (4.54). In this cdculation and in

the simulations described below, we chose a power spectrum P(u) = G, in order to

mimic the coIored vacuum near an isotropic band-edge. We see good agreement with

the correlation function (4.53). The agreement between our simulations and the =act

correlation function can be significantly improved by enlarging the size of the ensemble,

CHAPTER 4. NON-MARKOVLQN QUANTUM FLUCTUATIONS AND...

Figure 4.15: Comparison of the ensemble averaged atornic inversion, (J3(t))m, /!V, at an isotropic band-edge (6, = O) as calculateci by the methods of Sections 4.4 and 4.5. 2000 realizations of the superradiant system, Dashed line and long-short dashed line: inversion calculated using the computed polarization distribution at t = t fBG as initial conditions for a semiclassical evolution (Section 4.4) for N = 1000 and 10000 atoms respectively. Solid Iine and dotted Iine: inversion cdculated using the stochastic function of Section 4.5 for N = 1000 and 10000 atoms respectively.

at the expense of increased computation time for atom-field simulations.

The ensemble {c(t)} is used to simulate the effect of vacuum fluctuations in equations

(4.5) and (4.6). Written in terms of the dimensionless time variable T = N2I3fiitt these

equations for the isotropic band-edge become

with similar equations for the anisotropic gap. For both modeis, the noise term scaies as

CHAPTER 4. NON-MARKOVIAN QUANTUM FLUCTUATIONS AND... 98

1/n; this is the same dependence of the noise t e m on partide numbet exhibited in free

space [69J. In Fig. 4.15, we show the average inversion for an ensemble containing 2000

realizations of f(r) for N = 1000 and N = 10000 atoms. We fkd that our stochastic sim-

ulation scheme gives physical resdts only for systems of N > 500 atoms. The stochastic

simulations show good agreement with the atomic inversion obtained by the method of

Section 4.4. Ot her system properties, such as the ensembkaveraged poiarization and

the delay t h e distribution calculated by the present method also agree well with the

quantum calculations of the previous section. This suggests that Our stochastic approach

may be a valuable tool in the analysis of band-edge atom-field dynamics.

4.6 Conclusions

In this Chapter, we have treated the superradiant emission of two-levei atoms near a pho-

tonic band-edge. -4n analytic calculation of the atomic operator dynamics in the case of

low atomic excitation was given. The results demonstrate novel atomic emission spectra

and show the possibility of reducing atomic population fluctuations. This in tum sugges ts

that fluctuations in photon number may aiso be suppressed for light iocaiized near the

atoms. This raises the interesting question of whether squeezed light [75], antibunched

photons [76], and other f o m of non-classical light may be generated in a simple man-

ner from band-edge atom-field systems. For an initiaily inverted system prepared with

a mal1 macroscopic polarization, a mean field factorization was appiied to the atomic

quantum Langevin equations, giving a senii-classicd system evolution. We found that

the atoms exhibit fiactional population trapping and a macroscopic polarization in the

steady-state- Collective Rabi oscillations of the atomic popdation were found, and were

attributed to the interference of strongIydressed atom-photon States that are repded

from the band edge, both into and out of the gap. The degree of photon Iocdization, the

poIaization amplitude, and the phase angle of the polarization in the steadystate are ali

sensitive functions of the detuning of the atomic resonant frequency from the band-edge.

The steady-state atornic properties can thus be controlled by applying a d.c. Stark shift

to the atomic resonant frequency. In Section 4.3, we discussed the effect on band-edge

superradiance of inter-atomic and atom-phonon interactions (atomic dephasing). We

showed that such Iinewidth broadening effects cannot be treated by a phenomenological

decay constant as in free space, and that near the band-edge they will lead to the decay

of atomic polarization and inversion. Therefore, the steady-state properties of the super-

radiant system descnbed in this chapter will be limited by the time scale of the atomic

dephasing effects. The effect of dephasing mechanisms is important to the description of

almost ail band-edge atom-field systems.

The effect of quantum fluctuations for high initial excitation of the atoms was included

by distinguishing regimes of quantum and semi-classicai collective atomic evolution. We

f o n d that the early time quantum evolution mut be treated in detail, due to the non-

lvlarkovian electromagnetic reservoir correlations near a band-edge. This is in contrast

with free space, where the atomic system's evolution is insensitive to the treatment of the

full temporal evolution of the early, quantum regirne. F'ractional localization of light was

shown to persist under the influence of vacuum fluctuations. The atomic polarization

exhibits a non-zero amplitude with a randody distributed phase in the steady state.

This is much Like the steady-state of a conventional laser. Here, such lasing characteristics

are due only to the Braggscatteringof photons back to the atoms; there is neither externd

pumping nor a laser cavity in our system. The tirne scales for aii dynamical processes,

such as collective emission and the buildup of coiiective atomic polarization are strongly

modified from their free space d u e s due to the singuiar photon density of states near

a photonic band-edge. For an isotropie band-edge, the tirne scales as L V ~ / ~ @ ~ , while in

the more reaiistic anisotropic model, time scales as IV&. -4s a result, collective emission

phenomena can occur more rapidly near a band-edge than in free space. Throughout

our caIculations, we have employed an effective mass approximation to the band-edge

CHAPTER 4. NON-MARKOVIAN QUANTUM FLUCTUATIONS AND. .. 100

dispersion. For materials with a very srnaIl PBG, it may be important to include the

effects of both band edges. These issues are raised in Appendix D.

We have demonstrated that band-edge superradiance possesses many of the self-

organization and coherence properties of a conventionai laser. E'urthermore, we have

shown the possibility of the generation of novel emission spectra and atomic population

statistics. These results suggest that a laser operating near a photonic band-edge may

possess unusual spectral and statistical properties, as well as a low input power laskg

threshold due to the fractional inversion of the atoms in the steady-state. It may further

be possible to produce a PBG laser in a buik material without recourse to a defect-

induced cavity mode. Lending credence to this hypothesis, recent observations [771 and

theoretical studies [78] of lasing from a mdtiply-scattering random medium with gain

have demonstrated that one may obtain Iight with the properties of a laser field in the

absence of a cavity. A full description of the statistics of a band-edge laser Eeld will

require a fuli treatment of the non-Markovian nature of the electromagnetic reservoir.

