Non-Markovian Quantum Fluctuations and Superradiance Near a Photonic

PHYSICAL REVIEW A VOLUME 58, NUMBER 5 NOVEMBER 1998

Non-Markovian quantum ﬂuctuations and superradiance near a photonic band edge

Nipun Vats and Sajeev John Department of Physics, University of Toronto, 60 St. George Street, Toronto, Ontario, Canada M5S 1A7 ͑Received 9 June 1998͒ We discuss a point model for the collective emission of light from N two-level atoms in a photonic band-gap material, each with an atomic resonant frequency near the edge of the gap. In the limit of a low initial occupation of the excited atomic state, our system is shown to possess atomic spectra and population statistics that are radically different from free space. For a high initial excited-state population, mean-field theory suggests a fractionalized inversion and a macroscopic polarization for the atoms in the steady state, both of which can be controlled by an external dc field. This atomic steady state is accompanied by a nonzero expectation value of the electric field operators for field modes located in the vicinity of the atoms. The nature of homogeneous broadening near the band edge is shown to differ markedly from that in free space due to non-Markovian memory effects in the radiation dynamics. Non-Markovian vacuum fluctuations are shown to yield a partially coherent steady-state polarization with a random phase. In contrast with the steady state of a conventional laser, near a photonic band edge this coherence occurs as a consequence of photon localization in the absence of a conventional cavity mode. We also introduce a classical stochastic function with the same temporal correlations as the electromagnetic reservoir, in order to stochastically simulate the effects of vacuum fluctuations near a photonic band edge. ͓S1050-2947͑98͒04511-9͔ PACS number͑s͒: 42.50.Fx, 42.50.Lc, 42.70.Qs

I. INTRODUCTION photonic band edge ͓14,15͔ have shown that this non- Markovian system reservoir interaction gives rise to phe- In recent years, photonic band-gap ͑PBG͒ structures have nomena such as oscillatory behavior and a fractional steady been shown to lead to the localization of light ͓1͔ through state population for a single excited atomic state, as well as the carefully engineered interplay between microscopic scat- vacuum Rabi splitting and a subnatural linewidth for atomic tering resonances and the coherent interference of light emission. from many such scatterers ͓2͔. Since the initial proposal of We consider the Dicke model ͓16,17͔ for the collective photonic band gaps ͓3,4͔, PBG materials exhibiting photon emission of light, or superradiance, from N identical two- localization have been fabricated at microwave frequencies level atoms with a transition frequency near a photonic band ͓5͔ and more recently, large-scale two-dimensional PBG sys- edge. The study of superradiant emission is of interest not tems have been produced in the near-infrared ͓6͔. The ulti- only in its own right, but also because it provides a valuable mate goal for laser applications is a full three-dimensional paradigm for understanding the self-organization and emis- PBG at optical frequencies ͓7–10͔. A PBG comprises a sion properties of a band-edge laser. Of late, there has been a range of frequencies over which linear photon propagation is resurgence of interest in superradiance in the context of su- prohibited. Therefore, atoms with transition frequencies perradiant lasing action ͓18͔, and due to the experimental within the gap do not experience the usual fluctuations in the realization of a true Dicke superradiant system using laser- electromagnetic vacuum that are responsible for spontaneous cooled atoms ͓19͔. A low threshold microlaser operating decay. Instead, a photon-atom bound state is formed ͓11͔. near a photonic band edge may exhibit unusual dynamical, Unlike the suppression of spontaneous emission from an spectral, and statistical properties. We will show that such atom in a high-Q optical microcavity ͓12͔, the bound photon effects are already evident in band-edge collective spontane- may tunnel many optical wavelengths away from the atom ous emission. A preliminary study of band-edge superradi- before being reabsorbed. Near a photonic band edge, the ance for atoms resonant with the band edge ͓20͔ has shown photon density of states is rapidly varying, making it dra- that for an atomic system prepared initially with a small matically different from the ␻2 dependence found in free collective atomic polarization, a fraction of the superradiant space. This implies that the nature of vacuum fluctuations emission remains in the vicinity of the atoms, and a macro- and thus of spontaneous emission near a band edge is radi- scopic polarization emerges in the collective atomic steady cally different from the exponential decay found in free state. In addition to this form of spontaneous symmetry space ͓13͔. More fundamentally, the correlation time of the breaking, it has been demonstrated that superradiant emis- electromagnetic vacuum fluctuations near a band edge is not sion can proceed more quickly and with greater intensity negligibly small on the time scale of the evolution of an near a photonic band edge than in free space. In the absence atomic system coupled to the electromagnetic field. In fact, of an initial atomic polarization, the early stages of super- the reservoir exhibits long-range temporal correlations, mak- radiance are governed by fluctuations in the electromag- ing the temporal distinction between atomic system and elec- netic vacuum near the band edge. These fluctuations affect tromagnetic reservoir unclear. This renders the usual Born- the dynamics of collective decay and will determine the Markov approximation scheme invalid for band edge quantum limit of the linewidth of a laser operating near a systems. Studies of single atom spontaneous emission near a photonic band edge.

The organization of the paper is as follows. In Sec. II, we The Jij are collective atomic operators, defined by the rela- N present the quantum Langevin equations for collective tion Jijϵ͚kϭ1͉i͘kk͗j͉; i,jϭ1,2, where ͉i͘k denotes the ith atomic dynamics in band-edge superradiance. In Sec. III, we level of the kth atom. Using the Hamiltonian ͑2.1͒,wemay calculate an approximate, analytic solution for the equations write the Heisenberg equations of motion for the operators of that describe the N-atom system with low initial inversion of the field modes, a␭(t), the atomic inversion, J3(t)ϵJ22(t) the atomic population. We show that the atoms can exhibit ϪJ11(t), and the atomic system’s collective polarization, novel emission spectra and a suppression of population fluc- J12(t): tuations near a band edge. Sections IV and V treat the case of high initial inversion. In Sec. IV, the mean-field results of d a ͑t͒ϭϪi⌬ a ͑t͒ϩg J ͑t͒, ͑2.2͒ Ref. ͓20͔ are extended to the case of atoms with resonant dt ␭ ␭ ␭ ␭ 12 frequencies displaced from the band edge. It is shown that the phase and amplitude of the collective atomic polarization d can be controlled by an external field that Stark shifts the J3͑t͒ϭϪ2͚ g␭J21͑t͒a␭͑t͒ϩH.c. , ͑2.3͒ atomic transition relative to the band edge. The dissipative dt ␭ effect of dipole dephasing is also included in the framework of our non-Markovian system. Section V describes superra- d diant emission under the influence of vacuum fluctuations by J12͑t͒ϭ g␭J3͑t͒a␭͑t͒. ͑2.4͒ dt ͚␭ exploiting the temporal division of superradiance into quantum and semiclassical regimes. We find that the system ex- We may adiabatically eliminate the field operators by for- hibits a macroscopic steady-state polarization amplitude with mally integrating Eq. ͑2.2͒ and substituting the result into a phase precession triggered by band-edge quantum fluctua- Eqs. ͑2.3͒ and ͑2.4͒. The equations of motion for the collec- tions. In Sec. VI, we describe a method for generating a tive atomic operators are then classical stochastic function that simulates the effect of band- edge vacuum fluctuations. We show that, for a sufficiently d t large number of atoms, this classical noise ansatz agrees well J3͑t͒ϭϪ2 J21͑t͒J12͑tЈ͒G͑tϪtЈ͒dtЈ with the more exact simulations of Sec. V, and may thus be dt ͵0 useful in the analysis of band-edge atom-field dynamics. In Ϫ2J ͑t͒␩͑t͒ϩH.c. , ͑2.5͒ Appendix A, we give the details of the calculation of the 21 electromagnetic reservoir’s temporal autocorrelation func- d t tion for different models of the photonic band edge. This J12͑t͒ϭ J3͑t͒J12͑tЈ͒G͑tϪtЈ͒dtЈϩJ3͑t͒␩͑t͒. correlation function is central to determining the nature of dt ͵0 atomic decay. ͑2.6͒

Here, ␩(t)ϭ͚ g a (0)eϪi⌬␭t is a quantum noise operator II. EQUATIONS OF MOTION ␭ ␭ ␭ that contains the influence of vacuum fluctuations. G(t We consider a model consisting of N two-level atoms ϪtЈ) is the time-delay Green function, or memory kernel, with a transition frequency near the band edge coupled to the describing the electromagnetic reservoir’s average effect on multimode radiation field in a PBG material. For simplicity, the time evolution of the system operators. The Green func- we assume a point interaction, that is, the spatial extent of tion is given by the temporal autocorrelation of the reservoir the active region of the PBG material is less than the wave- noise operator, length of the emitted radiation. This is often referred to as the small sample limit of superradiance ͓17͔. We neglect the † 2 Ϫi⌬␭͑tϪtЈ͒ G͑tϪtЈ͒ϵ͗␩͑t͒␩ ͑tЈ͒͘ϭ g␭e . ͑2.7͒ spatially random resonance dipole-dipole interaction ͑RDDI͒ ͚␭ near the band edge, which may have a more important impact on atomic dynamics when the atomic transition lies † We have made use of the fact that ͗a␭(0)a␭(0)͘Ӎ0, as we deep within the PBG ͓20,21͔. Nevertheless, our simplified are dealing with atomic transition frequencies in the optical model should provide a good qualitative picture of band- domain ͓13͔. In essence, G(tϪtЈ) is a measure of the reser- edge collective emission. For an excited atomic state ͉2͘ and voir’s memory of its previous state on the time scale for the ground state ͉1͘, the interaction Hamiltonian for our system evolution of the atomic system. In free space, the density of can be written as field modes as a function of frequency is broad and slowly varying, resulting in a Green function that exhibits Markov- † † ian behavior, G(tϪtЈ)ϭ(␥/2)␦(tϪtЈ), where ␥ is the usual ϭ͚ ប⌬␭a␭a␭ϩiប͚ g␭͑a␭J12ϪJ21a␭͒, ͑2.1͒ H ␭ ␭ decay rate for spontaneous emission ͓13͔. Near a photonic band edge, the density of electromagnetic modes varies rap- † where a␭ and a␭ are the radiation field annihilation and cre- idly with frequency in a manner determined by the photon ation operators, respectively; ⌬␭ϭ␻␭Ϫ␻21 is the detuning dispersion relation, ␻k . We show that this results in long- of the radiation mode frequency ␻␭ from the atomic transi- range temporal correlations in the reservoir that affect the 1/2 tion frequency ␻21 . g␭ϭ(␻21d21 /ប)(ប/2⑀0␻␭V) e␭•ud is nature of the atom-field interaction. the atom-field coupling constant, where d21ud is the atomic In order to evaluate G(tϪtЈ) near a band edge, we first dipole moment vector, V is the sample volume, and e␭ make the continuum approximation for the field mode sum in ϭek,␴ , ␴ϭ1,2 are the two transverse polarization vectors. Eq. ͑2.7͒: 4170 NIPUN VATS AND SAJEEV JOHN PRA 58

