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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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