Spontaneous and induced atomic decay in photonic

JOURNAL OF MODERN OPTICS,

1994,

VOL .

41,

NO .

2, 3 5 3 -384

Spontaneous and induced atomic decay in photonic band structures A . G . KOFMAN, G . KURIZKI and B . SHERMAN Chemical Physics Department, Weizmann Institute of Science, Rehovot 76100, Israel (Received 21 May 1993 and accepted 2 July 1993)

Abstract . We present a comprehensive quantum electrodynamical analysis of the interaction between a continuum with photonic band gaps (PBGs) or frequency cut-off and an excited two-level atom, which can be either `bare' or `dressed' by coupling to a near-resonant field mode . A diversity of novel features in the atom and field dynamics is shown to arise from the non-Markovian character of radiative decay into such a continuum of modes . Firstly the excited atom is shown to evolve, by spontaneous decay, into a superposition of nondecaying single-photon dressed states, each having an energy in a different PBG, and a decaying component . This superposition is determined by the atomic resonance shift, induced by the spontaneously emitted photon, into or out of a PBG . The main novel feature exhibited by the decaying excited-state component is the occurrence of beats between the shifted atomic resonance frequency and the PBG cut-off frequencies, corresponding to a non-Lorentzian emission spectrum . Secondly the induced decay of a resonantly driven atom into such a continuum exhibits a cascade of transitions down the ladder of dressed states, which are labelled by decreasing photon numbers of the driving mode . Remarkably, this cascade is terminated at the dressed-state doublet, from which all subsequent transitions to lower doublets are forbidden because they fall within the PBG . This doublet then becomes an attractor state for the populations of higher-lying doublets . As a result, the photon-number distribution of the driving mode becomes strongly sub-Poissonian .

1.

Introduction Spectacular successes have been achieved in the study of quantum electrodynamic (QED) features of radiation processes in resonators, known as cavity QED, in which an atom interacts with an electromagnetic density of modes (DOM) that is spectrally modified ('coloured') compared with open space [1-3] . The advent of twoor three-dimensionally periodic dielectric structures that exhibit photonic band gaps (PBGs) [4-9] and incorporate defects, designed to form localized narrow-linewidth modes at PBG frequencies [4,10-12], opens up new perspectives in cavity QED . Such structures are expected to allow better control over the DOM spectral distribution and the spatial modulation of narrow-linewidth (high-Q) modes, in both microwave and optical domains . In cavities as well as in PBG structures, two distinct QED regimes are realizable . (a) In the regime of a single high-Q mode near resonance with an atomic

transition, reversible energy exchange is obtainable between the atom and the field . In the simple case of an atom travelling in a spatially uniform high-Q mode, this exchange is described by the Jaynes-Cummings model (JCM) [13-18] . Atomic motion through a periodically modulated high-Q mode, in a 0950-0340/94 $1000 ©C) 1994 Taylor & Francis Ltd .



354

A . G . Kofman et al . Fabry-Perot resonator or a defect at a PBG frequency, has been predicted to modify the JCM dynamics drastically, and thereby to allow better control over the generation of sub-Poissonian photon statistics in such a mode [19-21] . (b) In the regime characterized by resonant coupling of the atomic transition with a continuum of modes (an allowed photonic band), spontaneous as well as induced (resonance) fluorescence processes have been shown to exhibit new features because of vacuum `colouring' : suppression or enhancement of spontaneous emission rates [1, 2, 22-27], non-Lorentzian natural lines shapes [28], and elimination of both Mollow-triplet side bands in the classical (intense-pump) limit of resonance fluorescence when one of the side bands is within a PBG [29] .