Current techniques for treating the atom-field interaction in the absence of the Born and

Markov approximations [79, 801 are not directly applicable to externally driven atomic

systems. We briefly discuss our preliminary attempts at describing photonic band-edge

lasing in the concluding chapter.

Finally, we note that the steady-state atom-field properties described here are a result

of the effect of radiation localized in the vicinity of the active two-level atoms. This leads

to the question of how to pump energy into and extract energy out of these States, which

lie within the forbidden photonic gap. One possibility is to couple energy into and/or out

of the system through a third atomic level whose transition energy lies outside the gap

[34]. There is also the possibility of transmitting iight into the gap through high intensity

ultrashort pulses that locaüy distort the noniinear dieIectric constant of the materiai and

thus aiiow the propagation of light in the f o m of solitary waves within the forbidden

frequency range [81]. Such issues must be addressed in order to fully exploit the very

CHAPTER 4. NON-MARKOVIAN QUANTUM FLUCTUATIONS AND. ..

rich possibilities of quantum optical processes near a photoaic band-edge.

Chapter 5

Quantum information processing in

localized modes of light within a

phot onic band-gap mat erial

The field of quantum information has experienced an explosion of interest due in large

part to the creation of quantum dgonthms that are far more efficient at soiving certain

computationai problems than their classical counterparts 1821. However? despite the

development of error correction protocoIs designed to preserve a desired quantum state

[83] and severai promising proposais for the impiementation of quantum information

processing [84], experimental progress to date has been iimited to systerns of only 4 or

5 quantum bits (qubits) 1851. This is due to the difIiculties associated with the precise

preparation and manipulation of a quantum state, as well as decoherence; that isl the

degradation of a quantum state brought about by its inevitable coupiing to the degrees

of fieedom avaiiabie in its environment.

For "practical" quantum information processing, we require a collection of tm-state

quantum particies, or qubits, whose quantum states may be individudy modified. It is

preferable to choose a system in which the intnnsic interaction between proximal qubits

is rninimixed, so that single qubit errors do not become correlated with the states of

neighbouring qubits. We further require the ability to interact qubits pairwise in a con-

troilable fashion (a quantum gate operation), so that one may prepare desired, generally

entangled, states of a network of qubits through a series of gate operations. Finally,

the qubits should be weakiy coupled to their environments, so that their decoherence

time is much longer than the tirnescaie of a quantum gate operation. This allows for the

encoding of multiple operations in a quantum algorithm.

In this Chapter, we propose as a qubit the single photon occupation of a localized

defect mode in a threedimensional photonic crystai exhibiting a full photonic band-gap

(PBG) [86]. The occupation and entanglement of multiple qubits is mediated by the

interaction between these localized states and an atom with a radiative transition that

is nearly resonant with the localized modes. The atom passes between defects through

a matter-waveguide channel in the extensive void network of the PBG material. We

argue that such materials may provide independent qubit states, a long decoherence

time relative to the time for a quantum gate operation, and the potential for scalability

to a large number of qubits.

A PBG material amenabie to fabrication at microwave and optical wavelengths is the

stacked wafer or "woodpile" structure, constructed in andogy with the arrangement of

atoms in crystalline silicon [87]. An optical PBG materiai is the inverse opal structure

discussed in the introductory Chapter and pictured in Fig. 5.1 (a). The void regions in

both of these structures allow line of sight propagation of an atomic beam through the

crystai. For exampie, a Si inverse opal consists of approxiniately 75% connected void

regions, and c m exhibit a PBG spanning - 10% of the gap center frequency.

As outlined in Chapter 1, an atom with a transition in a PBG will be unable to

spontaneously emit a photon; instead, a long-lived photon-atom bound state is formed

[loi- By introducing isolated voids that are larger (air defect) or smaller (dielectcic

defect) than the rest of the array, strongly localized, high-Q single mode states of light

CHAPTER 5. QUANTUM INFORMATION PROCESSING IN LOCALIZED MODES ... 104

Figure 5.1: (a) Crosssectional diagram of the atomic trajectory (green) through a void channel in an inverted opal PBG material. A localized air defect (red) is made by enlarging one of the spherical voids dong the atom's path. (b) Defect arrangement for a CNOT operation. Local defects p, q, and p' are separated by line defects R1 and R2 (yellow), through which coherent fields are applied as the atom moves dong path A. Such a laser field in a line defect is also used to prepare a desired initial atomic state.

CHAPTER 5. QUANTUM INFORMATION PROCESSMG IN LOCALIZED MODES ... 105

can be engineered within the otherwise optically empty PBG [17,88, 891. Excited atoms

passing through a defect can then exchange energy coherently with this locaiized state,

thus preparing the state of our qubit. Furthemore, such "point defects" in PBG materials

can give rise to modal confinement to within the wavelength of the mode [89], giving an

enhancement of the cavity Rabi frequency of 5 -50 times over conventional rnicrocavities.

These facts make localized modes in PBG materials excellent candidates for the strong

coupling regime of cavity quantum electrodynamics (CQED) [14, 901.

Along with point defects, line defects (waveguides) can be engineered within a PBG

material, e.g., by modifying the initial templating mold of a single inverted opal crystal

1911, or by removing or modifjhg selected dielectric rods in a woodpile crystal [9]. Such

extended defects can be used to inject a coherent light field into the PBG of the system,

thereby controllably altering the Bloch vector of a two-level atom that passes through

the illuminated region without the generation of unwanted quantum correlations in the

systern. The information thereby input can then be transferred from the atom to the

localized modes. Our qubits are protected fiom the narrowly confined externally injected

fields by the surroundhg dielectric lattice.

