FLO/4-QO

Resonance Fluorescence in a Frequency Dependent Photonic Reservoir:

An Exact Multi-Photon Scattering Theory

Marian Florescu∗,† Sajeev John, and Valery Rupasov‡

Department of Physics, University of Toronto, Toronto, Ontario, Canada, M5S 1A7

We develop an exact theory of resonance fluorescence of a two-level atom embedded in a frequencydependent photonic reservoir exhibiting sharp features or gaps in the local density of states. Inour approach, the fluorescence is treated as scattering of incident photons on the atomic system.Within the framework of the dipole, rotating-wave approximation, we derive a solution of the multi-photon scattering problem without recourse to other approximations. In ordinary vacuum, both theoptical Bloch equations and the multi-photon scattering approach result in the well-known Mollowspectrum of fluorescence. Near a sharp feature in the photonic density of states, the non-Markoviancharacter of radiative decay invalidates the usual optical Bloch equations approach. The multi-photon scattering problem approach is still applicable in this case due to a hidden symmetry of thescattering problem. For a frequency dependent reservoir characterized by a step-like discontinuityin the photonic density of states, we find that, in the limit of weak driving field, the scattering crosssection becomes strongly non-Lorentzian, reflecting the non-exponential character of the atomicdipole decay. With increasing intensity of the driving field, the scattered field acquires new featuresassociated with the dynamical decoupling of the atomic system from the radiation reservoir. Thescattered spectrum suffers a global narrowing as the position of the sidebands components of thespectrum is shifted towards the central component, and a local narrowing as the width of theindividual components considerably decreases. The multi-photon scattering approach developedhere can be directly applied to systems characterized by a frequency-dependent isotropic dispersionrelation, as well as to realistic photonic crystal heterostructures characterized by an effective one-dimensional dispersion. We also derive the quantum version of the optical Bloch equations inthe ordinary vacuum, without making use of the Born-Markov approximation. They provide arelationship between the quantum statistical properties of incident light and the scattered light.

PACS numbers: 42.50.Ct, 42.50.Hz

PACS numbers: 42.50.Ct, 42.50.Hz

I. INTRODUCTION

The phenomenon of resonance fluorescence of a two-level atom in ordinary vacuum has been widely studied[1], [2]. In the case of an arbitrary strong incident field, ageneral solution of the problem has been found by Mollow[3]. In Mollow’s approach (as well in all other approachesbased on the optical Bloch equations (OBE) [1],[2]) res-onance fluorescence is treated as emission of photons byan atom driven by a classical laser field. Moreover, thequantum nature of the vacuum field is eliminated fromconsideration, and all physical characteristics of the emit-ted radiation are expressed only in terms of operators forthe atomic variables.

Another approach to the problem of resonance flu-orescence has been developed in Refs.[4], [6]. In thisapproach, the atomic system (in the framework ofthe dipole, rotating-wave approximation) coupled to an

∗Present address: Department of Physics, Princeton University,

Princeton, New Jersey 08544, USA‡Present address: ALTAIR Center, LLC. 1 Chartwell Circle

Shrewsbury, MA 01545, USA†Electronic address: florescu@princeton.edu

isotropic radiation reservoir is mathematically mappedto an effective one-dimensional model. This type of one-dimensionalization is characteristic to several problems(e.g., in condensed matter, the theory of magnetic alloys[7], [8], and in quantum optics, the theory of Dicke su-perradiance [4]) in which a bosonic field interacts witha point-like impurity. In quantum optical problems, therole of the impurity is played by the atom. In contrastto the condensed matter problems, where of interest arethe thermodynamic equilibrium properties of the wholeparticle-plus-impurity system, the physical formulationof the resonance fluorescence problem corresponds to amany-body scattering problem, which in the initial statecontains only laser photons, i.e., contains only field ex-citations. Consequently, it can be solved exactly eventhough the physical model is not completely integrable.

In this paper, we develop an exact multi-photonscattering theory of the resonance fluorescence in afrequency-dependent photonic reservoir (colored vac-uum). In this quantum approach, the resonance fluo-rescence consists in scattering of laser photons on theatomic system. We focus on the analysis of the observ-ables associated with the quantum degrees of freedomof the electromagnetic field. In particular, we describethe spectral density of the field of the scattered photons.We assume that while the electromagnetic dispersion isstrongly dependent on the frequency, it is essentially one-dimensional. A one-dimensional photonic reservoir canbe physically realized in a waveguide channel embedded

2

in a three-dimensional photonic band-gap (PBG)[5] ma-terial. By tuning the characteristics of the microstructure(geometry and index of refraction contrast), the lineardefect in the three-dimensional PBG may support onlytwo waveguide modes, one of which experiences a sharpcutoff in the middle of the three-dimensional PBG [9]. Inthis case, the sub-gap generated by the waveguide chan-nel has a true one-dimensional character, since there isonly one direction available for wave propagation. Foran infinite structure, there is a physical square-root sin-gularity in the photonic density of states near the cutoffof one of the waveguide modes. For a finite structure,the divergence is cutoff by the finite size effects. How-ever, the strong variation with frequency of the photonicDOS [46] remains. For this system, standard perturba-tion theory is highly questionable and the formalism wedevelop in this paper may constitute the only reliableway of calculating the scattered spectrum from active el-ements embedded in the dielectric microstructure.

To maintain generality and enable comparison withprevious theories of resonance fluorescence, we consideran isotropic radiation reservoir field with an optical den-sity of modes frequency dependent (colored vacuum). Inthis model system, which is formally equivalent to a sys-tem characterized by a one-dimensional dispersion rela-tion, the radiation is effectively described by a scalarwave vector k. For an isotropic system, the problempresents spherical symmetry and it is convenient to usethe spherical harmonic representation for the dynamicalvariables of the photonic field. The photon operators, ak,corresponding to a photon of momentum k are expanded

in terms of spherical functions Yjm(k),

ak =∑

jm

Yjm(k)cjm(k), (1.1)

where k = k/k and k = |k|. Since, in the dipole ap-proximation, only the electric dipole harmonic with an-gular momentum j = 1 is assumed to be coupled tothe atom, we can omit all higher harmonics. Then,the effective Hamiltonian contains only dipole operatorscj=1,m(k) ≡ c(k), where the magnetic quantum numberm = 0,±1 is fixed by the given polarization of the inci-dent field. Because the dipole operators depend only on asingle quantum number (the momentum modulus k), theeffective model is one-dimensional in terms of auxiliaryoperators

c(x) =∑

k

eikxc(k). (1.2)

and thus formally equivalent to an one-dimensional pho-tonic reservoir system.