In this paper we present a comprehensive QED analysis of the interaction between a mode continuum in a photonic band structure and an excited two-level atom (or an exciton, bearing in mind the differences [30, 31]), which can be either `bare' or `dressed' by coupling to a near-resonant field mode . A diversity of novel features in the atom and field dynamics is shown to arise when a bare or dressed atomic resonance is close to a PBG edge, that is the cut-off frequency of an allowed photonic band . These features stem from the non-Markovian character of radiative decay into a continuum of modes with an abrupt non-analytic cut-off . Two types of process are analysed here for atomic transitions within or near PBGs . (1) Spontaneous decay is investigated in section 2 and appendix A . In contrast with previous treatments, we explicitly analyse the time- and frequencydomain responses of an initially excited atom whose resonance lies either outside or inside a PBG, first for an arbitrary band structure and then for a model DOM obtained by the effective-mass approximation [28, 32, 33] . Earlier treatments of an excited two-level system coupled to a `coloured' reservoir included firstly numerical calculations of non-Lorentzian emission spectra, which may exhibit unusual double-peaked line shapes [28] and secondly prediction of the possibility of forming a discrete `dressed' state, whose energy lies outside the continuum and is stable against decay [34] . The latter effect is completely analogous to the stabilization of a discrete level against photoionization in an atom dressed by a strong field [35] . In the present case it can be interpreted as a shift of the atomic level by the emitted photon (an effect well known in cavities [1, 36, 37]) that pushes the level beyond cut-off . It has also been referred to as an `anomalous Lamb shift' [33, 36, 37] . Here the excited atom is shown to evolve, in general, into a superposition of a discrete, stable state with energy in a PBG (or several such states, each in a different PBG) and a decaying excited-state component . Analytic approximations are obtained for the non-Markovian timedependent decay and the corresponding non-Lorentzian emission spectrum . The main novel feature exhibited by the excited-state decay is the occurrence of beats between the reservoir-shifted atomic resonance frequency, the PBG cutoff frequencies, and/or the discrete dressed-state frequency . The previously calculated double-peaked spectrum [28] is shown to result from broadening of the discrete dressed state by residual DOM within the PBG, which is controllable by defects implanted in the structure .



Spontaneous and induced decay

355

(2) In section 3 we study the induced decay [38] of an atom with resonance frequency within a PBG, which is driven by a high-Q near-resonant field mode (e .g . a defect mode within the PBG) . As opposed to previous semiclassical treatments of the driving (pump) mode [29], we are concerned here with the evolution of its photon statistics, which is assumed to be initially broad (Poissonian or super-Poissonian, e .g . thermal) . We derive (appendix B) a non-Markovian master equation for the dressed-state populations of the atom-driving mode system, dissipated by the DOM reservoir. Analytical solutions of these equations describe the cascade of transitions down the ladder of dressed states labelled by decreasing photon numbers . Remarkably, this cascade is terminated at the dressed-state doublet associated with a photon number n o , such that all subsequent transitions to lower doublets (with n < n o ) are forbidden because they fall within the PBG . The n o doublet then becomes an attractor state for dressed-state populations, into which all higher-lying doublets (with n > no ) eventually decay . Consequently, the photon-number distribution of the driving mode becomes strongly sub-Poissonian with a narrow peak at n o . In section 4 we consider possible implications and realizations of the discussed effects . We also outline directions for further research in this area . 2. Spontaneous decay in a photonic band gap structure 2.1 . Equations of motion for the field and atom amplitudes The system Hamiltonian assumes the following form in the rotating-wave approximation (RWA) [39] : H=hwa le> and lg> are the excited and ground atomic states respectively, wa is the atomic transition frequency, at and a k are the creation and annihilation operators of a field mode with wave-vector k and frequency w k, and hK(w k) is the resonant coupling energy of this mode with the atomic dipole . The k summation will be replaced by an integral over the continuum DOM in the frequency domain,

Jo

dw p(w),

(2)

avoiding, for simplicity, the study of direction-(angle-)dependent DOM effects (cf. section 2 .4) . We consider an excited atom that is injected at t=0 into an empty periodic dielectric structure, so that the inital wavefunction of the system is IP(O)>=le, { 0 .}>

(3)

where {Ow} signifies the totality of field modes in the vacuum state . The subsequent evolution of the wavefunction has in the RWA the general form I'F(t)>=a(t)Ie, {Ow}>+ Jfl(t)I w g, 1 .>p(w)dw,

(4)



356

A . G. Kofman et al.

where Il,„ > denotes single-photon occupation of the co mode . The corresponding Schrodinger equation yields the following coupled equations [27, 28] :

a= -iwaa-1

~ fw (t)K(w)p(w)

dw,

(5 a)

J 0

-iwf W-ix*((O)a .

/u,=

(5 b)

These equations can be solved with the initial condition (3), that is a(0) = 1, # w (0) = 0, in the following Laplace-domain form : 1(s) _ [s+ iw a +J(s)] -1 ,

(6 a)

ix*(w)c4(s) s+iw

(6 b)

G((o) s+ico

(7)

G(w)=1 K(O)) I 2 p(w) .