Our basic scheme for the entanglement of localized modes is shown in Fig. 5.1. -4

two-level atom of excited state le) and ground state Ig) is prepared in an initial state

I$J) = ce le) + cg lg) in an iliuminated line defect as it passes through the crystal with

velocity v (path A). It then successively interacts with localized states p, q, and p'. The

state of the atom may be further modified in mid-flight as it passes though additional

line defects. Due to the absence of spontaneous emission in a PBG, the coherence of the

atom mediating the entanglement may be maintaineci over many defect spacings. This

aiiows for the entanglement of more than two defects or of pairs of more distant defects

with Iower gate error than is possible with conventional microcavity arrays. The lack of

spontaneous emission may also enable the use of atomic or molecular excitations that

would be too short-lived for use in conventional CQED. Measurements of the states of

C ~ T E R 5. QUANTUM INFORMATION PROCESSLNG IN LOCALIZED MODES ... 106

defect modes can be made, for example, via ionization measurements of the state of a

probe atom after it has interacted with individual defects (eg. path B of Fig. 5.1 b).

In principle, our proposal applies to systems with atomic transitions in the microwave

or in the optical/near-IR. However, to perform a precise sequence of many gate operations

w i i i require that single atoms of known velucities be sent through the crystal with known

trajectories at well-dehed times. In a microwave PBG material, the void channels

are sufficiently large (w 3mm diameter) that experiments c m currentiy be perforxied

using atoms velocity selected from a thermal source which initially emits atoms with a

Poissonian velocity distribution 1141. Due to uncertainties in atomic position, only simple

quantum algorithms are feasible using t his technique; the outcorne of t hese algorithms

must be evaluated by statistical meastires on an ensemble of appropriately prepared

atoms passing through the defect network [92].

Since optical PBG materiais have rnicron-sized void channels, atomic waveguiding,

analogous to atomic waveguiding in optical fibers [93], may be necessarÿ in order to pre-

vent the adhesion of the atoms to the dielectric surfaces of the crystai. To this end, a field

mode excited Çom the Lower photonic band edge resides aimost exclusively in the dielec-

tric fiaction of the crystai, producing an evanescent field at the void-dielectric interface.

If blueshifted from an atomic transition, this field acts as a repuIsive atomic potentid

at the dielectric surface [94]. For quantum information processing, such waveguiding re-

quires an auxiliary atomic transition that is detuned and completely decoupled from the

qubit transition, in order to avoid dephasing of the qubit state carried by the atom. Nev-

ertheless, using current technologies, few-qubit optical CQED and atom interferometry

experiments using thewaliy excited atoms or atoms dropped into a PBG material kom

a magnetmptical trap shodd presentiy be possible. We note that long-Iived localized

modes may *O be used to entangIe the electronic states of ions in ion trapsL [95] . in

contrast to atomic beams, trapped ions may be precisely manipulated into and out of

'Ion uapping frequencies Lie tar bdow the PBG required for opticai ionic transitions-

CHAPTER 5 . QUANTUM INFORMATION PROCESSING IN LOCALIZED MODES ... 107

a defect mode in a PBG material. Furthermore, unlike with a metaüic cavity, the ionic

trapping potential is not screened by the undopeci, linear dielectric substrate of a PBG

material at frequencies below the PBG. These facts may make the ionic system more

amenable to large scaie implementations of quantum information.

We first consider the dynamics of a two-Ievel atom passing through an isolated defect.

In the dipole and rotating wave approximations, the evolution of an atom at position r in

a localized defect mode is given by a position dependent Jaynes-Cummings Hamiitonian

[96, 971, tw, H(r) = -yz + k a t a + hG(r) (ao* + ata-) , (3.1)

where az, 8 = a, f ia, are the usual Pauli spin matrices for a two-level atom with

transition frequency wu, and a, ar are respectively the annihilation and creation operâtors

for a photon in a defect mode of frequency wd. The atom-field coupling strength may be

expressed as

where Qo is the peak atomic Rabi frequency over the defect mode, d21 is the orientation

of the atomic dipole moment, and ê(r) is the direction of the electric field vector at

the position of the atom. In general, the three-dimensionai mode structure wiil be a

compiicated function of the size and shape of the defect [17]. However, we need only

consider the one-dïmensional mode profile that intersects the atom's linear path, Le.

G(r) + G(r). The profile f(r) wiü have an exponential envelope centered about the

point in the atom's trajectory that is nearest to the center of the defect mode, ro. Within

this envelope, the field intensity wil l oscillate sinusoidally, and for fixed dipole orientation,

variations in the reIative orientation of the dipole and the electric field wüi also give a

sinusoidai contribution2 . For a PBG materiai with lattice constant a, we can thus set

2Rotation of dZ1 wiii result in a modification of the atom-defect interaction without the sporttanmus decay expected in a planar microcavity,

CHAPTER 5 . QUANTUM INFORMATION PROCESSING IN LOCALIZED MODES ... 108

Figure 3.2: Excited atomic state probability, lu12, after an initiaiiy excited Rb atom (w, = 2.4 x 1015rad/s) has interacted with a point defect of Rdef = a, 4 = 0; a = 0.8(2m/wa). We take Ro = 1.1 x 10LOrad/sec, which assumes the mode is confined within a single wavelength. b = w, - wd. Inset: corresponding mode profile, f (T ) .

d21 + ê ( ~ ) = 1 and mite (see Fig- 5.2):

Because we want to transfer energy between the atom and the localized mode, we wish

to use modes that are highIy symmetric about the atom's path. We therefore set 4 = 0.

ber defines the spatial extent of the mode, and is at most a few Iattice constants for a

strongly confined mode deep in a PBG [17].