For a sufficiently low intensity of the incident laserfield, such that the mean temporal inter-photon sepa-ration in the incident photon flux exceeds the charac-teristic time of the atom-photon interaction, fluorescencecan be treated as scattering of independent photons onthe atom. The elastic cross-section for the corresponding

one-photon scattering problem is given by the Weisskopfformula [10]

σ(ωL) ∼ γ2

(ωL − ωA)2 + (γ/2)2, (1.3)

where ωL is the frequency of the incident and the scat-tered photons and γ is the spontaneous decay constantof the atomic transition in vacuum. The inverse of thedecay rate constant, γ−1, represents the characteristictime scale of atom-photon interaction. If the intensity ofincident light increases, such that the mean inter-photonseparation becomes comparable to or smaller than thetime of atom-photon interaction, multi-photon processesbecome important and the atomic system is coherentlydriven between its bare states many times before spon-taneously emitting a photon.

In ordinary vacuum, the solution of the one-particlescattering problem for the effective one-dimensionalmodel leads to the Weisskopf result (Eq. (1.3)). Simi-lar to the quantum impurity problems in the theory ofmagnetic alloys [7, 8], at a sufficiently high intensity ofthe incident field, we need to solve a multi-photon scat-tering problem (MSP). In our model, the photon-atomscattering generates an effective photon-photon interac-tion [11], [6], and we deal with a strongly correlated sys-tem of dipole photons. Nevertheless, the effective one-dimensional model is diagonalized exactly [6] in an arbi-trary N -particle sector of the total Hilbert space of themodel Hamiltonian, by means of the Bethe ansatz tech-nique [7, 8]. Since a given multi-particle wave function ofthe incident photons is symmetric with respect to permu-tations of photon coordinates in the x-space (the auxil-iary x-space is generated by the Fourier transform of thek-space), we can limit our analysis to an ordered sectorof the N -particle wave function, say xN > . . . > x1. Thetwo-photon scattering processes are strictly “forbidden”,because, due to the conservation of the total excitationnumber, the appearance of two photons, say, in the re-gion (xi, xi + 1) would require no photons in the finalinterval (xN−1, xN ). However, the causal nature of thepropagation of the photons and the fact that a two-levelatom cannot be excited twice implies that there is a non-vanishing probability of finding a photon arbitrarily farfrom its point of creation. The absence of two-photonscattering processes has an important consequence: theorder of photons in an initial state of the scattering prob-lem is preserved in the scattering process [4]. As a result,the only manifestation of the multi-particle character ofscattering is a confinement of the wave function of thej-th scattered photon in the interval between the coordi-nates of the (j − 1)-th and (j + 1)-th scattered photons.

In ordinary vacuum, the solution of the multi-photonscattering problem results in the Mollow spectrum of thescattered field [12]. The solution obtained in the MSPformalism reproduces completely all the results of thetheory of resonance fluorescence derived in the frameworkof the OBE, without making recourse to any additional

approximations, such as the Born-Markov approximation

3

used to derive the OBE. The MSP approach shows thatthe OBE in vacuum are exact in the framework of thedipole, rotating-wave approximation, used to derive thestandard Hamiltonian of the problem.

The situation is drastically changed in the case of fre-quency dependent photonic reservoirs, such as confinedphotonic systems [13]. If the transition frequency of theatom, ωA, lies far enough from the photonic DOS discon-tinuity, the scattering problem can be linearized with agood accuracy in the vicinity of the transition frequency(on a frequency scale given by the Rabi frequency). Thisapproximation leads to the Mollow spectrum of fluores-cence. If the transition frequency lies close to a band-gapedge (or any other photonic DOS discontinuity) the lin-earization is in general impossible. Therefore, at firstglance, both OBE and MSP approaches would not workin the case of a frequency dependent reservoir. In the caseof OBE, the difficulty is related to the non-Markoviancharacter of radiative decay of the excited atomic state[14], [15], [16], whereas in the case of MSP, photons ofdifferent frequencies propagate with different velocities,and, as a result, the effective photon-photon correlationsgenerated by the photon-atom scattering must contributeessentially to the final state of the scattering problem.

Remarkably, the MSP approach is still applicable toa co loured vacuum, due to a hidden symmetry of theproblem. As in ordinary vacuum, the standard modelreduces to an effective one-dimensional model, but nowin an auxiliary operator τ -space, related to photon fre-quency ω through Fourier transform. We demonstratein this paper that the solution of the one-particle scat-tering problem leads to a non-Lorentzian cross-section ofscattering of the incident photon. For the multi-photonscattering problem, we show that, even if we deal witha strongly correlated system of dipole photons, since allphotons in the auxiliary τ -space propagate with the samevelocity, the order of photons in an initial state of thescattering problem in the auxiliary space is preserved inthe scattering process. This property allows us to find anexact expression for the final state of the multi-photonscattering problem and to compute the spectral densityof the scattered field.

This paper is organized as follows. In Sec. II we in-troduce a photonic crystal heterostructure [34] relevantto the formalism we develop. In Sec. III, we formulatethe problem of resonance fluorescence as a multi-photonscattering problem of an incident flux of photons on atwo-level atom. In Sec. IV, we analyze the one-photonscattering problem and generalize the Weisskopf solutionto the case of a frequency dependent reservoir. Section Vis dedicated to the analysis of the frequency dependenceof the atomic Lamb shift and the scattering cross sectionfor a model photonic density of states (DOS), presentinga step-like discontinuity in the photonic density of states.In Sec. VI, we find a solution of the multi-photon scat-tering problem and compute the spectral density of thescattered field for an arbitrary density of photon states.In Sec. VII, we study the ordinary-vacuum limit of the

resonance fluorescence problem and show that for a laserfield resonant with the atomic frequency, our general re-sults reduce to the standard Mollow spectrum of reso-nance fluorescence. We then apply the theory formu-lated to simple models of colored vacua. We also presentextended results for the ordinary vacuum case, by consid-ering arbitrary detunings of the laser field frequency withrespect to the atomic frequency. Section VIII presents asemi-classical interpretation of the resonance fluorescencein a frequency dependent photonic reservoir. We arguequalitatively that some features present in the spectrumof the scattered radiation can be associated with the pos-itive atomic inversion of the atomic system described in[16] and [17]. In Sec. IX, we derive the quantum opticalBloch equations in ordinary vacuum without making useof the Born-Markov approximation. Since in these equa-tions the incident laser field is represented by correspond-ing field operators rather than a classical field amplitude,they are valid for an arbitrary quantum state of incidentfield and allow the study of the dependence of physicalproperties of scattered light on quantum-statistical prop-erties of incident light. Finally, in Sec. X we present someconcluding remarks.