2 2 3 In both Eqs. ͑2.10͒ and ͑2.11͒, ␦ ϭ␻ Ϫ␻ is the detun- ␻21d21 d k c 21 c G͑tϪtЈ͒ϭ eϪi͑␻kϪ␻21͒͑tϪtЈ͒. ing of the atomic resonance frequency from the band edge, 2ប⑀ ͑2␲͒3 ͵ ␻ 0 k and is a constant that depends on the dimension of the ͑2.8͒ ␤␣ band-edge singularity. In particular, for the isotropic model, 3/2 7/2 2 3/2 3 In this paper, we use an effective-mass approximation to the ␤1 ϭ␻21 d21/12ប⑀0␲ c , while in the anisotropic model, 1/2 2 2 3/2 full dispersion relation for a photonic crystal. Within this ␤3 ϭ␻21d21/8ប⑀0␻c(A␲) . approximation, we consider two models for the near-band- edge dispersion. The details of the calculation of G(tϪtЈ) III. LOW ATOMIC EXCITATION: HARMONIC- for each model and a discussion of its applicability is given OSCILLATOR MODEL in Appendix A. In an anisotropic dispersion model, appro- priate to fabricated PBG materials, we associate the band In order to understand the effects of band-edge vacuum fluctuations, we begin by presenting a simplified model that edge with a specific point in k space, kϭk0 . By preserving the vector character of the dispersion expanded about k ,we permits an analytic solution, and is applicable to a system in 0 which only a small fraction of the two-level atoms are ini- account for the fact that, as k moves away from k0 , both the direction and magnitude of the band-edge wave vector are tially in their excited state. This discussion demonstrates how modified. This gives a dispersion relation of the form light emission near a photonic band edge can give rise to atomic dynamics, emission spectra, and photon-number sta- 2 ␻kϭ␻cϮA͑kϪk0͒ . ͑2.9͒ tistics that are very different from those expected in free space. We write the atomic operators in the Schwinger boson Here, Aϭ2c2/␻ , where ␻ is the frequency width of the gap gap representation ͓23͔: gap. The positive ͑negative͒ sign indicates that ␻k is expanded about the upper ͑lower͒ edge of the PBG, and ␻c is J ͑t͒ b†͑t͒b ͑t͒, ͑3.1͒ the frequency of the corresponding band edge. This form of 12 → 1 2 dispersion is valid for a gap width ␻gapӷc͉kϪk0͉, meaning J ͑t͒ b†͑t͒b ͑t͒Ϫb†͑t͒b ͑t͒, ͑3.2͒ that the effective-mass relation is most directly applicable to 3 → 2 2 1 1 large photonic gaps and for wave vectors near the band edge. subject to the constraint on the total number of atoms, Furthermore, for a large gap and a collection of atoms that † † † b1(t)b1(t)ϩb2(t)b2(t)ϭN. The operators bi (t) and bi(t) are nearly resonant with the upper band edge, it is a very then describe transitions of the system between the excited good approximation to completely neglect the effects of the state (iϭ2) and the ground state (iϭ1). In the limit of low lower photon bands. The band-edge density of states corre- atomic excitation, the state 1 has a large population at all 1/2 ͉ ͘ sponding to Eq. ͑2.9͒ takes the form ␳(␻)ϳ(␻Ϫ␻c) , ␻ times, meaning that we can replace the inversion operator by Ͼ␻ , characteristic of a three-dimensional phase space. The c the classical value J3(t)ϷϪN, and that b1(t) can be ap- resulting Green function for ␻c(tϪtЈ)ӷ1is proximated by b1(t)ϷͱN. In this case, the initially excited 1/2 two-level atoms behave like a simple harmonic oscillator ␤ ei͓␲/4ϩ␦c͑tϪtЈ͔͒ 3 coupled to the non-Markovian electromagnetic reservoir. In GA͑tϪtЈ͒ϭ 3/2 , ͑tϪtЈ͒ our model, the Heisenberg equations of motion ͑2.5͒ and 2.6 , reduce to tϾtЈ ͑anisotropic gap͒. ͑2.10͒ ͑ ͒ In addition to the anisotropic photon dispersion model, it d t b2͑t͒ϭϪN b2͑tЈ͒G͑tϪtЈ͒dtЈϪͱN␩͑t͒. ͑3.3͒ is instructive to consider a simpler isotropic model. In this dt ͵0 model, we extrapolate the dispersion relation for a one- dimensional gap to all three spatial dimensions. We thus as- Using the method of Laplace transforms, we can solve for sume that the Bragg condition is satisfied for the same wave b2(t) and find vector magnitude for all directions in k space. This yields an effective-mass dispersion of the form ϭ ϩA( k ␻k ␻c ͉ ͉ b2͑t͒ϭB͑t͒b2͑0͒ϪͱN A␭͑t͒a␭͑0͒, ͑3.4͒ 2 ͚␭ Ϫ͉k0͉) , which associates the band-edge wave vector with a sphere in k space, ͉k͉ϭk0 . Strictly speaking, an isotropic where PBG at finite wave vector ͉k0͉ does not occur in artificially created, face centered cubic photonic crystals. However, a B͑t͒ϭ Ϫ1͕˜B͑s͖͒, ͑3.5͒ L nearly isotropic gap near k0ϭ0 occurs in certain polar crystals with polaritonic excitations ͓22͔. A simple example of ˜B͑s͒ϭ͓sϩNG˜͑s͔͒Ϫ1, ͑3.6͒ such a crystal is table salt ͑NaCl͒, which has a polariton gap in the infrared frequency regime. The band-edge density of and states in the isotropic model has the form ␳(␻)ϳ(␻ g␭ Ϫ1/2 Ϫ1 ˜ Ϫ␻ ) , ␻Ͼ␻ , the square-root singularity being charac- A␭͑t͒ϭ B͑s͒ . ͑3.7͒ c c L sϩi⌬ teristic of a one-dimensional phase space. For the Green ͭ ␭ ͮ function we obtain ͑see Appendix A͒ Ϫ1 denotes the inverse Laplace transformation, and G˜ (s)is Lthe Laplace transform of the general memory kernel, G(t 3/2 Ϫi͓␲/4Ϫ␦c͑tϪtЈ͔͒ ␤1 e ϪtЈ). In this section, we consider the case of an isotropic G ͑tϪtЈ͒ϭ , tϾtЈ ͑isotropic gap͒. I ͑tϪtЈ͒1/2 band edge in the effective-mass approximation ͓Eq. ͑2.11͔͒, ͑2.11͒ for which G˜ (s) is written as PRA 58 NON-MARKOVIAN QUANTUM FLUCTUATIONS AND... 4171

FIG. 2. Collective atomic emission spectrum ͑␻͒͑arbitrary FIG. 1. Normalized population of the excited atomic state near S an isotropic photonic band edge for low initial atomic excitation. units͒ near an isotropic band edge for low initial atomic excitation. Various values of the detuning, ␦ ϵ␻ Ϫ␻ , of the atomic reso- Various values of the detuning, ␦cϵ␻21Ϫ␻c , of the atomic reso- c 21 c nant frequency ␻ from an isotropic photonic band edge at fre- nant frequency ␻21 from a band edge at frequency ␻c are shown. 21 quency ␻ are shown. Dotted line, ␦ ϭϪ1; dashed line, ␦ ϭ0; Dashed line, ␦cϭϪ0.5; solid line, ␦cϭ0; dotted line, ␦cϭ0.5. ␦c c c c 2/3 solid line, ␦ ϭ1. ␦ is measured in units of N2/3␤ . is measured in units of N ␤1 . c c 1 photonic band edge, resulting in dressed atomic states that ␤3/2eϪi␲/4 ˜ 1 straddle the band edge. Light emission from the dressed state GI͑s͒ϭ . ͑3.8͒ ͱsϪi␦c outside the gap results in highly non-Markovian decay of the atomic population, while the dressed state shifted into the For this isotropic Green function, we denote the inverse gap is responsible for the fractional steady-state population Laplace transform of Eq. ͑3.5͒ by BI(t). BI(t) was computed of the excited state. The consequences of this strong atom- in Ref. ͓15͔ in the context of single atom spontaneous emis- ﬁeld interaction are discussed in detail for single-atom spon- sion, and a detailed mathematical derivation may be found taneous emission in Ref. ͓15͔, and for superradiant emission therein. Here, it describes the mean or drift evolution of our in Secs. IV and V of this paper. We note that the degree of Heisenberg operator b2(t). The solution has the form steady-state localization is a sensitive function of the detun-

2/3 2 2/3 2 ing, ␦c , of the atomic resonance from the band edge. The N ␤1x1tϩi␦ct N ␤1x2tϩi␦ct 2/3 BI͑t͒ϭ2a1x1e ϩa2͑x2ϩy 2͒e decay rate scales as N ␤1t for the isotropic model. How- 3 ever, there is no evidence for the buildup of interatomic co- 2/3 2 2/3 2 N ␤1x j tϩi␦ct herence, as very few of the atoms are initially excited. Ϫ a jy j͓1Ϫ⌽͑ͱN ␤1x j t͔͒e , j͚ϭ1 Equation ͑3.4͒ also allows us to calculate the system’s emission spectrum into the modes ␻ for an atom with reso- ͑3.9͒ nant frequency ␻21 using the relation where ϱ Ϫi͑␻Ϫ␻21͒␶ † i␲/4 ͑␻͒ϭ e ͗b2͑␶͒b2͑0͒͘d␶ϩc.c. x1ϭ͑AϩϩAϪ͒e , ͑3.10͒ S ͵0