The atom-field state function afker an initiaily excited atom has passed through a

defect can be written in the form

u and w are obtained by replacing r -ro by ut-6 in the Hamiltonian (5.1) and integrating

the corresponding Schr6dinger equation fiom t = O to 2b/u. b is chosen so that the

Figure 5.3: Measure of entanglement of two identical defect modes of the type described in Fig. 5.2 as a function of the detuning, 62, of the second cavity from the defect resonance frequency; = O. The atomic velocity .v = 278m/s. Maximal entanglement occurs for [al2 / pl2 = l7 [-,12 = o.

interaction is negligible at t = O; we set b = lobr. Fig. 2.2 plots 1u12 for the 780nm

transition of an initially excited Rb atom traveling through a defect in an optical PBG

material (e.g., Gap) at thermally accessible velocities (v .Y 100 - 600m/s) for various

detunings of the atomic transition frequency from a defect resonance, d = w, - wd.

Such a detuning can be achieved by applying an external field to Stark shiEt the atomic

transition as the atom passes through selected defects. The final state of the atom is seen

to be a sensitive function of 6 and v. Similar velocity4ependent atomic inversions are

obtained for the above system with the atom in kee faii (u - .lm/s), and for the 3.9mm

Rydberg transition of Rb xsing a thermal beam of atoms passing through an appropriate

microwave PBG material. At thermal velocities, the inversion can be h e l y tuned within

the - 2m/s resolution of therrnaiiy generated atomic beams.

To show that our system is capable of encoding quantum dgorithms for reaIistic values

of system parameters, we demonstrate the viabiity of producing a mmimaliy entangled

state of two defect modes after an atom prepared in its excited state has passed through

CHAPTER 5. QUANTUM INFORMATION PROCESSING IN LOCALIZED MODES ... 110

both defects. The h a l state of this system after the two defect entanglement can be

written as

l*) =alg,I,O) + B I gAl ) +rle,O,O), (5.5)

where (i, j, k) refers to the state of the atom and the photon occupation number of the

first and second defect modes respectively, a, /.3 and y can be e.upressed in terms of

the probability amplitudes for an atom that has passed through only defect 1 or 2; i.e.,

= ~ ( l ) , f i = u(l)w(2), and 7 = u(l)u(2). Maximal defect entanglement is obtained

for la( = I f i l = l/fi, Iyl = O, which leaves the atom disentangled from the defect

modes. As an example, using the system of Fig. 5.2 with u = 278m/s and with the atom

on resonance with the first defect, maximally entangled states are obtained for atomic

detunings of J2/R0 = .07, .13 and .31 from the second defect mode (Fig. 5.3).

As discussed, by passing a 2-lever atom through an externally illuminated line de-

fect, the atom's Bloch vector c m be initialized or rnodified in-ûight without becoming

correlated with the state of a defect mode3 . For simplicity, we assume that an atom

sees a uniform mode profile as it crosses such an optical waveguide mode of width 2X.

For an injected singlemode field resonant with the atomic transition, the Bloch vector

wiU then rotate at the semi-classical Rabi frquency, & = dzl - E/2h, where E is the

amplitude of the applied electric field [68]. At a thermal velocity of 100m/s, the minimum

field strengeh, E, required to fiùly cotate the Bloch vector for the 5.9mrn transition in

Rb (dzl = 2.0 x ~ O - ~ ~ C . m) is E - lmV/m. For the 780nm optical transition of Rb

(dzl = 1.0 x 10-29C - ml, E - 9kV/m at v = 100m/s, whereas in free fall (v = .3m/s),

E N 30V/m. The weak RF field required in the microwave system implies that such

experiments must be conducted at low temperatures (5 .6K) in order to prevent the

modification of the atomic state by thermal photons. In the opticaI, the required field

strengths are attainable using a cw laser whose output is coupled into waveguide chan-

3Line defects should be engïneered such that they minünïze the enhancement of spontaneous ernission.

nels of X2 cross section, and are weli below the ionization field strengths of both the

messenger atoms and the PBG materiai.

For quantum computation, a universal 2 qubit gate can be constructed in anaiogy

with conventional CQED [82, 981. However, in contrast to virtually al1 such proposals,

in our case the photons are the qubits, and the atoms serve as the quantum "bus". This

fact allows us to exploit the significant benefits of using defect modes as units of quantum

information. We outline the construction of the associated controlled-NOT (CNOT) gate,

which conveys the flipped occupation state of the qubit in defect p to a target defect p',

conditioned on the occupation of a second control defect q (Fig. 5.1 b). The date of defect

p is k t transferred to an incident atom via a near-resonant atom-defect interaction,

such that (a Il), + IO),) 19) + (-ia le) + 19)) IO),. This is anaiogous to an integrated

~ / 2 interaction in a spatially uniform cavity, Le., one for which G(r) + no. The atomic

Bloch vector is then rotated by '&t = ~ / 4 by applying a classical field in line defect

R,, thus generating the transformation 7f le) = (le) + i lg))/&, 31 19) = (i le) + I9))/ f i .

N a t , a dispersive interaction [99] is created by detuning the atom far from the defect

resonance (but still within the PBG) as it passes through q. This is used to produce a

phase rotation of T on the excited state, conditioned on the presence of a photon in q! thus

causing the amplitude of le) to change sign if q is occupied. The Bloch vector rotation

X)I-' is then performed in R2, wtiich switches States le) and lg) relative to their initial

values only if a photon was present in q. Finally, the state of the atom is transferred to

defect p' by a near-resonant 3 ~ / 2 interaction, Ieaving the atom in its ground state. pi

then carries the result of the CNOT operation.

We now turn to the cruciai issue of the decoherence and energy loss of a photon in

a defect mode. The former is a resuit of the entanglement of a quantum state with its

environment, which can occur weil before the dissipation of energy [100]. It has however

been suggested that for the (Iinear) coupiing between a photon and a non-absorbing linear

dielectric, the coherence of a small nurnber of photons in a given mode is not destroyed by

CHAPTER 5 . QUANTUM INFORMATION PROCESSiNG IN LOCALIZED MODES ... 112

the interaction [1011. The likelihood of low decoherence in high-quality iinear dielectrics

is also supported by experiments in which the coherence of photons has been preserved

over long distances in optical fibers [102]. Therefore, the coherence of excited defect

modes in a high quality PBG material is essentially limited by energy loss. We note that

phonon mediated spontaneous Raman and BrilIouin scattering of photons out of a defect

mode are ineffective l o s mechanisrns due to the vanishing overlap between a localized

field mode and the extended States of the electromagnetic continuum [El. Unlike most

qubit elements, photons do not interact significantly with one another, preventing the

propagation of single bit enors through a network of defect modes. However, a photon

rnay "hop" from one defect to another either via direct tunneling or through phonon-

assisted hopping [l]. The likelihood of both of these processes decreases exponentially

with the spatial separation of the defects, and is negligible for a defect separation of - 10

lattice constants for a strongly confmed mode.