II. ONE-DIMENSIONAL PHOTON

DISPERSION RELATION IN A

THREE-DIMENSIONAL PHOTONIC CRYSTAL

HETEROSTRUCTURE

In this section, we introduce a three-dimensionalphotonic crystal heterostructure [34] that has an one-dimensional photon dispersion relation. This kind ofstructure is suitable for the multi-photon scattering the-ory developed in the following sections. The structureconsists of a two-dimensional PC slab sandwiched be-tween two three-dimensional PCs with large absoluteband-gaps. The cladding three-dimensional PCs can besquare spiral [35], woodpile [36], slanted pores [37], orinverse Si opal [38]. The two-dimensional PC slab con-sists of dielectric rods in a square lattice, which is latticematched with the three-dimensional claddings. One lineof rods in the two-dimensional slab is removed to createan on-chip waveguide. It is the waveguide that providesthe one-dimensional photon dispersion relation. Here,we use direct diamond:1-square spirals as the claddingsbecause this kind of square spirals has already been syn-thesized using glancing angle deposition techniques [39].In addition, this class of heterostructures exhibits excep-tional strong tolerance against the structure disorders[34] . The parameters of the direct diamond:1-squarespiral are [L, c, r] = [0.7a, 1.3a, 0.2a] [35] (a is the squarelattice constant). The square spiral of those parameterspossesses a three-dimensional PBG of ∼ 14% when therefractive index of the arms is 11.9.

The band structure of the waveguide is calculated byplane wave expansion method [40] and super cell method[41]. The result is depicted in Fig. 1. Plane waves with

4

0 0.1 0.2 0.3 0.4 0.5k

ya/2π

0.32

0.34

0.36

0.38

0.4a/

λ2D bands

3D bands

FIG. 1: The band structure of a waveguide in a three-dimensional heterostructure. The wave vector ky is along thewaveguide direction. The height of the two-dimensional slabis 0.3a. The rod radius is 0.2a..

reciprocal lattice vectors |G| < 2.5(2π/a) are used inour calculation, which corresponds to using 5975 planewaves. The size of the super cell is 7a × 1a × 5.5a. Thetwo-dimensional slab has three lines of rod on each sideof the waveguide (3a × 2 + 1a = 7a) and one unit alongthe waveguide (1a). Two units of square spiral structuresare used as the claddings (1.3a× 4 + 0.3a = 5.5a).

The details of the properties of the heterostructurecan be found in Ref. 34. The relevant features aresummarized below. When there is no on-chip waveg-uide inside the intercalated two-dimensional microchip,some two-dimensional planar waveguide bands (uppershaded regions in Fig. 1) will occupy the upper frac-tion of the original three-dimensional PBG. However,a complete on-chip three-dimensional PBG is retained(unshaded region between two-dimensional bands andthree-dimensional bands in Fig. 1) below these planarmodes. The size of this on-chip band-gap decreases withthe thickness of the two-dimensional microchip layer, dis-appearing completely when the two-dimensional slab isabout 1.1a. Meanwhile, removing one line of rods inthe two-dimensional microchip will create a linear air-waveguide mode inside the on-chip band-gap. Light lo-calization by the PBG materials surrounding this air-waveguide enables flow of light only along the waveg-uide. Consequently, the photon dispersion relation isone-dimensional as long as the light frequency is withinthe on-chip band-gap.

When the wave vector is 0.5(2π/a), the waveguidemode exhibits a cut-off frequency around 0.362(2πc/a),where c is the light velocity in the vacuum. The cut-off frequency obtained here is slight different from thatobtained by the finite-difference time-domain (FDTD)method [42] because the number of plane waves used inour calculation is limited by our computational abilities.This difference will disappear if enough number of planewaves are used [43].

Near the cut-off frequency ωc, the waveguide modedispersion relation can be approximated by a parabolicfunction, i.e., ω = ωc − A(ky − kc)

2, where kc is the

wave vector corresponding to ωc and A is a fitting con-stant. Consequently, the photon DOS will be ρ1(ω) =(1/2)A(ωc − ω)−1/2, which is divergent at ωc. However,in practice, the size of the hetero-structure (and lengthof the corresponding waveguide) is always finite. As aresult, the LDOS divergence is replaced by a large butfinite sharp peak near ωc.

0.35 0.355 0.36 0.365 0.37a/λ

0

400

800

1200

ρ/ρ 0

7a9a11a15a

FIG. 2: The local density of states of waveguides in a three-dimensional heterostructure. Here, ρ0 is the local density ofstates in the vacuum. The length of the waveguide is indicatedin the legend.

In order to calculate the LDOS within the microchip,a dipole is placed near the center of the rod adjacentto the waveguide half-way along the length of the waveg-uide. The dipole is perpendicular to the plane of the two-dimensional slab because the electric field of the waveg-uide mode is almost in this direction [34]. The finitestructure has three units of direct diamond:1-square spi-ral as the claddings above and below the two-dimensionalslab. The height of the two-dimensional slab is 0.3a.There are 5 lines of rods on each side of the air waveguide.The finite-size structure is placed in vacuum, which wesimulate with the second-order Mur absorbing boundarycondition [44]. Since the energy emitted from the dipoleis proportional to the LDOS at the dipole position [45],we integrate the Poynting vector over a closed surfaceoutside the structure to obtain the LDOS. The resultsare depicted in Fig. 2. In our numerical calculations, acubic FDTD grid [47] is used with 10 points per latticeconstant a, i.e., ∆x = ∆y = ∆z = ∆ = 0.1a.

As the length of the waveguide increases, the peakvalue of the LDOS increases rapidly. The LDOS in thewaveguide of 15a in length can be 1000 times than that ofthe vacuum. At the same time, the peak width decreases.Besides the peak near the cutoff frequency, there are somesmaller fringes in the LDOS due to finite-size effects [45].This is a Fabry-Perot effect arising from the heterostruc-ture to free space boundary at the edges of our sample.Near the cutoff frequency, the LDOS varies rapidly, form-ing a sharp peak. In this case, the non-Markovian char-acter of the radiative decay is crucial and invalidates thethe usual optical Bloch equations approach. To overcomethis obstacle, we develop an exact multi-photon scatter-ing theory.

5

III. GENERAL FORMALISM

We consider a physical system consisting of a two-levelatom interacting with a quantized radiation field and em-bedded in a frequency dependent photonic reservoir, de-scribed by the total Hamiltonian

H = HA +HF +HAF , (3.1)

where

HA =1

2ωA(σ3 + 1), (3.2)

HF =∑

k

ω(k)a†kak, (3.3)

are the Hamiltonians of the free atom and electromag-netic field in a medium characterized by an isotropic dis-persion ω(k), respectively, whileHAF describes the atom-field interaction Hamiltonian. For simplicity, throughoutthis paper we use a system of units in which h = c = 1.).Here, ω(k) represents the spectrum of elementary elec-tromagnetic excitations of the medium, which we referto loosely as “photons”(the spectrum of elementary ex-citations of the electromagnetic field in the presence of aphotonic crystal is strongly modified from its free spacecounterpart.). Their creation and annihilation operatorsobey the commutation relations:

[ak, a†k′ ] = δkk′ . (3.4)

The photon polarization is considered fixed by the exter-nal laser field.