Ϫi␲/6 i␲/6 Ϫi␲/4 ˜ x2ϭ͑Aϩe ϪAϪe ͒e , ͑3.11͒ ϳRe͕B*͓i͑␻Ϫ␻21͔͖͒, ͑3.15͒

i␲/6 Ϫi␲/6 i3␲/4 x3ϭ͑Aϩe ϪAϪe ͒e , ͑3.12͒ where ˜B(s) is deﬁned in Eq. ͑3.6͒. The spectrum for the isotropic model is then 3 1/2 1/3 1 1 4 ␦c A ϭ Ϯ 1ϩ , ͑3.13͒ Ϯ 2 2 ͫ 27 ␤3ͬ 0, ␻р␻c ͭ 1 ͮ

I͑␻͒Ӎ 3/2 ͱ␻Ϫ␻c 2 S N␤ , ␻Ͼ␻c. y jϭͱx , jϭ1,2,3. ͑3.14͒ 1 2 3 2 j ͭ N ␤ ϩ͑␻Ϫ␻21͒ ͑␻Ϫ␻c͒ 1 ͑3.16͒ x Ϫt2 The error function ⌽(x)ϭ(2/ͱ␲)͐0e dt. The probability of finding the atoms in the excited state is This spectrum is shown in Fig. 2, and differs significantly † 2 given by ͗b2(t)b2(t)͘ϭ͉BI(t)͉ , and is plotted in Fig. 1. We from the Lorentzian spectrum for light emission in free find that the excited-state population exhibits decay and os- space. In fact, the emission spectrum is not centered about cillatory behavior before reaching a nonzero steady-state the atomic resonant frequency, which is what one would ex- value due to photon localization. These effects are due to the pect for an atom decaying to an unrestricted vacuum mode strong dressing of the atoms by the radiation field near a density. We see that for an arbitrary detuning, ␦c ,of␻21 4172 NIPUN VATS AND SAJEEV JOHN PRA 58 from the band edge, the emission spectrum vanishes for frequencies at the band edge and within the gap, ␻р␻c . This is consistent with the localization of light near the atoms for electromagnetic modes within the PBG. As ␻21 is detuned further 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 dressed state is reduced. Conversely, as ␻21 is moved out of the gap, the emission profile becomes closer to a Lorentzian in form and the total emitted intensity increases. The spectral linewidth ratio between the isotropic band edge and free space is of the 1/3 order of ␤1 /(␥N ), while for an anisotropic band edge it is ϳN␤3 /␥. 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- FIG. 3. Fluctuations in the excited-state atomic population as oscillator model. Variances in the atomic population can be measured by the Mandel parameter, Q(t)ϭ͓͗n2(t)͘ 2 written in terms of the Mandel Q parameter ͓24͔, Ϫ͗n(t)͘ ͔/͗n(t)͘, for low initial excitation for an atomic resonant frequency tuned to an isotropic photonic band edge, ␦cϭ0. Dashed ͗n2͑t͒͘Ϫ͗n͑t͒͘2 line, Q(0)ϭ2; solid line, Q(0)ϭ0. Double-dashed line denotes Q͑t͒ϭ , ͑3.17͒ fluctuations for Poissonian population variance, Q(0)ϭ1. ͗n͑t͒͘ tions to the high excitation ͑superradiant͒ regime. In this where n(t)ϵb†(t)b (t) is the number operator for the occu- 2 2 case, the two-level nature of the atomic operators will be- pation of the excited state. Since both the free space and come important and will modify the quantum statistics from PBG solutions in our model can be written in the form of Eq. that of the harmonic-oscillator picture. This generalization is ͑3.4͒, we can write the Q parameter in the general form considered in the next two sections.

2 2 Q͑t͒ϭ͉B͑t͉͒ Q͑0͒ϩN ͉A␭͑t͉͒ . ͑3.18͒ IV. HIGH ATOMIC EXCITATION: ͚␭ MEAN-FIELD SOLUTION 2 Again, ͉B(t)͉ is the normalized probability of finding the When the atomic system is initially fully or nearly fully initially excited fraction of the atoms still in the excited state inverted, we expect interatomic coherences, transmitted via at time t. For an isotropic band edge, B(t)ϭBI(t) ͓Eq. the atomic polarizations, to have a strong influence on emis- ϪN t/2 ͑3.9͔͒, whereas in free space, B(t)ϳe ␥ , representing the sion dynamics. For such high initial atomic excitation, the exponential decay of the excited-state population. Using the quantum Langevin equations ͑2.5͒ and ͑2.6͒, paired with the 2 2 identity N͚␭͉A␭(t)͉ ϭ1Ϫ͉B(t)͉ , as derived in Appendix non-Markovian memory kernels ͑2.10͒ or ͑2.11͒, do not pos- B, we can write the population fluctuations as sess an obvious analytic solution. Moreover, conventional perturbation theory applied to these equations fails to recap- 2 Q͑t͒ϭ͉B͑t͉͒ ͓Q͑0͒Ϫ1͔ϩ1. ͑3.19͒ ture the influence of the photon-atom bound state ͓11͔, which plays a crucial role in band-edge radiation dynamics. How- For arbitrary initial statistics, atoms in free space decay to ever, when the superradiant system is prepared with a non- the vacuum state with Q(t)ϭ1; since the atoms decay fully, zero initial polarization ͓J12(0)Þ0͔, the average dipole mo- there are no meaningful atomic statistics in the long-time ment dominates the incoherent effect of the vacuum limit. Q(t) is plotted in Fig. 3 for the isotropic band edge fluctuations and the subsequent evolution is well described (␦cϭ0) for the cases Q(0)ϭ0, 1, and 2. Near the band edge, by a semiclassical approximation ͓17͔. In this case, it is pos- photon localization prevents the atomic system from decay- sible to factorize the atomic operator equations: ing to the ground state. We find instead that the steady-state statistics are sensitive to the statistics of the initial state and d t to the value of ␦ . A system initially prepared with super- ͗J ͑t͒͘ϭϪ4Re ͗J ͑t͒͘ ͗J ͑tЈ͒͘G͑tϪtЈ͒dtЈ , c dt 3 21 ͵ 12 Poissonian statistics ͓Q(0)Ͼ1͔ experiences a suppression of ͭ 0 ͮ 4.1 population fluctuations in the steady state. In a system that is ͑ ͒ initially sub-Poissonian ͓Q(0)Ͻ1͔, the fluctuations in- t crease, but are held below the Poissonian level by photon d ͗J12͑t͒͘ϭ͗J3͑t͒͘ ͗J12͑tЈ͒͘G͑tϪtЈ͒dtЈ. ͑4.2͒ localization. In both cases, the steady-state value of the dt ͵0 atomic population fluctuations is controlled by ␦c . Our harmonic oscillator model thus suggests that a PBG system may The brackets ͗ ͘ denote the quantum-mechanical average of exhibit atypical quantum statistics in the absence of a cavity the HeisenbergO operator over the Heisenberg picture atom- O or external fields. It is important to extend the analysis of field state vector, ͉⌿͘ϭ͉vac͘ ͉␺͘, where ͉vac͘ represents collective emission under the influence of vacuum fluctua- the electromagnetic vacuum state, and ͉␺͘ represents the ini- PRA 58 NON-MARKOVIAN QUANTUM FLUCTUATIONS AND... 4173

FIG. 4. Mean-field solution for the atomic inversion, ͗J (t)͘/N, 3 FIG. 5. Mean-field solution for the atomic polarization ampli- near an isotropic photonic band edge, starting with an infinitesimal tude, J (t) /N, near an isotropic photonic band edge, starting initial polarization, rϭ10Ϫ5. Various values of the detuning, ␦ ͉͗ 12 ͉͘ c with an infinitesimal initial polarization, rϭ10Ϫ5. Various values ϵ␻21Ϫ␻c , of the atomic resonant frequency ␻21 from a band edge of the detuning, ␦cϵ␻21Ϫ␻c , of the atomic resonant frequency at frequency ␻c are shown. ͑a͒ ␦cϭ1; ͑b͒ ␦cϭ0.5; ͑c͒ ␦cϭ0; ͑d͒ 2/3 ␻21 from a band edge at frequency ␻c are shown. ͑a͒ ␦cϭ1; ͑b͒ ␦cϭϪ0.5; ͑e͒ ␦cϭϪ1. ␦c is measured in units of N ␤1 . ␦cϭ0.5; ͑c͒ ␦cϭ0; ͑d͒ ␦cϭϪ0.5; ͑e͒ ␦cϭϪ1. ␦c is measured in 2/3 tial state of the atomic system. Clearly in this mean-field units of N ␤1 . approach, the quantum noise contribution is neglected, as sity is proportional to N5/3. This is to be compared with the ͗␩(t)͘ϭ0. Recently, Bay, Lambropoulos, and Mo”lmer ͓25͔ values N and N2 for the free space decay rate and peak found that, for a simpler Fano profile gap model, the dynam- radiation intensity, respectively. ics of superradiant emission are affected by the choice of As in single-atom spontaneous emission near an isotropic factorization applied to the full quantum equations. How- band edge ͓15͔, the dressing of the atoms by their own ra- ever, the complete factorization used here retains the quali- diation field causes a splitting of the band of collective tative features and evolution time scales of more elaborate atomic states such that the collective spectral density van- factorization schemes. Equations ͑4.1͒ and ͑4.2͒ were solved ishes at the band-edge frequency. The strongly dressed numerically in Ref. ͓20͔ for an atomic resonance frequency atomic states are repelled from the band edge, with some coincident with the band edge (␦cϭ0) and a small initial levels being pulled into the gap and the remaining levels collective polarization. The initial collective state was as- being pushed into the electromagnetic continuum outside the sumed to be of the form PBG. In the long-time ͑steady-state͒ limit, the energy con- N tained in the dressed states outside the band gap decays whereas the energy in the states inside the gap remains in the ͉␺͘ϭ ͑ͱr͉1͘ϩͱ1Ϫr͉2͘)k ͑4.3͒ k͟ϭ1 vicinity of the emitting atoms. It is the localized light asso- ciated with the gap dressed states that sustains the fraction- with rӶ1, so that initially the atoms are almost fully in- alized steady-state inversion and nonzero atomic polariza- verted. In this paper, we extend the previous analysis to tion. For the isotropic model, this splitting and fractional atomic frequencies detuned from the band edge. Despite localization persist even when ␻21 lies just outside the gap its neglect of vacuum fluctuations, mean-field theory illumi- (␦cϾ0), and the fraction of localized light in the steady state nates many of the interesting features of the system. The increases as ␻21 moves towards and enters the gap. In the relationship between mean-field theory and a more complete dressed-state picture, the self-induced oscillations in both the description including quantum fluctuations is discussed in inversion and the polarization that occur during radiative Sec. V. emission can be interpreted as being due to interference be- For clarity, we discuss separately the atomic dynamics in tween the dressed states. The oscillation frequency is propor- our isotropic and anisotropic dispersion models. Figures 4 tional to the frequency splitting between the upper and lower and 5 show the inversion per atom and the average polariza- collective dressed states. This is the analog of the collective tion amplitude per atom respectively for various values of ␦c Rabi oscillations of N Rydberg atoms in a resonant high-Q near an isotropic band edge. We see from Fig. 4 that a frac- cavity ͓26͔. From Fig. 4, we see that a dressed state outside tion of the superradiant emission remains localized in the the band gap decays more slowly for atomic resonant fre- vicinity of the atoms in the steady state, due to the Bragg quencies deeper inside the gap, causing the collective oscil- reflection of collective radiative emission back to the atoms. lations to persist over longer periods of time. Clearly, this This localized light exhibits a nonzero expectation value for decay is nonexponential and highly non-Markovian in na- the field operator, which in turn leads to the emergence of a ture. Figure 5 confirms that, as required, the polarization am- macroscopic polarization amplitude in the steady state. We plitude for large negative values of ␦c is constrained by the further note that the decay rate for the upper atomic state is condition, ͗J12(t)͘/Nр1/2. proportional to N2/3. Accordingly, the peak radiation inten- In Fig. 6, we plot the phase angle of the collective 4174 NIPUN VATS AND SAJEEV JOHN PRA 58