For an isolated point defect engineered well inside a large-scale PBG material, the

Q-factor wïil then be limited by impurity scattering and absorption from the dielectric

backbone of the crystal [103]. Large-scale, microwave crystals with few impurities c m

be fabricated using currently avaiiable techniques. The present generation of optical

inverse opal crystals are highly ordered over a distance of a hundred lattice constants.

and advances in fabrication techniques should soon see this value approach hundreds of

lattice constants [8]. In the opticd and microwave kequency regimes, away fiom the

etectronic gap and the Reststraiden absorption fiequencies of a high quality semiconduc-

tor materiai, the imaginary part of the dielectric constant, €2, c m be as low as IO-' at

room temperature, and may be reduced at lower temperatures [104]. One can further

reduce absorptive losses by miniminng the fraction of the mode in the dielectric (e.g., a

strongly localized mode in an air defect). Judicious defect fabrication in a high quality

dielectric materiai should then give Q-factors of 101° (assriming 10% of the mode is in

the dielectric) or higher, which corresponds to photon lifetimes of 10-'sec and 104sec at

CHAPTER 5. QUANTUM INFORMATION PROCESSING IN LOCALIZED MODES ... 113

rnicrowave and optical frequencies respectively. This Q-value is comparable with present

microwave CQED experirnents, and is in excess of that currently used in optical CQED.

Assuming defects are separated by 10 lattice constants, we obtain a decoherence to gate

time ratio upper bound of N 200, even at room temperature, showing that, with im-

proved atomic manipulation through the crystal, our system is potentially capable of

encoding complex quantum algorithms. Using present technologies, this ratio should be

comparable to or better than those of current quantum information irnplementations.

Chapter 6

Conclusions and future direct ions

In t h i thesis, we have investigated a number of radiative phenomena in photonic crys-

tals. The descriptions we have developed for these systems motivate certain theoretical

extensions and avenues of experimental research.

Chapter 2 presented a formalism for the description of fluorescence in a PC. We

explicitly took into account the spatial dependence of the field modes available to the

active materiai. In the process, we aiso provided a practical means for incorporating

realistic photonic band structure computations into the description of fluorescent ernission

in a PC. tVe then appiied our formalism to simple models of the density of states in a PC.

These modeis give a strong qualitative understanding of the atom-field interaction in a

PC, however their Limitations motivate an accurate description of the density of states in

a real system in order to obtain a more precise description of radiative emission. When

interfaced with more reaiistic caicuiations, our fonnalisrn should provide a guide for the

realization of geometries that produce a desired modification of spontaneous emission

both for basic research and technologicai applications. Finaily, we have shown that

our approach dows us to caIculate the autoconelation function of the eIectromagnetic

vacuum. This permits the accurate incorporation of the effect of spontaneous emission

into descriptions of quantum opticai phenomena in photonic crystals. As we discussed

CHAPTER 6. CONCLUSIONS AND FUTURE; DIRECTIONS 115

in Section 2.6, our approach contains certain idealizations. In particdar, we have not

considered the effects of fabrication defects, local fields, and absorption, each of which

rnay have a significant effect on the nature of fluorescence in a PC. It would therefore be

useful to extend the present work to account foc these complicating factors.

In Chapter 3, we devebped a classicd description of the radiation dynamics of an

initidy =cited dectric dipole oscillator in a PC. We fomuiated a classical field theory

for the electromagnetic modes in a PC interacting with the system dipole. Ive then

represented the electromagnetic reservoir by a large but finite number of h m o n i c os-

cillators whose spectral distribution is given by the local density of states in the crystal.

We were able to eiiminate spurious revivals of the excitation of the system oscillator

coupled to this finite mervoir by assuming a random distribution of reservoir modes.

This cIassical calcuiation provides a description of the dynamics of radiating antennae

in the microwave regime. It also provides a good description of atomic emission when

atomic saturation effects are negligible. The applicability of our description to microwave

systems motivates experiments involving emission from active elements in the microwave

regime, in which PCs exhibithg a full PBG are more readiiy realizable than at opticai

length scales.

Chapter 4 treated superradiant emission £rom a dense collection of two-level atoms

near a photunic band-edge. A mean field analysis of the atomic operator equations

showed that superradiant emission may praceed more rapidly near a band-edge than in

free space. We also demonstrated the existence of a macroscopic collective atomic polar-

ization in the steady state. This coilective state is analogous to the macroscopic atomic

coherence experienced in a conventional laser. However, near a photonic band-edge this

state self-organizes in the absence of an extenial cavity mode. We have also studied

the influence of electromagnetic vacuum fluctuations on the initiation of superradiant

emission. We found that the evolution of the atomic system leads to a spontaneous

coiiective polarbation and a laser-üke state of the associated field - a macroscopic signa-

CHAPTER 6. CONCLUSIONS AND FUTURE DIRECTIONS 116

ture of the non-Markovian temporal correlations of the electromagnetic reservoir. This

demonstrates that one must explicitly treat the non-Markovian vacuum fluctuations near

the band-edge in order to accurately characterize superradiant emission. Furthemore,

we introduced a classical noise ansatz that, under certain circumstances, simulates the

influence of band-edge vacuum fluctuations. This "coloured" noise function may ûnd

applications to stochastic descriptions of quantum optical phenomena near a photonic

band-edge.