The two-level atom with the transition frequency ωA

between the ground (1) and excited (2) levels is describedby the Pauli matrices σi = (σx, σy, σz), with the commu-tator

[σi, σj ] = 2ieijkσk, (3.5)

where eijk is the unit anti-symmetric tensor.In the continuum limit, when the quantization volume

becomes very large, V → ∞, the free field Hamiltoniantakes the form

HF =

∫d3

k

(2π)3ω(k)a†(k)a(k′), (3.6a)

with

[a(k), a†(k′)] = (2π)3δ(k − k′). (3.6b)

For the purpose of this paper, it is convenient to expandthe field operators in terms of spherical harmonics,

a(k) =∑

j,m

∫

C

dω

2πΨωjm(k)cjm(ω), (3.7)

where Ψωjm(k) are a complete set of eigenfunctions ofthe free field Hamiltonian, HF ,

[ω(k) − ω]Ψωjm(k) = 0. (3.8)

The operator cjm(ω) corresponds to a photon of fre-quency ω, angular momentum j = 1, . . . ,∞, and pro-jection thereof m = −j, . . . ,+j. The integration contourC contains all allowed photon states in the respectivemedium. The functions Ψωjm(k) are chosen to be

Ψωjm(k) = B(ω)δ[ω − ω(k)]Yjm(n), (3.9)

with the normalization coefficient

B(ω) =(2π)2

k(ω)

1√ρ(ω)

(3.10)

specified by the condition

[cjm(ω), c†j′m′(ω′)] = 2πδjj′δmm′δ(ω − ω′). (3.11)

Here, ρ(ω) = dk(ω)/dω is the density of states for thephotonic reservoir.

The radiation field Hamiltonian becomes

HF =∑

jm

∫

C

dω

2πω c†jm(ω)cjm(ω), (3.12)

In what follows, we will assume that the photon spec-trum, ω(k), of the medium has a frequency pseudo-gap(or a fill-gap), where the density of states is much lessthan outside a pseudo-gap (or vanishes inside a full-gap).

In deriving the coupling operator in the spherical har-monic representation, we take into account that, in theoptical spectral range, the length scale of the atomic sys-tem, a, is much less than the characteristic wavelengthω−1

A , aωA ≪ 1. As a result, for a dipole allowed atomictransitions, we account only for the coupling of the tran-sition to electric dipole photons (j = 1), while all higherelectric harmonics and all magnetic ones can be omitted[10]. Thus

HAF =∑

m=0,±1

∫

C

dω

2π

√z (ω) [cm(ω) + c†m(ω)] [σ+ + σ−],

(3.13)

where the operators cm(ω) ≡ cj=1,m(ω) correspond tothe electric dipole photons, and σ± = (σx ± iσy)/2. Inthe resonance approximation, we assume that the maincontribution to the coupling parameter γ(ω) comes fromthe neighborhood of the resonance transition ωA and wereplace γ(ω) by the constant value 4

3ω3Ad

2, where d isthe dipole matrix element of the transition. The atomicfactor z(ω) = 1/(dω/dk) [18] accounts for corrections ina frequency dependent reservoir and is proportional tothe photon density of states ρ(ω):

z(ω) = 1/(dω/dk) = 2π2 ρ(ω)

k(ω)2(3.14)

The operators of incident plane wave photons propa-gating along an arbitrary axis passing through the atomicsystem are also represented in terms of operators cjm(ω),

a†kL

=1√Va†(kL) = A†

kL+ β1/2 c†(ωL), (3.15)

6

where the operator AkLis the sum of higher spherical

harmonics with j > 1 (not coupled to the atomic transi-tion), and β ≡ σ1/SLρL. Here, ρL = ρ(ωL) is the densityof photon states at the frequency of incident light. S andL are the cross-section and the length of the incident flux(V = SL), and σ1 = 3π/k2(ωL) is the impact area of thedipole harmonic [19].

If the polarization of the incident laser pulse is fixed,we can fix the magnetic quantum number m = 0,±1of dipole photons. For example, if the incident planewave pulse is linearly polarized, we can choose the axisof quantization of atomic states directed along the vec-tor of the polarization of the incident field. Then, onlyelectric dipole photons with m = 0 are coupled to theatomic transition. In this case, the operator c(ωL) inEq. (3.15) corresponds to the electric-dipole photon offrequency ωL, angular momentum j = 1 and projectionm = 0.

The Hamiltonian of the atom and dipole photons sys-tem can be cast in the form

HD =1

2ωA(σ3 + 1) +

∫ ∞

−∞

dω

2πω c†(ω)c(ω)

−√γ

∫ ∞

−∞

dω

2π

√z(ω)

[c(ω)σ+ + σ−c

†(ω)],

(3.16)

where c(ω) ≡ cj=1,m=0(ω). In deriving Eq. (3.16),we made use of the rotating-wave and resonance ap-proximations. That is to say, we have (i) omitted theterms, which contain c†σ+ and cσ (ii) neglected the fre-quency dependence of the coupling parameter γ(ω) andset γ = γ(ωA) = 4

3ω3Ad

2, and (iii) extended the lowerlimit of integration to −∞.

We emphasize the the Hamiltonian (3.16) can also beused to describe a one-dimensional system, in which atwo-level atom interacts with the radiation reservoir of aphotonic crystal heterostructure presenting an effectiveone-dimensional dispersion relation, such as the one in-troduced in Sec. II. The only difference from the case ofan isotropic reservoir is that the “non-dipolar” contribu-

tion (the A†kL

term) and the√β coefficient in front of the

“dipolar” contribution in Eq. (1.1) would have differentvalues. However, these changes will have only a minorinfluence on our formalism and will result in an overallnormalization constant for the scattering cross-sectionsevaluated in Sec. IV and Sec. VI.

The main purpose of our study is to solve the scatteringproblem for a general initial state and then to computeexpectation values of the scattered field in the limit N →∞, L→ ∞, but N/L = constant.

IV. ONE-PARTICLE SCATTERING PROBLEM

The operator of the number of excitations of the totalsystem (atom plus dipole photons),

N =1

2(σ3 + 1) +

∫

C

dω

2πc†(ω)c(ω), (4.1)

commutes with the model Hamiltonian (Eq. (3.16)), andthe number of excitations is a good quantum number andwe can analyze separately the sectors of Hilbert space cor-responding to different numbers of excitations. We startby considering the one-particle scattering problem withan initial state containing a single plane-wave photon

|In, 1〉 = a†k|0〉 = A†

k|0〉 + β1/2 |ψin〉, (4.2a)

where

|ψin〉 = c†(ωL)|0〉 =

∫ ∞

−∞dτ eiωLτ c†(τ)|0〉. (4.2b)

Here, and throughout the paper, |ψ . . .〉 is used to labelthe dipole component in the frequency representation ofa given state | . . .〉.