(8)

where J(s) =

dco

J o"0 is defined by means of the reservoir spectral response

The ensuing analysis of equations (6) and (7), with G((o) appropriate for photonic band structures, is aimed at revealing the prominent features of the atomic decay a(t) and the corresponding emission spectrum . 2.2 . Incomplete decay and the photon-bound state In what follows we consider a photonic band structure with several PBGs, separated by allowed bands . Each PBG is labelled by index j and has lower and upper cut-off frequencies WLj and 0U j respectively . Then G((o)=0

for

W Li 0 . The opposite limit of large c, which models the case D=0, is briefly considered below . The Laplace transform of the excited-state amplitude is obtained in this model on substituting equation (45 b) into equation (6 a) and integrating over w, which yields (section A .2) Q(S)= S+lwa +

C

1

if 1/2 +(iS_WU) 1/2

(46)

The spectrum of the emitted light is obtained from equation (25), with (cf. equations (A 2) and (A 6)) C(w-wu) 11z

(47 a)

Y(w)= Co -wu+E 8(w-w u )

and dw)

_

E 1 / 2 /(w wu +e), -C [(wu-w)112+E1i2]-1

w>wu , w0 of at least one of the

Laplace-transformed decay rates Ri,n- j,,,-1(s) in the denominator of equation (83) is non-vanishing . Since Mji OO, this implies (equations (77) and (79)) that lim[fi,n-j,n-1(s)]aclim[ i(s,Sij,n-1)]= s-o S-0 be non-zero . Then Pi , n (t-> 00)

=

lim [spin(s)] = 0,

7CC( 3 'ij,n-1)

(84)

n > no ,

(85)

S-0

which signifies eventual complete decay of the initial population at the dressed states I i, n> . (b) As soon as G($;j,n _ 1) = 0, since oij,n _ 1 enters the PBG for all i, j and n < n o (figure 6 (b)), the vanishing of all decay rates in equation (84) yields the extraordinary result Pi .n(t-oo)=Pi,n(t=0)+limZ Pj,n+1(s)Rj,n+1-i,ns)) s-0 (= 1,2 Pi.n(t->oo)=Pi,n(t=0), n or Jg> state, we project out the In o> or the In o + 1 > Fock-state contributions to the doublet (71) respectively . This can lead to the reduction in the defect-mode field to nearly a Fock state, if the lower-lying doublets with n < n o correspond to photon numbers at the far tail of the initial (e.g. Poissonian) distribution (figure 6 (c)) . 4.

Discussion The present QED treatment has revealed a variety of unusual properties, some of them hitherto unknown, exhibited by spontaneous and stimulated atomic decay in photonic band structures . Our general analysis is based on the use of the directionally-averaged spectral response G(w) of a `coloured' DOM distribution with PBGs (equation (9)) . This directional averaging, which has been used in all QED studies of photonic band structures thus far [28, 32, 33], is expected to be adequate near wide PBGs in three-dimensionally periodic structures (interacting with arbitrarily oriented dipoles) or in two-dimensionally periodic structures



Spontaneous and induced decay

377

(interacting with dipoles oriented perpendicular to the periodicity plane) . Nevertheless, the adaptation of the present treatment to specific photonic structures requires detailed calculations involving the angular averaging of G(O),) = IK(wk)12p((ok) which allows for the orientation dependence of the dipole, the DOM p( (O k ), and the polarization . The resulting angle-averaged response function should then be compared with the standard [28, 33] model of section 2 .4 . Such calculations are warranted by the intriguing implications of the results presented here . (a) The dependence of the resonance frequency shifts, into or out of a PBG, on