Our study of superradiance rnay assist in the analysis of radiative emission experi-

ments from PBG materials in which we have a dense collection of active elements. The

results of this work may be improved by considering a more realistic description of the

density of States near a band-edge using the techniques of Chapter 2. It would also be

interesting to move beyond a point mode1 to explicitly consider the spatial dependence

of the emission and propagation of radiation in a superradiant event. Following our

suggestions, a number of authors have considered the connection between Our operator

description of a band edge harmonic oscillator and the non-Markovian output couphg

of an atom laser [105]. Our work has also been of interest to those involved in treating

the non-Markovian dynamics of open quantum systems [106].

The laser-like state produced by band edge superradiance suggests that lasing near

a photonic band-edge may exhibit novel properties as a result of the non-Markovian

coupling between the lasing medium and the electromagnetic vacuum. For example,

there have recently been a number of experïments that have suggested the presence of

low threshold lasing action from distributed gain media in three dimensionai PCs that do

not exhibit a full PBG [107]; it is reasonable to expect this effect to be enhanced for lasing

near a photonic band-edge. We have carrïed out a preliminary study of photonic band-

edge lasing [108] by making use of a simple perturbative approach suggested by Florescu

and John in the context ofresonance fluorescence near a photonic band-edge [421. To this

level of approximation, we find that the threshold and steady state properties of a band-

CHAPTER 6. CONCLUSIONS AND FUTURE DIRECTIONS 117

edge laser can be described using the conventional Laser equations with the inclusion of

a frequency-dependent spontaneous decay rate that is proportional to the band-edge

density of states. Near an anisotropic band-edge, this gives Y ( W ~ ~ ) OC d m , where

~ 2 1 is the frequency of the lasing transition of the active medium and wu is the band-

edge frequency. This suggests that we rnay obtain extremely low lasing thresholds as the

transition frequency of the lasing medium approaches the band-edge, where incoherent

decay £rom the lasing transition is no longer a factor. However, the accuracy of our

description of band-edge lasing rnay be strongly limited by our use of a lowest-order

approximation scheme to describe the system very close to the band-edge. For example,

an improved treatment rnay reveal that the 'Sntracavitf laser field is itself sufficiently

strong to split a degenerate lasing level of the active atoms that lies very close to the

band-edge, forcing one level into the band-gap. This rnay result in a jump in the steady

state field intensity when a critical pump value is reached, thereby causing a switching

behaviour in the output characteristics of the laser field. By making a higher order

approximation in the atom-field coupling, we should be able to investigate the possibility

of such effects.

We also expect that the non-Markovian coupling near a band-edge may affect the

fluctuation properties of the laser field. The baseline fluctuations in the phase of the laser

field in a conventional laser are determined by the quantum fluctuations associated with

cavity losses, dephasing, and spontaneous emission [71, 1091. The Markovian nature of

each of these fluctuation sources in free space gives rise to a difisive behaviour of the

phase of the laser field, which in tum determines the quantum-limited linewidth of the

laser. Near a photonic band-edge, the non-Markovian fluctuations in the EM vacuum

that are responsible for spontaneous decay may modify this diffusive behaviour [111]. In

the limit of a high finesse mode and low dephasing, the spontaneous emission contribution

will dominate, and rnay give rise to a sub or super-diffusive evolution of the field, which

rnay then mod@ the laser Imewidth. The study of such possibilities in band-edge lasing

CHAPTER 6. CONCLUSIONS AND FUTURE DIRECTIONS 118

are the subject of our continuing research.

Finaliy, in Chapter 5 we considered the feasibility of using the occupation states of

single photons in localized defect modes in PBG materials as units of quantum informa-

tion. In our proposai, the entanglement of defect modes is mediated by near-resonant

atoms interacting with these localized states of light as the atoms traverse the connected

void network of a PBG materiai. We showed that photons in defect modes serve as qubit

states that are highly insensitive to the states of neighbouring qubits. These qubits may

exhibit very low decoherence, as required For quantum information processing. Our p r e

posai is distinct from proposais for quantum information processing in conventionai cavity

quantum electrodynamics, in which atoms are used as qubits, and a single microcavity

plays the role of the quantum bus. Our use of localized states of photons as qubits has

the advantage of scalabiiity to a large number of qubits due to the robust nature of qubits

in defect modes. Simple calculations show that the controllable entanglement of atoms

and defect modes is possible in both the microwave and opticai regimes using experimen-

taiiy achievable system parameters. However, cunent technologies suggest that simple

experiments at microwave frequencies would provide the most immediate reaiization of

our proposai. Extensions to opticai systems will require a more careful consideration of

atomic waveguiding by externaliy exciteci fieid modes in the crystal. A preliminary study

of this possibility has been carried out in Ref. [112]. Recently, Giovanetti et al. [Il31

have demonstrated how a system such as ours may be used to carry out complex quantum

algorithms. Additionaliy, the recent work of Angelakis and Knight [Il41 has extended

our idea to develop a proposal for an extremely sensitive test of Bell's inequaiities. We

therefore see that defect modes in PBG materials may prove to be a fruitN avenue of

research for quantum information applications.

In summary, we have demonstrated how simple radiative systerns in photonic crys-

tals may give rise to novel effects of considerable fundamental and applied interest. A

combination of quantitative theoretical caiculations, precise materials fabrication, and

high resolution spectroscopy are essential to realize the considerable potentid of these

materials in the field of photonics-

Appendix A

Outline of photonic band structure

calculat ions

Here, we outline the concepts of photonic band-structure and the associated Bloch modes

of t he electromagnetic field. We develop our theory in terms of the magnetic field fi rat her

than in terms of the electric or displacement fields because (i) V H = O and, (ii) the

transverse and longitudinal components of the magnetic field are continuous across the

dielectric boundaries. In practice, this leads to more rapid convergence of the relevant

Fourier series expansions.