We introduce the one-particle eigenstates in the form

|λ〉 =

[ξ(λ)σ+ +

∫

C

dω

2π

√z(ω)φ(ω|λ)c†(ω)

]|0〉, (4.3)

where λ is the eigen-energy of the |λ〉 state. Then, theSchrodinger equation, (HD − λ)|λ〉 = 0, becomes

(ω − λ)φ(ω|λ) −√γξ(λ) = 0 (4.4a)

(ωA − λ)ξ(λ) −√γ

∫

C

dω

2πz(ω)φ(ω|λ) = 0. (4.4b)

Inserting the general solution of Eq. (4.4a),

φ(ω|λ) = 2πδ(ω − λ) +

√γ

ω − λ− i0ξ(λ), (4.5)

into Eq. (4.4b), we find a closed system of equations foratomic wave functions,

[ωA − λ− Σ(λ)]ξ(λ) =√γz(λ), (4.6)

where the atomic self-energy is given by

Σ(λ) = γ

∫

C

dω

2π

z(ω)

ω − λ− i0. (4.7)

The imaginary part of the self-energy, Σ′′(λ), can be com-puted for an arbitrary function z(ω),

Σ′′(λ) =iγ

2z(λ), (4.8)

whereas its real part is formally represented as

Σ′(λ) = γ P

∫

C

dω

2π

z(ω)

ω − λ, (4.9)

7

where P stands for the principal part of the integral.Thus, the photon eigenfunction is found to be

φ(ω|λ) = 2πδ(ω − λ) − iϕ(λ)

ω − λ− i0, (4.10a)

where

ϕ(λ) =−iγ

h(λ) + iγ/2, (4.10b)

h(λ) =λ− [ωA − Σ′(λ)]

z(λ). (4.10c)

The solution of the one particle scattering problem canbe cast into a more familiar form by introducing the aux-iliary τ -space. In the auxiliary τ -space, the Hamiltonian

HD =1

2ωA(σ3 + 1) − i

∫ ∞

−∞dτ c†(τ)

∂

∂τc(τ)

+√γ

∫ ∞

−∞dτ w(τ)[c(τ)σ+ + σ−c

†(τ)]

(4.11)

describes particles which propagate in the positive direc-tion of the τ axis and are scattered by the atom locatedat the point τ = 0. The atomic “potential”,

w(τ) =

∫ ∞

−∞

dω

2π

√z(ω) e−iωτ , (4.12)

tends to zero as τ → ±∞, because z(ω) → 1 as ω → ±∞.Negative (positive) values of the coordinate τ correspondto incoming (outgoing) dipole photon.

In order to find the solution of the one-particle scatter-ing problem in the τ -space, we construct first a eigenstateof the Hamiltonian HD, which asymptotically coincideswith the initial state given by Eq. (4.2a) as τ → −∞, andthen study the asymptotics of this state as τ → +∞, toobtain the scattered field [19].

The photon wave function in the auxiliary τ−space isthe Fourier transform of the direct space wave function(Eq. (4.5)), and is given by [20]

Φ(τ |λ) =h(λ) − i(γ/2) sgn(τ)

h(λ) + iγ/2eiλτ . (4.13)

In solving the scattering problem, we are interested onlyin the asymptotic behavior of the photon wave functionsas τ → ±∞. In these limits, the corresponding Fourierimages take the form

Φ(ω|λ) = 2πδ(ω − λ) ·

1 , (τ → −∞)h(λ)−iγ/2h(λ)+iγ/2 , (τ → +∞)

.

(4.14)The state c†(τ)|0〉 (of central importance in evaluat-ing the dipole component of the incident photon statesEq. (4.2a)) can be represented as a linear superposition of

one-particle eigenstates, |λ〉, (see Eq. (4.3)) of the Hamil-tonian HD:

c†(τ)|0〉 =

∫ ∞

−∞

dλ

2πA(τ |λ)|λ〉. (4.15)

where the expansion coefficients are given by

A(τ |λ) =1

z(λ)

∫ ∞

−∞

dω

2π

√z(ω) Φ∗(ω|λ)e−iωτ . (4.16)

As τ → ∞, we find the following expansion for the statec†(τ)

c†(τ)|0〉 τ→∞=

∫ ∞

−∞

dλ

2π

h(λ) − iγ/2

h(λ) + iγ/2

×∫ ∞

−∞dτ ′eiλ(τ ′−τ)c†(τ ′)|0〉. (4.17)

Therefore, in the auxiliary τ−space the out-state of theone-particle scattering problem is

|ψout〉 =

∫ ∞

−∞dτ ψout(τ) c

†(τ)|0〉, (4.18a)

where

ψout(τ) =

∫ ∞

−∞dτ ′ s(τ |τ ′) eiωLτ ′

, (4.18b)

and the scattering matrix

s(τ |τ ′) =

∫ ∞

−∞

dλ

2π

h(λ) − iγ/2

h(λ) + iγ/2eiλ(τ−τ ′) (4.19)

relates the photon wave functions in the in- and out-states.

As a result of the scattering on the “atomic poten-tial” w(τ), the photon experiences a delay in propagationand the photon wavefunction in the auxiliary τ−space,acquires a scattering phase,

h(λ) − iγ/2

h(λ) + iγ/2= 1 + ϕ(λ). (4.20)

This relates the scattering matrix (s−matrix)

s(τ |τ ′) = δ(τ − τ ′) + t(τ |τ ′), (4.21a)

to the t−matrix, given by

t(τ |τ ′) =

∫ ∞

−∞

dλ

2πϕ(λ) exp [iλ(τ − τ ′)]. (4.21b)

In Eq. (4.21a), the first term corresponds to the trans-mitted dipole photon, while the second one representsthe scattered dipole photon. Since the scattering am-plitude, ϕ(λ), is analytic (this property of the functionϕ(λ) is also evident from the analytical properties of theself-energy Σ(λ) (Eq. (4.7))) in the upper half-plane of

8

the complex variable λ, the t−matrix must vanish forτ − τ ′ > 0.