FIG. 6. Mean-field solution for the phase angle ͑in radians͒ of FIG. 7. Mean-field solution for the atomic inversion, ͗J3(t)͘/N, the atomic polarization, ␪(t), near an isotropic photonic band edge, near an anisotropic photonic band edge, starting with an infinitesi- Ϫ6 starting with an infinitesimal initial polarization, rϭ10Ϫ5. Various mal initial polarization, rϭ10 . Various values of the detuning, ␦ ϵ␻ Ϫ␻ , of the atomic resonant frequency ␻ from a band values of the detuning, ␦cϵ␻21Ϫ␻c , of the atomic resonant fre- c 21 c 21 edge at frequency ␻ are shown. Dashed line, ␦ ϭ0.1; solid line, quency ␻21 from a band edge at frequency ␻c are shown. ͑a͒ ␦c c c ␦ ϭ0; dotted line, ␦ ϭϪ0.3. ␦ is measured in units of N2␤ . ϭ0.5; ͑b͒ ␦cϭ0; ͑c͒ ␦cϭϪ0.75; ͑d͒ ␦cϭϪ1. ␦c is measured in c c c 3 2/3 units of N ␤1 . in contrast with the isotropic model, we see that photon lo- atomic polarization in the isotropic model, ␪(t) calization is lost for even a small detuning of ␻21 into the Ϫ1 ϭtan ͕Im͗J12(t)͘/Re͗J12(t)͖͘. Prior to atomic emission, continuum of field modes outside the band gap. Therefore, this phase angle rotates at a constant rate, and in the vicinity while we find a macroscopic steady-state polarization and of the decay process ␪(t) exhibits the effects of collective precessional dynamics of the Bloch vector for ␦cр0 ͑Fig. 8͒, Rabi oscillations. When the emission is complete, the rate of for ␦cϾ0 the polarization dies away after collective emission change of phase angle, ␪˙ (t), attains a new steady-state value, has taken place. Photon localization from an atomic level lying just outside the gap in a three-dimensional PBG mate- ␪˙ (t ), that depends sensitively on the detuning frequency s rial may, however, be realized through quantum interference ˙ ␦c . ␪(ts) is a measure of the energy difference between the effects if there is a third atomic level lying slightly inside the bare atomic state and the localized dressed state, ប(␻21 gap ͓28͔. These results point to the greater sensitivity of the Ϫ␻loc). Such a polarization phase rotation implies that the atomic dynamics to the more realistic anisotropic band edge. collective atomic Bloch vector of the system exhibits preces- Because the isotropic model overestimates the momentum sional dynamics in the steady state. Unlike the conventional space for photons satisfying the Bragg condition, photon lo- precession ͓27͔ of atomic dipoles in an ordinary vacuum driven by an external laser field, Bloch vector precession in a PBG occurs in the absence of an external driving field. In- stead, 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. 6 that for values of ˙ ␦c such that ␻21Ϫ␻locϽ0, ␪(ts) is negative, while for ␻21 ˙ Ϫ␻locϾ0, ␪(ts) is positive, i.e., the phase is rotating in the opposite direction. At a detuning corresponding to a constant ˙ phase in the steady state ͓␪(ts)ϭ0͔, the dressed and bare states are of the same energy; this occurs for a detuning 2/3 value of ␦cϭϪ0.644N ␤1 . At this value of ␦c , we also find that ͗J3(ts)͘ϭ0, implying that there is no net absorption of light by the atomic system. This is, in essence, a collective transparent state ͓27͔. Collective emission dynamics near an anisotropic band edge are pictured in Figs. 7 and 8. For ␻21 coincident with the band edge or slightly within the gap (␦cр0), we again FIG. 8. Mean-field solution for the atomic polarization ampli- find a fractional atomic inversion in the steady state Fig. 7 . ͑ ͒ tude, ͉͗J12(t)͉͘/N, near an anisotropic photonic band edge, starting Rabi oscillations in the atomic population are much less pro- with an infinitesimal initial polarization, rϭ10Ϫ6. Various values nounced than in the isotropic model, even for ␻21 detuned of the detuning, ␦cϵ␻21Ϫ␻c , of the atomic resonant frequency into the gap. This demonstrates that the dressed atomic states ␻21 from a band edge at frequency ␻c are shown. Dashed line, ␦c outside a physical photonic band gap decay much more rap- ϭ0.1; solid line, ␦cϭ0; dotted line, ␦cϭϪ0.3. ␦c is measured in 2 idly than the isotropic model would suggest. Furthermore, units of N ␤3 . PRA 58 NON-MARKOVIAN QUANTUM FLUCTUATIONS AND... 4175 calization effects and vacuum Rabi splitting are exaggerated in the isotropic model relative to an engineered photonic crystal. In the anisotropic model, the phase space available for propagation vanishes as the optical frequency approaches the band edge. As 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 N2 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 mirrorless, low-threshold mi- crolasers exhibiting ultrafast modulation speeds. From polarization phase and amplitude results, we con- clude that ͑i͒ unlike in free space, the atoms near a photonic band edge attain a fractionally inverted state with constant FIG. 9. Mean-field solution for the atomic inversion ͑solid line͒ polarization amplitude and rate of change of phase angle. and polarization amplitude ͑dashed line͒ under the influence of col- This corresponds to a macroscopic atomic coherence in the lision broadening for an atomic resonant frequency at an isotropic steady state analogous to that experienced in a laser. In our photonic band edge, ␦cϭ0. The system is given an infinitesimal case, however, ‘‘lasing’’ occurs in the band-edge continuum initial polarization, rϭ10Ϫ5. The simulated Stark shift is a Gauss- rather than into a conventional cavity mode. ͑ii͒ By varying ian random distribution with zero mean and standard deviation 0.5N2/3␤ . the value of ␦c , one may control the direction and rate of 1 change of the steady-state polarization phase angle. This may be realized by applying a small external dc field to the derstood by noting that the random frequency shifts are sym- sample that Stark shifts the atomic transition frequency of metrically distributed about the mean resonant frequency. the atoms. This type of control over the collective atomic Frequency shifts into the gap promote photon localization, Bloch vector may be of importance in the area of informa- while those away from the gap cause further decay of the tion storage and optical memory devices ͓29,30͔. atomic inversion. Over time, the net result is that the fre- The above analysis makes it clear that collective sponta- quency shifts away from the gap encourage the decay of the neous emission dynamics in a PBG are significantly different atomic population. This is true even in atomic systems for from those in free space. In a real PBG material, the dephas- which the mean resonant frequency lies within the gap. From ing of atomic dipoles due to interatomic collisions or the above considerations, it is clear that dephasing is a sig- phonon-atom interactions may also have a significant effect nificant perturbation on photon localization near a photonic on the evolution of our system over a large range of tempera- band edge. As in the case of a conventional laser, the effects tures. In the free space Markov approach, dipole dephasing is of dephasing may be partially compensated for by external described by a phenomenological polarization decay con- pumping. stant ͓31͔. Since the Markov approximation does not apply Although a superradiant system can be prepared in a co- near a band edge, one cannot account for dephasing by sim- herent initial state of the type described by Eq. ͑4.3͓͒27͔, ply adding a phenomenological decay term to Eq. ͑4.2͒. collective emission is typically initiated by spontaneous However, we expect that the atomic resonant frequency will emission, a random, incoherent process. Over time, sponta- experience random Stark shifts due to atom-atom or atom- neous emission leads to the buildup of macroscopic coher- phonon interactions. This effect can be included in the de- ence in the sample. The effect of vacuum fluctuations is then scription of our system by adding a variation ⌬ to the detun- of considerable importance in the full description of superradiance, both from a fundamental point of view, and for ing frequency ␦c at each time step in a computational simulation of Eqs. ͑4.1͒ and ͑4.2͒. ⌬ is chosen to be a Gauss- potential device applications, such as the recently proposed ian random number with zero mean. The width of the Gauss- superradiant laser ͓18͔. In the next section, we present a ian distribution is determined by the magnitude of the ran- more detailed description of PBG superradiance that takes dom Stark effect. Such a simulation in free space would into account the role of quantum fluctuations. include a random ⌬ only in the equation for the atomic polarization. This is because the slowly varying photon density V. BAND-EDGE SUPERRADIANCE of states seen by the atoms at the frequency ␻21ϩ⌬ does not AND QUANTUM FLUCTUATIONS change significantly with typical homogeneous line broaden- In order to describe the evolution of the superradiant sys- ing effects. In contrast, we have seen that near a photonic tem’s collective Bloch vector under the influence of quantum band edge, slight variations in ␦ may drastically change the c fluctuations, we consider atomic operator correlation func- atomic inversion. Therefore we include ⌬ in both system tions of the form ͓32͔ equations. In Fig. 9, we plot the evolution of the collective inversion and polarization under the simulated collision pq p q g ϭ͗͑J12͒ ͑J21͒ ͘. ͑5.1͒ broadening described above. The random Stark shifts lead to the loss of macroscopic polarization and the loss of atomic Here the operators are evaluated at equal times. As in free inversion in the long-time limit. The latter effect can be un- space, we expect vacuum fluctuations to drive the system 4176 NIPUN VATS AND SAJEEV JOHN PRA 58 from its unstable initial state with all atoms inverted to a d t ͗J ͑t͒͘ ϭN ͗J ͑tЈ͒͘ G͑tϪtЈ͒dtЈϩN␩͑t͒. new stable equilibrium state. Such fluctuations are particu- dt 12 A ͵ 12 A larly relevant prior to the buildup of macroscopic atomic 0 ͑5.6͒ polarization. Indeed, they provide the trigger for superradiant emission. In the early-time, inverted regime, we may set This is a linear equation that has lost its operator character J3(t)ϭJ3(0) in Eqs. ͑2.5͒ and ͑2.6͒, giving over the atomic variables but not over the electromagnetic reservoir, as evidenced by the presence of the quantum noise d t operator, ␩(t). Equation ͑5.6͒ can be solved by the method J12͑t͒ϭ dtЈJ3͑0͒J12͑tЈ͒G͑tϪtЈ͒ϩJ3͑0͒␩͑t͒. dt ͵0 of Laplace transforms. The solution has the form ͑5.2͒