In a three-dimensional PC, we can mite the eigenvalue equation for the magnetic

field a s

with 17p(7') the inverse of the periodic dielectric permittivity,

The medium is assumed to consist of a background material with bulk permittivity q, and

a set of scatterers, with bulk permittivity ea- The shape of the scatterers is described by

APPENDIX A. OUTLINE OF PHOTONIC BAND STRUCTURE CALCULATIONS 121

the function S, i. e.,, S(3 = 1 if F Les inside the scatterer and zero eIsewhere, distributed

periodicaily at positions

The notation of Eq. (A.2) is obtained by d e h g the matrix A = (& Ü2 Ü3) and Z3 =

Z@Z@Z. The dielectric permittivity is spatiaily periodic modula C.A. The assumption

of a scalar permittivity is reasonable for bulk materials which are not birefringent but in

no way restricts the considerations below. Chromatic dispersion effects are considered to

be negligible, thus allowing the time-dependence of the permittivity to be ignored. Let -# - *

us define the dual matrix B = ~ T ( A - ' ) ~ . For B = (bl b3), this definition leads to the

orthogonality relation

Whereas the points Z - A are the real space lattice vectors, the points f i B, for f i E Z3

are the reciprocai lattice vectors. The inverse permittivity c m be expanded in the duai

basis as

The differential equation (Al) has periodic coefficients. By the Bloch-Floquet theo-

rem we can expand the magnetic field as

where üE is spatiaiiy periodic modulo A; that is,

üY(fl = Ük'(F+ fi. A). (A.7)

The set {i} labeling the solutions can be restricted to lie within in the irreducible part

of the 6rst Bciiiouin zone (BZ), since any value of can then be obtained through a

APPENDK A- OUTLME OF PHOTONIC BAND STRUCTURE CALCULATIONS 122

combination of group transformations with respect to an operation £rom the point group

of the crystal and translations with respect to a reciprocal lattice vector. We can thereiore

express each wavevector as

where k. is an element of the irreducible part of the 1. BZ and T an element of the

crystal's point group.

Applying the Bloch-Floquet theorem, Eq, (A.6), the magnetic field can be expanded

as

Here X is the index of poiarization and the vectors

form an orthonormal right-handed tnad. This expansion inserted into Eq (-4.1) yields

an infinite eigenvaiue problem which is then solved numericaily by a suitable truncation.

Typically the cardinality of the set {5} is on the order of IO3 [12]. For any given t we

obtain a discrete set of eigenfrequencies w,, and corresponding eigenfunctions H,, which

we label by the band index n E N. It is important to note that the expression for the

electric field can be recovered fiom the magnetic field via

In addition, the Bloch waves obey the foiiowing orthogonality relations:

APPENDLX A. OUTLINE OF PHOTONIC BAND STRUCTURE CALCULATIONS

where the integration is over al1 Wace in both cases-

Appendix B

Classical field t heory for a radiating

In this Appendix, we develop the Hamiltonian describing the coupling of a classical

radiating dipole coupled to the Bloch modes of a PC, as described in Chapter 3.

B. 1 Free-field Hamiltonian

Based on the considerations of Appendix A, we denve general expressions for the scalar

and vector potential, #(T, t) and A(?, t) respectively. for the classical Hamiltonian of the

free radiation field. These expressions are particularly transparent in the Dzyaloshinsky

gauge, Le., when #(?, t) O. Then,

and the gauge condition V- (r,(r)À(F, t)) = O, reveals that in a PC the natural modes of

the radiation field are no longer transverse. This is of importance when quantizing the

field theory [Ils, 291. Given Eqs. (,4.1), (A.11), (B.1) and (B.2), it is now straightforward

APPENDIX B. CLASSICAL FIELD THEORY FOR A RADUTING DiPOLE

to derive the following expansion of the vector potential x(r', t)

where the tirne evolution of the free field is described by Bni(t) = &E(0)e-Wn~t. The field

which is the same equation as that for the electric field modes Enz(?') oE Eq. (A.11). We

choose the normalization of XnE such that

This also 6xes the normaiization in Eqs. (-4.12) and (A.13). As a consequence, the total

electric and magnetic field are given by

APPENDIX B. CLASSICAL FIELD THEORY FOR A RADIATING DiPOLE 126

where we have reintroduced the elecnic and magnetic field mades. Ëni(fl = (wd/c) -id(T')

and &(T) = V x Ad(?), respectively. Eqs. (3.7) and (B.8) ha l iy lead us to the free

field Hamiltonian

B.2 Radiating dipole embedded in a Photonic Crys-

We consider the insertion of a point dipole into a PBG structure at a prescribed location

Fo. The free dipole oscillator is described by the Hamiltonian Hdip

(B. 10)

where the dipole's natural frequency is wo = l/LC and the comple~ osciliator ampli-

tude rr is given in terms of the charge p and "cunent" Lq as a(t) = q( t ) JLwa/Sw + z ( L q ( t ) ) / d E , with Poisson brackets {a, CI') = z/[. The point dipole couples to the

electnc field via its dipole moment d(t ) = aq(t) with orientation d, which yields the

interaction energy

Bi, = -a*@) (2- E(6, t ) ) . (B.11)

Using the rotating wave approximation to the interaction term, the minimal couphg

Hamiltonian for a radiating dipole in a PC is

CoIIecting ali the above results we obtain

APPENDIX B. CLASSICAL FIELD THEORY FOR A RADIATING DIPOLE 127

Here, we have introduced the composite index p E (ni) and the coupling constants g,

In addition, in Eq. (B.12) we have introduced a mass renormalization counter term,

He, = -(A lai2 in order to cancel unphysical UV-divergent terms of our non-relativistic

theory. The nonzero Poisson brackets for an initiaiiy excited radiating dipole coupled to

the Bloch waves of a PC are:

where a(0) = 1 and &(O) = O for aii p. This, together with the Hamilton function H in

Eq. (B.12) completely defines our problern,

Appendix C

Isotropic dispersion and photon

density of states

A particularly stringent test of the accuracy of the discretized reservoir approach to

radiative dynamics in a PC cornes from its application to a dipole coupled to a 3D

isotropic band-edge electromagnetic reservoir. In this model, the coherent scattering

condition that characterizes the photonic band edge is assumed to occur a t the same

frequency for ail directions of propagation. CIearly this is not the case in a reai crystal,

whose Brillouin zone cannot have full rotational symmetry. As a result, the isotropic

model overestimates the electromagnetic modes available at a band-edge. In particdar,

near the upper photonic band edge a t frequency wu, the corresponding DOS exhibits a

divergence of the form N(w) a 1/Jw-W,. For large frequencies (w » wu) we choose

our DOS such that N(w) a u2, as is the case for free space.