The out-state of the one-particle scattering problem isexpressed as a sum of states corresponding to the trans-mitted and scattered dipole photon components,

|ψout〉 = |ψin〉 + |ψscatt〉, (4.22a)

with

|ψscatt〉 =

∫ ∞

−∞dτ ψscatt(τ) c

†(τ) |0〉, (4.22b)

and

ψscatt(τ) =

∫ ∞

−∞dτ ′ t(τ |τ ′) eiωLτ ′

. (4.22c)

For example, in ordinary vacuum (z(ǫ) = 1) h(λ) =λ− ωA, and

t(vac)(τ |τ ′) = −γθ(τ ′ − τ) exp [i(ωA − iγ/2)(τ − τ ′)],(4.23)

and the integral in Eq. (4.22c) is computed in an explicitform,

ψ(vac)scatt (τ) =

−iγωL − ωA + iγ/2

exp (iωLτ). (4.24)

In a frequency dependent reservoir, the t-matrix can-not, in general, be expressed in a simple analyticalform. Nevertheless, we can evaluate the wave functionof scattered photons for an arbitrary isotropic or one-dimensional density of states z(λ). Inserting the integralrepresentation for the t-matrix given in Eq. (4.21b) intoEq. (4.22c), and computing first the integral over τ ′, weobtain

ψscatt(τ) = eiωLτ

∫ ∞

−∞

dλ

2πi

ϕ(λ)

λ− ωL + i0. (4.25)

Since the scattering amplitude is analytical in the upperhalf-plane, the λ−integration yields

ψscatt(τ) = ϕ(ωL) exp (iωLτ). (4.26)

Therefore, the out-state of the one-particle scatteringproblem takes the form

|Out, 1〉 = a†k|0〉+β|ψscatt〉 =

[a†k

+ β1/2 ϕ(ωL)c†(ωL)]|0〉,

(4.27)where the first term represents the transmitted planewave component, and ϕ(ωL) is the scattering amplitudeof incident dipole photon of frequency ωL.

The spectral density of the scattered field,

G(ω) = 〈Out, 1|c†(ω)c(ω)|Out, 1〉, (4.28)

can be expressed as

G(ω) = 2πδ(ω − ωL)σ(ω), (4.29)

where the scattering cross section is given by:

σ(ω) =σ1

Sρ(ω)

γ2

h2(ω) + (γ/2)2. (4.30)

In general, this function may have a non-Lorentziancharacter and may exhibit several peaks depending onthe behavior of the atomic form-factor z(ω) and the den-sity of photon states ρ(ω). We note that by settingz(ω) = 1, Eq. (4.30) reduces to the Weisskopf result (1.3).

V. LAMB SHIFT AND SCATTERING

CROSS-SECTION FOR A MODEL DOS

In this section, we consider a specific model of fre-quency dependent photonic reservoir and evaluate theLamb shift and the scattering cross section. Our toymodel is based on a step-like discontinuity in the pho-tonic density of states, which is present in a number ofrealistic band structure calculations [21], [22].

The model density of states considered is assumed togives rise to an atomic form factor

z(ω) = 1 − h

2

tanh

[ω − ωl

γc

]− tanh

[ω − ωr

γc

],

(5.1)where the frequencies ωl and ωr identify the (left andright) positions of the step discontinuities (surroundinga gap or a pseudo band gap) and the constants h and γc

determine the magnitude of jump in the density of statesand the steepness of the discontinuity, respectively.

FIG. 3: The “atomic form factor” for various choices of the hand γc parameters. The pseudo-gap “band-edges” are givenby ωl = ωA − 0.05 ωA, ωr = ωA + 0.05 ωA. The continuouscurve corresponds to γc = 0.01 ωr and h = 0.99, the dashedcurve corresponds to γc = 0.1 γ and h = 0.99, the dot-dashedcurve corresponds to γc = 0.1 γ and h = 0.5, respectively.The inset shows a close-up (on a γ scale) of the right “band-edge” region.

9

In Fig. 3, we plot the “atomic form factor”, for somespecific values of the parameters entering its definition.For frequencies located deep into the pseudo-gap, ω ∈(ωl, ωr), z(ω) ≈ 1 − h, whereas for frequencies far awayfrom the pseudo-gap, z(ω) ≈ 1. The free space case is re-gained for h = 0, while the case of a two dimensional pho-tonic band gap (a true step discontinuity) corresponds toh = 1, and γc → 0. This model of the “atomic form fac-tor” allows us to include in our study a broader class ofphotonic reservoirs, including those associated with di-electric structures that do not posses a full band gap inthe density of states, but their density of states exhibitsrapid variation with frequency on specific spectral ranges(photonic crystals).

The Lamb shift is defined as the real part of the selfenergy (Eq. (4.9)). In the resonance approximation, weassume that the main contribution to the ω-space inte-grals comes from frequencies in the neighborhood of theatomic frequency and the integration contour can be thenextended to C = (−∞,∞). It follows that in free space,the principal part in Eq. (4.9) vanishes. If we had notextended the integration contour to −∞, the real partof the self energy would become divergent. This diver-gence can be removed by introducing a cutoff in the fre-quency integration at the electron Compton’s frequency,ωe = mec

2/h, with me the electron rest mass (higher en-ergies of the photonic reservoir excitations would probethe relativistic structure of the electronic wave-packetand can therefore be neglected in our analysis). This sim-plification together with the pole approximation, Σ′(ω) ≈Σ′(ωA), gives the usual Wigner-Weisskopf result for thefree space Lamb shift, Σ′(ωA) ≈ γ ln(ωe/ωA)/2π. Inthe visible spectral range, γ ≈ 108 Hz, ωA ≈ 1015 Hz,such that the Lamb shift is approximatively given byΣ′

FS(ω) ≈ Σ′FS(ωA) = 2.15 × 108 Hz. Since the Lamb

shift is approximately constant, it can be easily incorpo-rated in the definition of the atomic state energy.

In Fig. 4 and 5 we plot the relative difference betweenthe Lamb shifts evaluated for the pseudo-gap modeland the free space Lamb shift, δLamb(ω) = −(Σ′(ω) −Σ′

FS(ωA))/Σ′FS(ωA). Clearly, the frequency dependence

of Lamb shift is inherited from the DOS, i.e., importantcorrections from the free space values are obtained onlyif the photonic DOS varies with frequency on the scale ofγ. As shown in Fig. 4, the magnitude of the Lamb shift isstrongly dependent on both the depth and the sharpnessof the pseudo-gap. For a sufficiently strong pseudo-gap(h = 0.99 and γc = 0.1 γ), the maximum departure fromthe free space Lamb shift can be as high as ±125%. Thisvalue corresponds to the “band edge” frequencies forwhich the photonic DOS exhibits the greatest asymme-try. However, if the “band edges” of the pseudo-gap are“smooth” and defined on spectral regions comparableto the atomic frequency ωA (rather than γ), the pseudo-gap emission phenomena can be treated by means of amodified Wigner-Weisskopf approximation involving thedensity of states at the atomic frequency position, ρ(ωA).