͗J12͑t͒͘AϭD͑t͒͗J12͑0͒͘AϩN C␭͑t͒a␭͑0͒, ͑5.7͒ The resulting equation remains nonlinear, and involves prod- ͚␭ ucts of atomic and reservoir operators. We may simplify ex- pressions containing operators in this inverted regime by where considering operator averages over only the atomic Hilbert space. For an arbitrary Heisenberg operator (t), we denote D͑t͒ϭ Ϫ1͕D˜ ͑s͖͒, ͑5.8͒ the atomic expectation value for an initial fullyO inverted state L Ϫ1 ͉I͘ by ͗ ͘Aϵ͗I͉ ͉I͘. We denote by the set ͕͉␭͖͘ a complete D˜ ͑s͒ϭ͓sϪNG˜͑s͔͒ , ͑5.9͒ set of 2ON normalizedO basis vectors for the atomic Hilbert space including ͉I͘, such that ͗␭͉I͘ϭ␦␭,I , where ␦␣,␤ is the and Kronecker delta function. Clearly, ͗I͉J3(0)͉␭͘ϭN␦I,␭ . g Since J3(0) acts as a source term for J12(t) in Eq. ͑5.2͒,we Ϫ1 ␭ ˜ C␭͑t͒ϭ D͑s͒ . ͑5.10͒ also have the property ͗I͉J12(t)͉␭͘ϭ0 for ␭ÞI in the in- L ͭ sϩi⌬␭ ͮ verted regime. This can be shown by considering the equa- Ϫ1 tion of motion for ͗I͉J12(t)͉␭͘: Again, denotes the inverse Laplace transform. The LaplaceL transformation of the memory kernel for an isotropic t ˜ d band edge, GI(s), is given in Eq. ͑3.8͒. Despite the fact that ͗I͉J12͑t͉͒␭͘ϭ dtЈ ͗I͉J3͑0͉͒␮͗͘␮͉J12͑tЈ͉͒␭͘ dt ͵0 ͚␮ ͗J12(0)͘Aϭ0, we retain the first term in Eq. ͑5.7͒ for later notational convenience. ϫG͑tϪtЈ͒ϩ͗I͉J3͑0͉͒␭͘␩͑t͒ The early-time quantum fluctuations in a superradiant system prevent us from predicting a priori the evolution of any t single experimental realization of the atoms. Instead, we can ϭN dtЈ͗I͉J12͑tЈ͉͒␭͘G͑tϪtЈ͒ϩN␦I,␭␩͑t͒, ͵0 only determine the probability of a particular trajectory of the collective atomic Bloch vector. In order to obtain the ͑5.3͒ statistics of a band-edge superradiant pulse, we first deter- where ␮ labels a complete set of atomic states. This inte- mine the statistics of the collective Bloch vector for a set of grodifferential equation satisfies the initial condition identically prepared systems after each has passed through the early-time regime governed by vacuum fluctuations. The ͗I͉J (0)͉␭͘ϭ0, since J (0) acts as a raising operator on 12 12 relevant time scale will be referred to as the quantum to the fully inverted bra vector ͗I͉. For ␭ÞI, the source term in semiclassical evolution crossover time, tϭt0 . Our approach Eq. ͑5.3͒ is also absent, leading to the solution ͗I͉J12(t)͉␭͘ ϭ0. Using this property, we may replace the atomic average is to calculate the phase and amplitude distributions of the over products of atomic operators with products of atomic polarization at the crossover time quantum mechanically. averages, provided that J (t)ϭJ (0). For example, The subsequent (tϾt0) evolution of the ensemble is then 3 3 obtained by solving the semiclassical equations ͑4.1͒ and ͑4.2͒ using the polarization distribution function at t0 .In other words, the distribution of values of J (t ) obtained ͗J12J21͘ϭ͚ ͗vac͉ ͗I͉J12͉␮͗͘␮͉J21͉I͘ ͉vac͘ ͗ 12 0 ͘ ␮ from the early-time quantum fluctuations provide the initial conditions for subsequent, semiclassical evolution. In order ϭ͗͗J12͘A͗J21͘A͘R . ͑5.4͒ to implement this approach, we must first identify t0 for our system ͓17,33͔. One expects such a transition to occur in the Here, ͗ ͘ ϵ͗vac͉ ͉vac͘ denotes an expectation value over R high atomic inversion regime, ͗J (t)͘ӍN. It is natural to the reservoirO variables.O For an arbitrary moment g pq,we 3 define t such that for tϾt the expectation value of the have 0 0 commutator of the system operators J21(t) and J12(t) be- pq p q comes very small compared to the expectation value of their g ϭŠ͗J12͘A͗J21͘A‹R . ͑5.5͒ product ͓17͔. This gives the condition

We note that such a factorization is valid only for an anti- ͗J21͑t͒J12͑t͒͘ӷ͓͗J21͑t͒,J12͑t͔͒͘, tϾt0 . ͑5.11͒ normal ordering of the polarization operators, since ͗I͉J21(t)͉␭͘ does not vanish in general. Evaluating the above commutator, we have Taking the atomic expectation value of Eq. ͑5.2͒,weob- ͓͗J21(t),J12(t)͔͘ϭ͗J3(t)͘, which is equal to N for full tain atomic inversion. From Eqs. ͑5.5͒ and ͑5.7͒, we ﬁnd that PRA 58 NON-MARKOVIAN QUANTUM FLUCTUATIONS AND... 4177

pϪ1 ͗J21͑t͒J12͑t͒͘ϭ͗J3͑t͒ϩJ12͑t͒J21͑t͒͘ This expression has corrections of order N , meaning that it is asymptotically valid for large N. Equating Eqs. ͑5.14͒ 2 2 and ͑5.16͒, we solve for the distributions P(␬) and Q(␾)to ϭN 1ϩN͚ ͉C␭͑t͉͒ ϭN͉D͑t͉͒ . ͫ ␭ ͬ obtain the desired initial polarization distribution for the semiclassical superradiance equations. The early time distri- ͑5.12͒ butions for free space and the band edge differ only in the The last equality is obtained by use of the identity form of the function D(t), as the above analysis makes no 2 2 other distinction between the two cases. Thus in the band- N͚ ͉C (t)͉ ϭ͉D(t)͉ Ϫ1, as derived in Appendix B. In free ␭ ␭ edge system, as in free space, the entire effect of the early space, ͉D(t)͉2ϭeN␥t, giving the crossover time, tfree 0 time atomic evolution can be recaptured using the distribu- Ӎ1/N␥. One can solve for the crossover time near a band PBG tion of initial conditions given at tϭt0 . The phase of the edge, t0 , computationally. In the isotropic model, for ␦c PBG 2/3 polarization is given by the relation ϭ0 we find that t0 Ӎ1.24/N ␤1 . The crossover time maintains this 1/N2/3␤ dependence for ␻ displaced from 2␲ 1 21 i͑pϪq͒␾ the band edge. The corresponding time scale for the aniso- ͵ d␾e Q͑␾͒ϭ␦ p,q . ͑5.17͒ 2 0 tropic gap is 1/N ␤3 . The buildup of a macroscopic polarization then occurs more slowly near an isotropic band edge This shows that Q(␾) is uniformly distributed between 0 and more quickly near an anisotropic band edge than in free and 2␲. The initial polarization amplitude distribution is space. found from the relation Using a semiclassical approach, we may write the value ϱ of the polarization at any time tуt0 in terms of an amplitude 2p 2 p Cl d␬P͑␬͒␬ ϭp!͓N͉D͑t0͉͒ ͔ . ͑5.18͒ ␬ and a phase angle ␾, ͗J12(t)͘ ϵJ(␬,␾,t). The super- ͵0 script Cl refers to the fact that the expectation value ͗͘Cl is 2 taken in the semiclassical regime tуt0 . We define P(␬)d␬ The result is a Gaussian distribution of width N͉D(t0)͉ cen- as the probability of finding the amplitude between ␬ and tered at zero, ␬ϩd␬, and Q(␾)d␾ as the probability of finding the phase angle between ␾ and ␾ϩd␾. We may then write the mo- 1 Ϫ␬2 P͑␬͒ϭ exp . ͑5.19͒ ments of the macroscopic polarization distribution as 2 ͫN͉D͑t ͉͒2ͬ ͱ␲N͉D͑t0͉͒ 0

p q Cl It has been shown via density matrix methods ͓17͔ that in ͗„J12͑t͒… „J21͑tЈ͒… ͘ ϭ d␬ d␾P͑␬͒Q͑␾͒ ͵ ͵ free space one may choose the crossover time anywhere in ϫ͓J͑␬,␾,t͔͒p͓J*͑␬,␾,tЈ͔͒q. the inverted regime, the simplest choice being t0ϭ0. This is due to the absence of temporal correlations of the reservoir ͑5.13͒ for tÞtЈ. Figure 10 shows the ensemble-averaged collective emission in free space and at an isotropic band edge (␦c For tϭt0 , we assume that the polarization has the form ϭ0) for Nϭ100 atoms. Both the free space and band edge i␾ J(␬,␾,t)ϭ␬e , giving for the moments systems are shown for two choices of initial polarization dis-