Consider a 1D photonic dispersion relation in the extended zone scheme. in order to

describe a PBG at wave number ko with centra1 fiequency w, = cko = (wu + w1)/2 and

upper and lower band edge at wu and wl, respectively, we use the foilowing h a t z

APPENDK C. ISOTROPIC DISPERSION AND PHOTON DENSITY OF STATES 129

Using the requirernents w(k = 0 ) = O , w(k = ko - O + ) = Ur, w ( k = ka + O+) = WU,

akw(k = 0) = bkw(k + oo) = c, and bkw(k = ko - O+) = &w(k = ka f O+) = 0,

the unknown parameters in (C.1) can easily be expresseci in t e m of a single parameter

11 = wr/wc, 112 < 77 5 1 that describes the size of the photonic bandgap. This yields:

w+ = wc, C+ = C, y+ = ko(l - q ) , W - = wc(q2)/(2q - l ) , C- = q/J2r]-lr]-l, and

-,- = ka (1 - 7 ) ) / d m - .

n o m the dispersion relation ( C l ) , the correspondhg DOS, i. e,: N ( w ) = J d3k b(w -

w ( k ) ) is given by

for O 5 w 5 wi

For sufficiently large gaps (7 5 0.9) and bare eigedrequencies wo of the radiating dipole

close to the upper band edge, it is an excellent approximation to ignore the lower branch

of the photon dispersion relation, i. e., for k 5 ko. The resulting DOS for this so-called

three-dimensional isotropic, one-sided bandgap mode1 is shown in Fig. 3.1 for a value of

gap width parameter = 0.8 and the gap center frequency w,a/2nc = 0.5. The square-

root singularity at the band edge as weli as the W divergence Nm(w) a w2 as w + 3u

are clearly visible.

Appendix D

Calculation of the memory kernel

We first present the calculation of the memory kemel Gr(t - t') used in Chapter 4 for

the isotropic model in the effective mass approximation. Starting from equation (4.8)

and the isotropic dispersion relation near the upper band edge! w ç - w, + 4(lk( - l k ~ l ) ~ !

GI(t - t') can be expressed as

Here, 6, = ~ 2 1 - W, is the detuning of the atomic resorant frequency from the band edge-

h = mclh is a cutoff in the photon wavevector above the electron rest mass. Photons of

energy higher than the electron rest mass probe the relativistic structure of the electron

wave packets of our resonant atoms[37]. Because the isotropic model associates the band

edge with a sphere in k-space, there is no an&r dependence in the expansion of wk

about the band edge. We may thus separate out the angular integration over solid angle

R in (D.1). We may ais0 make a stationary phase approximation to the integral, as the

non-exponential part of the integrand will only contribute significantly to the integrai

for k = ko. The redt ing integraI is

APPENDIX D. CALCULATION OF THE MEMORY KERNEL 13 1

As A is a large number, we extend the range of integration to infinity in order to obtain

a simple analytic expression for Gr(t - t') [1161:

Because the relevant fiequencies in equation (D.3) are roüghly of the same order of

magnitude near a band edge, we may rewrite the prefactor as fi:/2 2 uzl '128 11 / 12he0~3/2~

, in agreement with the value given in Section II. We emphasize that the stationary phase

method yields the correct asymptotic behavior for the memory kernel for Iarge It - t'l. Irt

short times, the integral must be evaluated more precisely using the full photon dispersion

relation, as discussed below for the anisotropic model.

For an anisotropic band gap model, we must account for the variation in the mag-

nitude of the band edge wavevector a s k is rotated throughout the Brïiiouin zone. We

associate the gap with a specific point in k-space that satisfies the Bragg condition,

k = i ~ . Zn the effective mass approximation, the dispersion relation is expandeci to

second order in k about this point, t ~ k = t ~ , & A(k - b)2. Making the substitution q =

k - ka and performing the angular integration, GA(t - t)) is expresseri as

Extendhg the wavevector integration tu infll i i~, the Green function is [Il61

P (z) = 5 18 eWEdt. For oc(t - <) > 1, (w, - 10iSs-l for opticai transitions), taking

APPENDIX D. CALCULATION OF THE MEMORY KERNEL

the asymptotic expansion of 9(z) to second order gives

As t - t' + O,, (D.5) reduces to

~ ~ ( t - t') = 8 t i ~ ~ ~ * A ~ / * ";14' [,/T- iwC(t - t') , t - d + O + (D.7)

GA (t - t) possesses a weak (square root) singulacity at t = f . This is an integrable

singularity and can thus be treated numericaliy [117].

Appendix E

Evaluat ion of r~ 1 A~ (t) 1

We outline the evaluation of G 1AA(t)l2, used to obtain the low excitation population

fluctuations in Section 4.2, Eq. (4.30). A suniIar procedure is used to arrive at Eq. (4.45)

in Section 4.4. Starting kom the Laplace transform &(s) (equation (4.18)), we may use

the properties of a convolution of Laplace transforms in order to mite

Therefore, we have

with G(t - t') defined as in equation (4.7). We may rewrite this double integral in the

fom:

where Il and I2 are the f i t and second double integrals respectively. By changing the

order of the integrations in 12, we obtain

Thedore, CA 1&(t)l2 = 2Re ( I l } , and we need only explicitly evaluate 1,. The Laplace

transform of B ( t ) , ~ ( s ) (equation (4.17)), is equivaient to the Laplace transforrn of the

equation

Substituting (E.5) into IL and its complex conjugate, we obtain

as Il is red. Substituting the initial condition 18(0)(* = 1 into (E.6) gives the resuit

quoted in Section 4.2.

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