Further insight is gained by analyzing the scattering

FIG. 4: The relative difference, δLamb(ω) = −(Σ′(ω) −Σ′

FS(ωA))/Σ′FS(ωA), between the pseudo-gap Lamb shift and

the free space Lamb shift. The continuous curve corre-sponds to a “sharp” pseudo-gap characterized by ωl − ωA =−0.05 ωA, ωr − ωA = 0.05 ωA, γc = γ/10, h = 0.99, whereasthe dashed curve corresponds to a “smooth” pseudo-gap,with the same parameters as above except γc = 0.01ωA. Theinset presents an expanded view of the right peak in the mainplot.

FIG. 5: The relative difference δLamb(ω) between the pseudo-gap Lamb shift and the free space Lamb shift a function ofthe parameters characterizing the pseudo-gap. In the figureon the left, the continuous curve corresponds to a pseudo-gap characterized by γc = γ/20, while γc = γ/10 for thedashed curve. In the figure on the right, the continuous curvecorresponds to a pseudo-gap characterized by h = 0.99, andh = 0.5 for the dashed curve. For all curves, ωr − ωA =0.05 ωA, ωl − ωA = −0.05 ωA.

cross-section (4.30). Clearly, the zeros of the functionh(ω) determine the position of the peaks in the scatteringcross section, while the widths of these individual peaksare determined the by value of the “atomic form fac-tor” z(ω) at the peak position. The relative magnitude ofthe individual peaks is determined by the “envelope” fac-

10

FIG. 6: The scaled Lamb shift function eΣ′

(x + ωr)/γ forωl − ωA = −0.05 ωA, ωr − ωA = 0.05 ωA. The figure on theleft corresponds to a large depth pseudo-gap, h = 0.99. Dif-ferent curves correspond to different values of the parameterγc, which varies from γc = γ/20 for the continuous curve toγc = γ for the double-dotted-dashed curve. The figure on theright corresponds to a relatively moderate depth pseudo-gap,h = 0.5. The straight line represents the function y(x) = −xand the intersection points with the bell-shaped curves rep-resent solutions of Eq. (5.4).

tor 1/ρ(ω) (it can be shown [18] that ρ(ω) ∝ z(ω)). Find-ing the solutions of the equation h(ω) = 0 is equivalent toidentifying the dressed (by the real and virtual photons)atomic frequencies. The zeros of the function h(ω) arefound from the equation

ωA − ω = Σ′(ω) (5.2)

Using the fact that the real part of the self energy ismaximal for the “band edge” frequencies, ωl, ωr, weintroduce

ωA = ωA − Σ′(ωr) (5.3a)

Σ′(ω) = Σ′(ω) − Σ′(ωr) (5.3b)

and hereafter treat ωA as the transition frequency of theatom in the medium. The analysis is further simplifiedby assuming “resonant” conditions, i.e., ωA = ωr. ThenEq. (5.2) becomes

Σ′(x+ ωr) = −x (5.4)

where we introduced the variable x ≡ ω − ωr. Equa-tion (5.4) may have one, two or three distinct solu-tions, depending of the characteristics of the photonicDOS. Clearly, in the Wigner-Weisskopf approximation(in which the frequency dependence of the self-energyis neglected), Eq. (5.4) has a unique solution ωA =ωA−Σ′(ωA), which reproduces the usual free space Lambshift of the atomic frequency. As shown in Fig. 6, thereis a direct correlation between the frequency dependent

character of the photonic DOS and features of the scat-tering cross-section. The most important influence is re-lated to the scale over which the photonic DOS varies.Even if the depth of the pseudo-gap is not very large(Fig. 6, the plot on the right), for well defined “bandedges” (γc = γ/20, for the continuous bell shaped curve),Eq. (5.4) possesses three distinct solutions, leading to thescattering cross-section exhibits three peaks. Since thesedistinct peaks occur in the scattering of just a single pho-ton, this effect can be interpreted as vacuum Rabi split-ting.

FIG. 7: The scattering cross section, σ(ω), for a strongpseudo-gap, h = 0.99 and for various choices of the parameterγc. For all curves ωl − ωA = −0.05 ωA, ωr − ωA = 0.05 ωA.

FIG. 8: The scattering cross section, σ(ω), for a relativelymodest pseudo-gap, h = 0.5 and for various choices of the γc.For all curves ωL − ωA = −0.05 ωA, ωr − ωA = 0.05 ωA.

The scattering cross-section is plotted in Fig. 7 andFig. 8. As expected, for important deviations of thephotonic DOS from the free space value, the spec-

11

trum becomes non-Lorentzian, leading to a strong non-exponential decay of the atomic system [21]. For the spe-cific choices of parameters used here, some of the spec-tral features predicted by Eq. (5.4) overlap, and even forstrong pseudo-gaps (large values of h and small values ofγc), the spectrum presents at most two distinct peaks.We also note the important reduction of the radiationemitted in the spectral range of the pseudo-gap.

VI. MULTI-PARTICLE SCATTERING

PROBLEM

We now consider the scattering on the atomic systemof a multi-particle state describing an incident pulse con-taining N plane wave photons. The choice of an initialstate containing N plane wave photons has a two-foldmotivation. From a formal point of view, it is very prac-tical, since, as seen in the single-particle excitation prob-lem, it allows for a direct separation of the dipole compo-nent of the field from the contribution of higher sphericalharmonics, which are not influenced by the interactionwith the atomic system (3.15). On the other hand, itis a natural extension of the single-particle problem pre-sented in the preceding section, and the calculations aregreatly simplified by making use of the results obtainedin Sec. IV. More specifically, we start with an initial(t → −∞) state containing N plane-wave photons (seeAppendix A)

|In, N〉 =1√N !

[ak]N |0〉, (6.1)

(with ak defined by Eq. (3.15)), which is scattered on theatomic system, and we then seek its asymptotic behavior,

|Out, N〉 = limτ→∞

[exp (−iHτ) |In, N〉] . (6.2)

Once the out-state of the scattering problem is found,analogous to the single photon case, we can evaluate thespectral properties of the scattered radiation from theatomic system.

Since only the dipole photons interact with the atomicsystem, the problem reduces to the scattering of n dipolephotons in an initial symmetrized state,

|Ψin, n〉 =√n!

∫d~τ ′θ(τ ′1 > . . . > τ ′n)Ψin(~τ ′)

n∏

j=1

c†(τ ′j)|0〉.

(6.3)Here, the integration is carried out over the ordered par-ticle coordinates τ ′1 > . . . > τ ′n. As shown in Appendix A,the out-state of the n scattered dipole photons, |Ψout, n〉,is obtained from Eq. (6.3) by replacing Ψin(~τ ′) by

Ψout(~τ) =

∫d~τ ′S(~τ |~τ ′)Ψin(~τ ′), (6.4a)

where the integration range is specified by τj−1 < τ ′j <

τj , and the multi-photon S-matrix is given by

S(~τ |~τ ′) =n∏

j=1

s(τj |τ ′j). (6.4b)

Here, the single particle s-matrix s(τj |τ ′j) is expressedthrough a t-matrix, as in Eq. (4.21).