p q Cl tribution. The solid lines correspond to the choice of t0ϭ0in ͗„J12͑t0͒… „J21͑t0͒… ͘ the amplitude distribution ͑5.19͒ for both free space and the band edge. The dashed lines correspond to the choice t0 ϭ d␬ d␾P͑␬͒Q͑␾͒␬ pϩqei͑pϪq͒␾. free PBG ͵ ͵ ϭt0 and t0ϭt0 for the free-space and band-edge systems, respectively. As per Eq. ͑5.17͒, the initial phase of the ͑5.14͒ polarization in all cases is chosen from a uniform random distribution. As expected, Fig. 10 demonstrates that the The quantum analog, ͗͘Q, of Eq. ͑5.14͒ can be written in the choice of t0 is unimportant in free space, so long as it is form of Eq. ͑5.5͒ evaluated at tϭt0 . Substituting Eq. ͑5.7͒ chosen in the inverted regime. Near a photonic band edge, and its adjoint into Eq. ͑5.5͒ yields we see that the choice of t0 affects the later evolution of the p q Q system. In particular, it affects the onset time for collective ͗„J12͑t0͒… „J21͑t0͒… ͘ emission. It is clear from these simulations that the details of p q the non-Markovian evolution in the quantum regime play a pϩq * † ϭN ͚ C␭͑t0͒a␭͑0͒ ͚ C␭ ͑t0͒a␭͑0͒ . crucial role in the subsequent semiclassical evolution of the ͳ ͫ ␭ ͬ ͫ ␭ ͬ ʹ R band edge superradiance. The long-range temporal correla- ͑5.15͒ tions of the reservoir require that we treat the vacuum fluctuations explicitly throughout the quantum evolution of the As the reservoir expectation value is taken over the the op- system. A similar picture holds in the case of an anisotropic erators a␭ , which satisfy a Gaussian probability distribution, PBG material. In our anisotropic model, memory of the ini- Wick’s theorem ͓13͔ is applied in order to reduce the opera- tial state is expressed through the Green function ͑2.10͒.In tor averages of products of field operators to averages over this case, superradiance is also highly sensitive to early stage products of pairs of field operators. We then have quantum fluctuations. Since ensemble averages of atomic observables are ex- p q Q p 2p ͗„J12͑t0͒… „J21͑t0͒… ͘ Ӎ␦ pqN p!͉D͑t0͉͒ . ͑5.16͒ perimentally measurable quantities, we consider these in 4178 NIPUN VATS AND SAJEEV JOHN PRA 58

FIG. 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 (␦cϭ0). Solid lines, result for initial polarization distribution at tϭ0 for each system; dashed lines, result for initial polarization distribution at tϭt0 for each system.

some detail. We use the notation ͗͘ens to denote an ͗J3(t)͘ens for a given atomic detuning is unchanged from the ensemble-averaged quantum expectation value. For illustra- mean-field result, ͗J3(ts)͘. Since the steady state is deter- tion, we focus on the ␦cϭ0 and zero dephasing case for a mined by the atom-field coupling strength, and not by the system of 100 atoms in the isotropic effective-mass model. dynamics of the system, it is insensitive to initial conditions. The extension to non-zero detuning and finite dipole dephas- Fluctuations in the excited-state atomic population may be ing follows from the discussion of Sec. IV. From Fig. 11, it expressed in terms of the delay time for the onset of super- is evident that the ensemble exhibits a fractional population radiant emission, defined as the time at which the system is inversion in the steady state. The steady-state value of exactly half excited, i.e., ͗J3͘ϭ0. Vacuum fluctuations result in a distribution of delay times for the ensemble, asymmetri- cally centered about a peak value, as pictured in Fig. 12. The delay time distribution is qualitatively similar to that ob-

FIG. 11. Ensemble-averaged atomic inversion, ͗J3(t)͘ens /N, and atomic polarization amplitude, ͉͗J12(t)͘ens͉/N ͑dot-dashed line͒, for a system of Nϭ100 atoms near an isotropic photonic band edge. The ensemble average is taken over 2000 initial polarization FIG. 12. Distribution of delay times for a system of 100 atoms values. Inversion: long-dashed curve, ␦cϭϪ0.5; solid line, ␦c at an isotropic band edge (␦cϭ0) for 4000 realizations of the su- 2/3 ϭ0; short-dashed line, ␦cϭ0.5. ␦c in units of N ␤1 . perradiant system. PRA 58 NON-MARKOVIAN QUANTUM FLUCTUATIONS AND... 4179

FIG. 13. Atomic polarization distribution for a system of 100 atoms at an isotropic band edge (␦cϭ0), subject to quantum fluctuations PBG 2/3 at early times. 5000 realizations of the superradiant system. ͑a͒ tϭt0 ; ͑b͒ tϭ5; ͑c͒ tϭ11; ͑d͒ steady state. t in units of 1/N ␤1 . tained in free space ͓33͔. However, the width of the distri- random phase. This behavior is reminiscent of the fluctua- bution scales with the relevant time scale for the isotropic tions of the order parameter in the vicinity of a phase and anisotropic gaps, showing that, near a photonic band transition. In the steady state, the polarization amplitude col- edge, atomic population fluctuations during light emission lapses to a very well-defined nonzero value. This amplitude can be reduced from their free space value. Because of the is again accompanied by a random phase that is uniformly variation in initial conditions, the Rabi oscillations in ͗J3(t)͘ distributed between 0 and 2␲. We may interpret our steady- for the isotropic gap are much less pronounced than in mean- state result in the following manner: A fraction of the field simulations. The differences in emission times due to photons emitted near the photonic band edge remain local- fluctuations cause the ensemble average inversion to smear ized in the vicinity of the atoms, causing both the atomic out these oscillations. Therefore, one can no longer directly dipoles and the electromagnetic field to self-organize into a relate the amplitude and period of the oscillations to the en- cooperative steady state. However, vacuum fluctuations ergies of the collective dressed states. cause this cooperative quantum state to have a random More striking is the nature of the ensemble’s collective phase, resulting in a zero ensemble average polarization am- polarization under the influence of vacuum fluctuations. Fig- plitude, ͉͗J12(t)͘ens͉ϭ0, as shown in Fig. 11. Measurements ures 13͑a͒–13͑d͒ show the evolution of the polarization dis- of the degree of first- and second-order coherence of the tribution from the initial distribution given by Eqs. ͑5.17͒ electromagnetic field in a band-edge superradiance experi- and ͑5.19͒ to the steady-state distribution. Initially, the dis- ment would provide a probe of the nature of this self- tribution is sharply peaked about zero. In the decay region, organized state of photons and atoms near a band edge. We the polarization amplitude is broadly distributed and has a further note that this state—well defined in amplitude but 4180 NIPUN VATS AND SAJEEV JOHN PRA 58 with random phase—is similar to the steady state of a con- where again ␣ϭ1 and 3 for isotropic and anisotropic band ventional laser ͓34͔ with a well-defined electric field ampli- edges, respectively. Problems in band-edge atom-field dy- tude and random phase diffusion. namics, such as the present superradiant problem, often in- volve nonlinear equations under the influence of colored VI. SIMULATED QUANTUM NOISE NEAR A BAND quantum noise. It is interesting to note that nonlinear prob- EDGE lems involving classical colored noise are of considerable interest in classical statistical physics ͓35͔. In what follows, We have shown that the statistical properties of a band- edge superradiant system can be determined because the col- we use the method first introduced by Rice ͓36͔ and elabo- lective behavior of the constituent atoms leads to a semiclas- rated on by Billah and Shinozuka ͓37͔ in order to generate sical system evolution, triggered by early-time quantum colored noise satisfying Eq. ͑6.1͒. For noise with a power fluctuations. However, a seamless quantum description of spectrum P(␻), defined as the Fourier transform of the au- band-edge quantum optical systems is extremely difficult to tocorrelation function ͗␰(t)␰(tЈ)͘, their algorithm gives 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 include the N 1/2 effects of quantum fluctuations. Unlike the semiclassical ␰͑t͒Ӎ2 ͓P͑␻n͒⌬␻͔ cos͑␻ntϩ⌽n͒, nϭ1,2, ...,N, n͚ϭ1 simulations of Sec. IV, which neglected the effect of the ͑6.2͒ quantum noise operator, as ͗␩(t)͘ϭ͗␩†(t)͘ϭ0, we propose to replace ͗␩(t)͘ in our semiclassical equations by a complex classical stochastic function with the same mean and with equality obtained for N ϱ. Here, ␻ ϭn⌬␻, ⌬␻ two-time correlation function as its quantum counterpart. n ϭ␻ /N, and ␻ is a cutoff→ frequency above which the This noise function then simulates the quantum noise in our max max power spectrum can be neglected. Each ⌽ is a random system throughout the entire system evolution. We may test n phase uniformly distributed in the range 0,2 . By use of a the validity of our simulated noise ansatz for band-edge su- ͓ ␲͔ perradiance by comparing the results obtained to those cal- particular set of random phases ͕⌽n͖ to generate the noise culated in Sec. V. values at each time step, we obtain a single ‘‘experimental’’ The classical noise function required to simulate quantum realization of the quantum noise in our system. Since we noise near a photonic band edge involves a real stochastic cannot predict a priori the specific form of the quantum fluc- function ␰(t) possessing the underlying temporal autocorre- tuations in a particular experiment, we again average over lation of our non-Markovian quantum noise operator, ␩(t). many realizations of the superradiant system, each governed In the effective-mass approximation, this means that ͓see by a different ␰(t), in order to obtain distributions and en- Eqs. ͑2.10͒ and ͑2.11͔͒, semble averages of relevant quantities. We note that Eq. ͑6.2͒ clearly gives ͗␰(t)͘ensϭ0, as desired, since the random 1 ⌽ cause the ensemble to average to zero. To show that Eq. ͗␰͑t͒␰͑tЈ͒͘ϭ , ͑6.1͒ n ͑tϪtЈ͒␣/2 ͑6.2͒ also gives the correct autocorrelation function, we write