the structure parameters and wa has been revealed (equations (11) and (12)) . The important finding that PBGs can be too narrow to block spontaneous emission for any wa (equations (15)) imposes restriction on the design of structures intended for the suppression of electron-hole radiative recombination or excitonic decay [4-9, 30] . On the other hand, the possibility of incomplete decay for co a either inside or outside PBGs may allow extra flexibility in the matching of the structure with the emission band that is to be suppressed . (b) The evolution of an initially excited atom into a superposition of states, which includes one or several discrete photon-dressed states (equations (16)(18)) provides a novel scheme for generation of atomic coherences by spontaneous decay . The following potential applications of this scheme can be the subject of further studies : firstly it should be possible to perform quantum non-demolition (QND) measurements of photon-number distributions in defect modes, by detecting the dispersive phase shifts of the atomic dressed-state coherences, in complete analogy with the Ramseyfringes QND measurements suggested thus far [48] ; secondly the generation of atomic coherences by spontaneous decay may serve as a basis for novel schemes of lasing without inversion [49] or correlated emission lasers [50] in photonic band structures . (c) The spectral and time-resolved features of nearly-exponential decay with beats deduced in section 2 .3, namely the generalized form of WeisskopfWigner solutions (equations (38) and (41)) and the criteria for their validity (equations (36)) are of principal theoretical importance . The ability to control the emission line shape and time dependence by the detuning of CU a from cut-off (illustrated by equations (54)-(62) and figures 2 and 3 for a model DOM) may have applications, in the design of radiation sources as well as in `diagnostics' of the photonic band structure . (d) The ability of induced (dressed-atom) decay to convert the initial Poissonian or super-Poissonian pump photon statistics into a strongly sub-Poissonian distribution (section 3) may prove to be the most important practical application discussed here . The formalism used here for stimulated decay may be further extended to account for the competing reservoir-induced spontaneous shifts of equation (12) . Such shifts can affect the location of the photon-number peak in figure 6, by one photon at most .

Acknowledgment This work has been supported by a grant from the Minerva Foundation, Germany .



A . G . Kofman et al.

378

Appendix A : Calculation of the excited-state amplitude in the Laplace and time domains A .1 . Analytical continuation of a(s)

According to the theory of the Laplace transform [44, 45], the finiteness of I z(t)I at t-> co implies that i(s) (equation (6 a)) is an analytic function of s for Re s > 0, if the integral (7) is convergent, that is if G(w) has at most weak singularities at finite co and vanishes for w->oo . Hence, in this case, the inverse Laplace transform is [45] a(t) =

°+i°° 1 6i(s) exp (st) ds 2xi f a - j .

(o > 0) .

(A 1)

The use of contour integration theorems A similar equation connects fl (t) with for the calculation of equation (Al) requires the analytic continuation of c4(s), and thus of J(s) (equations (6 a) and (7)), into the region Re s 0 . (b) Integration contour for equation (A 12) . The cut is shown by the horizontal bold line .

since [45] yf(y)-+O for IyI-+oo . The integral over C 1 tends to J(s) with increase in the size of C1 . The right-hand side of equation (A 5) follows from the residue theorem [45] . Hence

(ip) -

1/2 1 0/2 J(s) = C ip+E

C (lp)

1/2

(A 6)

+ ic 1/2 .

Inserting equation (A6) into equation (6a) yields equation (46) .

A .3 . Calculation of a(t) for the model density of modes with cut-off The inverse Laplace transform of 2(s) (equation (46)) can be obtained by two methods, as follows . First, we rewrite equation (46) as a ratio of two polynomials in 1/2 : v=(s+lwu) &(s) (v

ic 1 J 2 + exp ('i1) v 2 + iz1 )(ic 1 / 2

+ exp

(411[) v)

P 1 (v) + C

(A 7)

P3(v)

The third-order polynomial in the denominator of this expression becomes proportional to the polynomial in equation (50) upon making the substitution



A . G . Kofman et al.

380

v = exp(4in)u . Therefore its roots v j =exp(4i7t)uj are expressed by up the roots of equation (50) . The expression (A 7) can be expanded in elementary fractions :

dj

3

&(s) i

(s+ico) 112 -exp (ain) uj

(A 8)

where [45] P1(v,)

dj _ P,3 (

.

(A 9)

v, )

Bringing the fractions in equation (A 8) to a common denominator, we obtain a ratio of polynomials in v . Noting that the polynomial in the numerator should be of first order (cf. equation (A 7)) yields the relation

d 1 +d2 +d 3 =0 .