1/2 ͗␰͑t͒␰͑tЈ͒͘ensϭ 4⌬␻͚ ͚ ͓P͑␻k͒P͑␻l͔͒ cos͑␻ktϩ⌽k͒cos͑␻ltЈϩ⌽l͒ ͳ k l ʹ ens

1/2 ϭ 2⌬␻͚ ͚ ͓P͑␻k͒P͑␻l͔͒ ͕cos͑␻ktϪ␻ltЈϩ⌽kϪ⌽l͒ϩcos͑␻ktϩ␻ltЈϩ⌽kϩ⌽l͖͒ ͑6.3͒ ͳ k l ʹ ens

ϭ 2⌬␻͚ P͑␻k͒cos͓␻k͑tϪtЈ͔͒ . ͑6.4͒ ͳ k ʹ ens

In Eq. ͑6.3͒, only the kϭl components, in which the random tion of phases ⌽k as input, rather than requiring the phases ⌽k and ⌽l cancel each other, survive the ensemble computation of a Gaussian stochastic function as an interme- average. As N ϱ, Eq. ͑6.4͒ becomes the Fourier transform diate step. This decreases the likelihood of spurious correla- → of P(␻), which equals ͗␰(t)␰(tЈ)͘ens . Studies have shown tions between our random numbers. Figure 14 shows the that for values of N as small as 1000, the desired autocorre- two-time correlation of ␰(t) for ␣ϭ1, for an ensemble of lation may be obtained with as little as 5% error ͓37͔, mak- 2000 realizations of the noise function generated by the al- ing this a computationally feasible technique. Furthermore, gorithm of Eq. ͑6.2͒. In this calculation and in the simula- unlike other methods for the generation of stochastic functions described below, we chose a power spectrum P(␻) tions ͑see Ref. ͓35͔͒, the present method computes the de- ϭͱ␲/2␻, in order to mimic the colored vacuum near an sired function, ␰(t), using only a uniform random distribu- isotropic band edge. We see good agreement with the corre- PRA 58 NON-MARKOVIAN QUANTUM FLUCTUATIONS AND... 4181

FIG. 15. Comparison of the ensemble averaged atomic inver- FIG. 14. Solid line: ensemble-averaged autocorrelation function, sion, J (t) /N, at an isotropic band edge (␦ ϭ0) as calculated ͗␰(␶)␰(␶Ј)͘ , of the classical colored noise function ␰͑␶͒ corre- ͗ 3 ͘ens c ens by the methods of Secs. V and VI. 2000 realizations of the super- sponding to vacuum fluctuations near an isotropic band edge. The radiant system. Dashed line and double dashed line, inversion cal- dashed line is a plot of the exact autocorrelation function in the culated using the computed polarization distribution at tϭtPBG as effective-mass approximation, (␶Ϫ␶Ј)Ϫ1/2. 0 initial conditions for a semiclassical evolution ͑Sec. V͒ for N ϭ1000 and 10 000 atoms, respectively. Solid line and dotted lines, lation function ͑6.1͒. The agreement between our simulations inversion calculated using the stochastic function of Sec. VI for N and the exact correlation function can be significantly im- ϭ1000 and 10 000 atoms, respectively. proved by enlarging the size of the ensemble, at the expense of increased computation time for atom-field simulations. averaged polarization and the delay time distribution calcu- The ensemble ␰(t) is used to simulate the effect of ͕ ͖ lated by the present method also agree well with the quantum vacuum fluctuations in Eqs. ͑2.5͒ and ͑2.6͒. Written in terms calculations of the previous section. This suggests that our of the dimensionless time variable ␶ϭN2/3␤ t, these equa- 1 stochastic approach may be a valuable tool in the analysis of tions for the isotropic band edge become band-edge atom-field dynamics. d ͗J ͑␶͒͘ d␶ 3 VII. CONCLUSIONS In this paper, we have treated the collective spontaneous

͗J21͑␶͒͘ ␶ ͗J21͑␶Ј͒͘ emission of two-level atoms near a photonic band edge. An ϭϪ4Re eϪi␲/4 ͵ ei␦c͑␶Ϫ␶Ј͒d␶Ј analytic calculation of the atomic operator dynamics in the ͭ ͱ␲ 0 ͱ␶Ϫ␶Ј case of low atomic excitation was given. The results demon- strate highly atypical atomic emission spectra and show the Ϫi͑␲/8Ϫ␦c␶͒ ͗J21͑␶͒͘e possibility of reducing atomic population fluctuations. This ϩ ␰͑␶͒ , ͑6.5͒ in turn suggests that fluctuations in photon number are like- ͱNͱ␲ ͮ wise suppressed for light localized near the atoms. This raises the interesting question of whether squeezed light ͓38͔, antibunched photons 39 , and other forms of nonclassical d ͗J3͑␶͒͘ ␶ ͗J21͑␶Ј͒͘ ͓ ͔ Ϫi␲/4 i␦c͑␶Ϫ␶Ј͒ ͗J12͑␶͒͘ϭe e d␶Ј light may be generated in a simple manner from band edge d␶ ͱ␲ ͵0 ͱ␶Ϫ␶Ј atom-field systems. For an initially inverted system prepared with a small macroscopic polarization, a mean field factor- Ϫi͑␲/8Ϫ␦c␶͒ ͗J3͑␶͒͘e ization was applied to the atomic quantum Langevin equa- ϩ ␰͑␶͒, ͑6.6͒ tions, giving a semiclassical system evolution. We found that ͱNͱ␲ the atoms exhibit fractional population trapping and a macroscopic polarization in the steady state. Collective Rabi os- with similar equations for the anisotropic gap. For both mod- cillations of the atomic population were found, and were els, the noise term scales as 1/ͱN; this is the same depen- attributed to the interference of strongly dressed atom-photon dence of the noise term on particle number exhibited in free states that are repelled from the band edge, both into and out space ͓32͔. In Fig. 15, we show the average inversion for an of the gap. The degree of photon localization, the polariza- ensemble containing 2000 realizations of ␰͑␶͒ for Nϭ1000 tion amplitude, and the phase angle of the polarization in the and Nϭ10 000 atoms. We find that our stochastic simulation steady state are all sensitive functions of the detuning of the scheme gives physical results only for systems of N atomic resonant frequency from the band edge. The steady- Ͼ500 atoms. The stochastic simulations show good agree- state atomic properties can thus be controlled by applying a ment with the atomic inversion obtained by the method of dc Stark shift to the atomic resonant frequency. In Sec. IV, Sec. V. Other system properties, such as the ensemble- we discussed the effect on band-edge superradiance of inter- 4182 NIPUN VATS AND SAJEEV JOHN PRA 58 atomic and atom-phonon interactions ͑atomic dephasing͒. states encountered at a photonic band edge, nor are they all We showed that such linewidth broadening effects cannot be directly applicable to externally driven atomic systems. treated by a phenomenological decay constant as in free Finally, we note that the steady state atom-field properties space, and that near the band edge they will lead to the decay described here are a result of the effect of radiation localized of atomic polarization and inversion. Therefore, the steady- in the vicinity of the active two-level atoms. This leads to the state properties of the superradiant system described in this question of how to pump energy into and extract energy out paper will be limited by the time scale of the atomic dephas- of these states, which lie within the forbidden photonic gap. ing effects. The effect of dephasing mechanisms is important One possibility is to couple energy into and/or out of the to the description of almost all band edge atom-field systems. system through a third atomic level whose transition energy As a particular example, dephasing determines the threshold lies outside the gap ͓15͔. There is also the possibility of external pumping required to achieve atomic inversion in a transmitting light into the gap through high intensity ul- laser operating near a photonic band edge. It also facilitates trashort pulses that locally distort the nonlinear dielectric the emission of laser light from a photonic crystal. constant of the material and thus allow the propagation of The effect of quantum fluctuations for high initial excita- light in the form of solitary waves within the forbidden fre- tion of the atoms was included by distinguishing regimes of quency range ͓44͔. Such issues must be addressed in order to quantum and semiclassical collective atomic evolution. We fully exploit the very rich possibilities of quantum optical found that the early time quantum evolution must be treated processes near a photonic band edge. in detail, due to the non-Markovian electromagnetic reservoir correlations near a band edge. This is in contrast with ACKNOWLEDGMENTS free space, where the atomic system’s evolution is insensitive to the treatment of the full temporal evolution of the We are grateful to Dr. Tran Quang for a number of useful early, quantum regime. Fractional localization of light was discussions. N.V. acknowledges support from the Walter shown to persist under the influence of vacuum fluctuations. Sumner Memorial Foundation. This work was supported in The atomic polarization exhibits a nonzero amplitude with a part by Photonics Research Ontario, the Natural Sciences randomly distributed phase in the steady state. This is much and Engineering Research Council of Canada, and the New like the steady state of a conventional laser. Here, such lasing Energy and Industrial Technology Development Organiza- characteristics are due only to the Bragg scattering of pho- tion ͑NEDO͒ of Japan. tons back to the atoms; there is neither external pumping nor a laser cavity in our system. The time scales for all dynamical processes, such as collective emission and the buildup of APPENDIX A: CALCULATION collective atomic polarization are strongly modified from OF THE MEMORY KERNEL their free space values due to the singular photon density of We first present the calculation of the memory kernel for states near a photonic band edge. For an isotropic band edge, the isotropic model in the effective-mass approximation, the time scales as N2/3␤ , while in the more realistic aniso- 1 GI(tϪtЈ). Starting from Eq. ͑2.8͒ and the isotropic disper- tropic model, time scales as N2␤ . As a result, collective 3 sion relation near the upper band edge, ␻kϭ␻cϩA(͉k͉ emission phenomena can occur more rapidly near a band Ϫ͉k ͉)2, G (tϪtЈ) can be expressed as edge than in free space. Throughout our calculations, we 0 I have employed an effective-mass approximation to the band- 2 2 2 ⌳ ϪiA͑kϪk0͒ ͑tϪtЈ͒ edge dispersion. For materials with a very small PBG, it may ␻21d21 e G ͑tϪtЈ͒ϭ ei␦c͑tϪtЈ͒ d⍀ dk . be important to include the effects of both band edges. These I ͵ ͵ ϩ Ϫ 2 4ប⑀0␲ k0 ␻c A͑k k0͒ issues are raised in Appendix A. ͑A1͒ We have demonstrated that band-edge superradiance possesses many of the self-organization and coherence properties of a conventional laser. Furthermore, we have shown the Again, ␦cϭ␻21Ϫ␻c is the detuning of the atomic resonant possibility for the generation of unusual emission spectra and frequency from the band edge. ⌳ϭmc/ប is a cutoff in the photon statistics. These results suggest that a laser operating photon wave vector above the electron rest mass. Photons of near a photonic band edge may possess unusual spectral and energy higher than the electron rest mass probe the relativ- statistical properties, as well as a low input power lasing istic structure of the electron wave packets of our resonant threshold due to the fractional inversion of the atoms in the atoms ͓45͔. Because the isotropic model associates the band steady state. It may further be possible to produce a PBG edge with a sphere in k space, there is no angular depen- laser in a bulk material without recourse to a defect-induced dence in the expansion of ␻k about the band edge. We may cavity mode. Lending credence to this hypothesis, recent ob- thus separate out the angular integration over solid angle ⍀ servations ͓40͔ and theoretical studies ͓41͔ of lasing from a in Eq. ͑A1͒. We may also make a stationary phase approxi- multiply scattering random medium with gain have demon- mation to the integral, as the nonexponential part of the in- strated that one may obtain light with the properties of a laser tegrand will only contribute significantly to the integral for field in the absence of a cavity. A full description of the kӍk0 . The resulting integral is statistics of a band-edge laser field will require a treatment of the non-Markovian nature of the electromagnetic reservoir. 2 2 ⌳ ␻21d21 k0 2 Current techniques for treating the atom-field interaction in i␦c͑tϪtЈ͒ ϪiA͑kϪk0͒ ͑tϪtЈ͒ GI͑tϪtЈ͒Ӎ e ͵ dke . the absence of the Born and Markov approximations ͓42,43͔ 4ប⑀0␲ ␻c k0 cannot account for the van Hove singularity in the density of ͑A2͒ PRA 58 NON-MARKOVIAN QUANTUM FLUCTUATIONS AND... 4183

As ⌳ is a large number, we extend the range of integration to For an anisotropic band-gap model, we must account for infinity in order to obtain a simple analytic expression for the variation in the magnitude of the band-edge wave vector GI(tϪtЈ) ͓46͔: as k is rotated throughout the Brillouin zone. We associate the gap with a specific point in k space that satisfies the 2 2 Ϫi͓␲/4Ϫ␦c͑tϪtЈ͔͒ ␻21␻cͱ␻gapd21 e Bragg condition, kϭk0 . In the effective-mass approxima- G ͑tϪtЈ͒ϭ I 12ប⑀ ␲3/2c3 tion, the dispersion relation is expanded to second order in k 0 ͱtϪtЈ 2 about this point, ␻kϭ␻cϮA(kϪk0) . Making the substitu- Ϫi ␲/4Ϫ␦ tϪt ͒ ͓ c͑ Ј ͔ tion qϭkϪk0 and performing the angular integration, GA(t 3/2 e ϭ␤1 . ͑A3͒ ϪtЈ) is expressed as ͱtϪtЈ

Because the relevant frequencies in Eq. ͑A3͒ are roughly of 2 2 2 ⌳ 2 ϪiAq ͑tϪtЈ͒ the same order of magnitude near a band edge, we may re- ␻21d21 q e G ͑tϪtЈ͒ϭ ei␦c͑tϪtЈ͒ dq . 3/2 7/2 2 3/2 3 A 2 ͵ ϩ 2 write the prefactor as ␤1 Ӎ␻21 d21/12ប⑀0␲ c , in agree- 4ប⑀0␲ 0 ␻c Aq ment with the value given in Sec. II. We emphasize that the ͑A4͒ stationary phase method yields the correct asymptotic behavior for the memory kernel for large ͉tϪtЈ͉. At short times, the integral must be evaluated more precisely using the full Extending the wave-vector integration to inﬁnity, the Green photon dispersion relation, as discussed below. function is ͓46͔

2 2 ␻21d21 ␲ ␲ ␻c i␦c͑tϪtЈ͒ i␻c͑tϪtЈ͒ GA͑tϪtЈ͒ϭ 2 e ͱ Ϫ ͱ e ͓1Ϫ⌽„ͱi␻c͑tϪtЈ͒…͔ . ͑A5͒ 8ប⑀0␲ ͭ i␻c͑tϪtЈ͒ 2 A ͮ

x Ϫt2 The double-valued nature of ␻ at k is made explicit by the ⌽(x) is the probability integral, ⌽(x)ϭ(2/ͱ␲)͐0e dt. k 0 function sgn(kϪk )ϭ1 for kϾk , and sgn(kϪk )ϭϪ1 for k For ␻c(tϪtЈ)ӷ1, a condition satisfied for all but the tЈ t 0 0 0 15 Ϫ1 → Ͻk . This gives a gap of width ␻ ϭ2␥c, centered about limit ͑as ␻cϳ10 s for optical transitions͒, taking the 0 gap 2 2 asymptotic expansion of ⌽(x) to second order gives the midgap frequency ␻0ϭcͱk0ϩ␥ . Note that Eq. ͑A8͒ gives the correct linear dependence in k for both large posi- 2 2 i͓␲/4ϩ␦c͑tϪtЈ͔͒ ␻21d21 e tive and negative k, and gives the effective-mass dispersion G tϪt ϭ , A͑ Ј͒ 3/2 3/2 for kϳk0 . Like the effective-mass model, Eq. ͑A8͒ gives a 8ប⑀0͑␲A͒ ␻c ͑tϪtЈ͒ singular density of states at the band edges, ␻cϭ␻0Ϯc␥. ␻c͑tϪtЈ͒ӷ1. ͑A6͒ The full dispersion relation allows us to evaluate the influence of both band edges for arbitrary gap width and atomic As tϪtЈ 0ϩ , Eq. ͑A5͒ reduces to resonant frequency. Preliminary numerical calculations show → a stronger reservoir memory effect than demonstrated in the 2 2 effective mass model for the isotropic band edge. This may ␻21d21 ␲ G ͑tϪtЈ͒ϭ Ϫ␲ͱ␻ , have a significant effect on theoretical predictions regarding A 8ប⑀ ␲2A3/2 ͱi␻ ͑tϪtЈ͒ c 0 ͫ c ͬ the atom-field interaction in the vicinity of a PBG. A further tϪtЈ 0 . ͑A7͒ simplification has been made in the anisotropic model, in → ϩ that we have not included the dependence of ␻k on the symmetry of a specific photonic crystal. In a real three- GA(tϪtЈ) possesses a weak ͑square root͒ singularity at t ϭtЈ. This is an integrable singularity and can thus be treated dimensional PBG material, the Bragg condition is satisfied numerically ͓47͔. for different values of k as the wave vector changes direction The effective-mass dispersion relation used in the evalu- in k space. This directional dependence may lead to a much ation of G(tϪtЈ) is, strictly speaking, valid only near the stronger dependence of the localization of light on the detun- photonic band edge, as it fails to give the required linear ing of ␻21 away from the band edge. The impact on the atom-field interaction in a PBG of both the full isotropic photon dispersion relation for large ͉kϪk0͉ ͑far away from the gap͒. Therefore, the integration of the effective-mass dis- dispersion model and more realistic dispersions for three- persion for large wave vector in Eqs. ͑A1͒ and ͑A4͒ intro- dimensional photonic crystals will be treated elsewhere. duces a spurious contribution to G(tϪtЈ). This difficulty may be overcome for an isotropic gap model by using a 2 dispersion relation that has the correct wave-vector depen- APPENDIX B: EVALUATION OF ͚␭ͦA␭„t…ͦ dence for all k. The simplest model dispersion with the cor- 2 We outline the evaluation of ͚␭͉A␭(t)͉ , used to obtain rect form is the low excitation population fluctuations in Sec. III, Eq. ͑3.19͒. A similar procedure is used to arrive at Eq. ͑5.12͒ in ␻ /cϭ k2ϩ␥2ϩsgn͑kϪk ͒ͱ͑kϪk ͒2ϩ␥2. ͑A8͒ ˜ k ͱ 0 0 0 Sec. V. Starting from the Laplace transform A␭(s) ͓Eq. 4184 NIPUN VATS AND SAJEEV JOHN PRA 58

͑3.7͔͒, we may use the properties of a convolution of Laplace t tЉ transforms in order to write I2ϭ dtЉ dtЈB͑tЉ͒B*͑tЈ͒G͑tЈϪtЉ͒ϭI1*. ͑B4͒ ͵0 ͵0 t A ͑t͒ϭg dtЈB͑tЈ͒eϪi⌬␭t. ͑B1͒ ␭ ␭ ͵ 2 0 Therefore, ͚␭͉A␭(t)͉ ϭ2Re͕I1͖, and we need only explic- ˜ itly evaluate I1 . The Laplace transform of B(t), B(s) ͓Eq. Therefore, we have ͑3.6͔͒, is equivalent to the Laplace transform of the equation t t A ͑t͒ 2ϭ dtЈ dtЉB͑tЉ͒B*͑tЈ͒G͑tЈϪtЉ͒, ͚ ͉ ␭ ͉ ͵ ͵ d t ␭ 0 0 B͑t͒ϭϪN dtЈG͑tϪtЈ͒B͑tЈ͒. ͑B5͒ ͑B2͒ dt ͵0 with G(tϪtЈ) deﬁned as in Eq. ͑2.7͒. We may rewrite this double integral in the form Substituting Eq. ͑B5͒ into I1 and its complex conjugate, we obtain t tЈ 2 * ͚ ͉A␭͑t͉͒ ϭ ͵ dtЈ͵ dtЉB͑tЉ͒B ͑tЈ͒G͑tЈϪtЉ͒ ␭ 0 0 1 t d 2 t t 2I1ϭϪ dtЈ ͉B͑tЈ͉͒ N ͵0 dtЈ ϩ͵ dtЈ͵ dtЉB͑tЉ͒B*͑tЈ͒G͑tЈϪtЉ͒ 0 tЈ 1 2 2 2 ϭ ͓͉B͑0͉͒ Ϫ͉B͑t͉͒ ͔ϭ ͉A␭͑t͉͒ , ͑B6͒ ϭI1ϩI2 , ͑B3͒ N ͚␭ where I1 and I2 are the ﬁrst and second double integrals, 2 respectively. By changing the order of the integrations in I2 , as I1 is real. Substituting the initial condition ͉B(0)͉ ϭ1 we obtain into Eq. ͑B6͒ gives the result quoted in Sec. III.

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