(A10)

Since equation (A 8) is reducible to a tabulated Laplace transform [51], one obtains with the help of equations (A 7), (A 9) and (Al 0), 3 u' (u~ + u a(t )=ex P (-iwut ) J=1 ex p erfc [-exP (4lilt ) u •t 112 ]> 3uj +2E 1 ZUj+A~ P ( , )

( A11 )

where erfc [ ] is the complementary error function [51] . This equation is identical with equation (19) from [35] with the accuracy to the sign of the error function argument . A more explicit expression can be obtained in the following way . Consider the integral

I= 271 1 f Cd(s) exp (st) ds,

(A 12)

where the contour C = C1 + . . . + C4 is shown in figure A 1 (b) . The integrals over the contours C 3 and C4 can be shown to vanish when the radii of C 3 and C4 tend to zero and infinity respectively . The integral over C 1 in this limit is a(t) (cf. equation (A 1)) . Therefore a(t) =I-Ic ,

(A 13)

where Ic is the integral over C 2 . The integral (A 12) equals the sum of residues of &(s) for the poles of I(s) which reside on the Riemann sheet defined by the cut shown in figure A 1 (b), that is for - 7E < arg (s + iw o) < it . This yields the first term on the righthand side of equation (A 17) . To calculate the integral Ic, we note that equation (A 7) can be rewritten as I(S) -

exp(Iiit)p 1 / 2 +is 112 _ Ap 112 +B eXp(4in)(p+iA c )p 1"2 +iE112 (p+iA j +C C(p)p112+D(p) .

(A14)

Multiplying the numerator and the denominator of the latter fraction by C(p)p 1 / 2 -D(p) and multiplying equation (A 14) by exp(st) yields I(s) exp (st)=p112BC(P) AD(p) exp [(p - icou)t] +g(p)=p1J2f(p)+g(p), (A 15) pC (p) - D (p) where f(p) and g(p) have no branching points as functions of p . The explicit form of g(p) is not shown here, since the integration of a function with no branching points



3 81

Spontaneous and induced decay

over the contour C 2 yields zero . Taking into account the fact that, at the upper edge of the cut, argp=7c while, at the lower edge, argp= -it, one obtains

]c = 2xi

exP (1'7C) ( p) 1

1 - exp(zlx) 2t[i

=1 f R

°°

exp (- zix) ( - p) 112f(p) dp)

2f(p) dp +

o e l /2f(C )dC

J

-exp(-zits) J

~yl 2f( S 0

-C )

d~)

dC .

4 f( - ~)

(A 16)

o

Combining equations (A 13)-(A 16) yields [~, 2u (u +c hI2) a(t)=exp(-iwut) exp(iujt)

3uj+2a112uj+dr

C l1z exp ( - fit) d~ +exp 4t[i i(24 . -s)~2+[2s(d~-y~)-d~]~y .)2 3 / C J °y3_ "0

C

(A 17)

where y, =C/E112 and the prime near the sum symbol means that the summation involves only those roots u j of equation (50) which satisfy the condition -4tt < arg uj < 41t .

(A 18)

For sufficiently long times, only very small contribute to the integral . Then one can neglect all the terms in denominator of the integrand, except for the term of the lowest order in 4, so that the long-time behaviour of a(t) is exp[1(atc-wet)] a(t)=c°exp(-iwot)- y` -yc)2(.mct 3)112 2(dc

(d

C

=A Y~)

(A19a )

or

[ (a-w u t)]

\

,

exp 1

1/2 (d . =y~) .

xt

Yc

(A19b)

Appendix B: Non-Markovian master equation for dressed-state populations In this appendix we derive the rate equation for dressed-states populations (69) and (76) without resorting to the Markovian approximation . Starting from the Liouville equation, which is governed by the Hamiltonian H=H ° +H 1 (equations (67) and (68)), the standard projection operator technique [47] leads to the following equation of second order in the weak perturbation H 1 : t

atPo=- h2

f

[H I (0), [Hl(t-2),

PD(T)11 dT

(B 1)

0

and H 1 (t) is taken in the interaction picture :

H 1 (t)=exp(= fi1y_ Kkak

k

i

H°t

b

)

H1 (0)exp(iHot)

exp (- l(Okt)a + (t) + h .c.

(B 2)



382

A . G . Kofman et al.

We adopt the standard assumption that the diagonal density operator factorizes into the direct product of the reservoir and the JCM density operators [52] : PD(t) ;Zt~ P(t)OO PR

(B 3)

PR - Y_IOk> . Using the completeness of the dressed-states set and bearing in mind the diagonal form of p(t), one finally arrives at the following expression : IKkI 2

exp (-i(O kt)-

k

Oil

Pi ,,,(t) = - Jdt Y {[