As discussed in the introduction, the characteristicfeature of the interaction between the photons and theatomic system is that the two-photon scattering pro-cesses are “forbidden”, and, as a consequence, the orderof incident particles on the τ axis is preserved in the scat-tering process. Furthermore, making use of the solutionof the scattering problem for n dipole photons presentedin Appendix A, we can isolate in the out-state of thescattering problem with N incident plane wave photons

|Out, N〉 =N∑

n=0

√Cn

N βn/2 |Ψscatt, n〉

⊗ 1√(N − n)!

[a†k]N−n|0〉, (6.5)

the state

|Ψscatt, n〉 =√n!

∫d~τΨscatt(~τ)

n∏

j=1

c†(τj)|0〉, (6.6a)

which contains only scattered photons. Here,

Ψscatt(~τ) =n∏

j=1

ψ(τj−1, τj) exp (iωLτj), (6.6b)

and the wave function

ψ(τj−1 − τj) =

∫ ∞

−∞

dλ

2πi

ϕ(λ)

λ− ωL + i0

×

1 − exp [−i (λ− ωL) (τj−1 − τj)]

(6.6c)

vanishes for negative argument, τj−1 − τj < 0.Therefore, despite the factorization of the multi-

particle scattering into a product of one-particle scatter-ings, the multi-particle wave function of scattered dipolephotons is not factorized into a product of one-particlewave functions. This strongly correlated state of the scat-tered field explains multi-photon effects in resonance fluo-rescence found by Mollow in the case of ordinary vacuum[3].

All information about the medium is contained inψ(τj−1−τj) through ϕ(λ), and its knowledge fully solvesthe multi-particle scattering problem. In ordinary vac-uum and on-resonance case (ωL = ωA), the scatteringamplitude reduces to the simple form

ϕvac(λ) =−iγ

λ+ iγ/2. (6.7)

12

Then, the integral over λ in Eq. (6.6c) yields

ψvac(τ) =[1 − exp

(−γ

2τ)]θ(τ). (6.8)

In a frequency dependent reservoir, such a simple ex-pression does not in general occur. Nevertheless, thisformal difficulty can be overcame by noticing that theexpectation values of the scattered field are expressed interms of the Fourier transform of the function ψ(τ),

ψ(ω) =

∫ ∞

0

dτψ(τ)eiωτ =i

ω + i0ϕ(ω + ωL). (6.9)

This functional is analytical in the upper half-plane (sinceψ(τ < 0) = 0) and can be evaluated for an arbitraryfrequency dependent photonic reservoir.

In order to make closer contact with experimental rel-evant situations (usually, the resonance fluorescence ex-periments involve strong incident fields) and to compareour results to previous calculations for the resonance flu-orescence spectrum (done, in general, in the approxima-tion that the incident field is strong enough to be treatedas a classical variable), we rewrite the multi-particle scat-tering problem in terms of coherent states of the incidentfield (as shown in Appendix A, this involves just a changeof basis in the photonic Hilbert space).

An initial coherent state pulse with a mean photonnumber N of frequency ωL and wave vector kL = k(ωL)is described by

|In〉 = e−N/2 exp (√Na†

kL)|0〉. (6.10)

After scattering from the atomic system, the final (τ →∞) state of the field is given by

|Out〉 =

∞∑

n=0

(ρ0)n/2

√n!

|Ψscatt, n〉 ⊗ |In〉, (6.11)

where

ρ0 ≡ β N =σ1

S

N

L

1

ρ(ωL)(6.12)

is the linear density of dipole photons in the incidentpulse (see Appendix A for a full derivation of Eq. (6.11)).

As usual in the scattering theory [19], we assume thatthe incident beam has a finite cross section and that themeasurements of the scattered field are carried out out-side this cross section. For this reason, the out-state ofthe multi-particle scattering problem has to be renor-malized and, as shown in Appendix B, the renormalizedeffective out-state of the scattered field is given by

|Out〉 =

∞∑

n=1

(ρ0)n/2

∫ ∞

−∞d~τ v(τn + τ0)

×n∏

j=1

u(τj−1 − τj) c†(τj)|0〉, (6.13)

where the Fourier transforms of the functions u(τ) andv(τ) (evaluated in Appendix B) are found to be

v(ω) =i[h(ω + ωL) + iγ/2]

ω[h(ω + ωL) + iγ/2] − γρ0, (6.14a)

u(ω) =γ

ω[h(ω + ωL) + iγ/2] − γρ0. (6.14b)

The spectral density of the scattered field per unit oftime, G(ω), is given by the Fourier image over the differ-ence of coordinates of the correlation function

G(τ, τ ′) =1

N 〈Out|c†(τ)c(τ ′)|Out〉, (6.15)

where N is the normalization constant (C2).In Appendix C, we show that for an arbitrary fre-

quency dependent radiation reservoir, the spectral den-sity of the scattered field can be cast in the form

G(ω) =ρ30

F0

[ρ−10 Γ(2)(ω) +

Γ(1)(ω)Γ(1)∗(−ω)

1 − ρ0U(ω)

+Γ(1)(−ω)Γ(1)∗(ω)

1 − ρ0U(−ω)

]. (6.16)

with F0, U(ω), Γ(1)(ω), Γ(2)(ω) defined by Eqs. (C3),(C14b) and (C16b). We emphasize that, in contrast tothe analysis of the one-particle scattering problem [seeEq. (4.28)], we define here the function G(ω) in such amanner that dN(ω) = G(ω)dω/2π represents the numberof photons scattered by the atom per unit of time in thefrequency interval dω.

VII. SPECTRAL DENSITY OF THE

SCATTERED FIELD

A. Ordinary Vacuum: Analytical Results

In this section, we demonstrate the ability of our ap-proach to recapture the ordinary vacuum results. Toavoid more complicated expressions, we restrict our con-sideration to the case of exact resonance, ωL = ωA.

The information about the dispersion law of photonsin the frequency dependent reservoir is contained only inthe function h(ω) and the linear density of dipole photonsin the incident pulse, ρ0, given by Eq. (6.12). In ordinaryvacuum, z(ω) = 1, and in exact resonance, the effectivewave function of scattered photon, u(ω), takes a simpleform:

uvac(ω) =γ

(ω − ω+)(ω − ω−), (7.1)

where

ω± = −iγ4±

√γρ0 − (γ/4)2. (7.2)

Using Eqs. (7.1) and (C3) , the function U(ω) (the func-tion U(ω) is defined as a Fourier transform of U(τ) =

13

|u(τ)|2, which represents the probability of having a pho-ton scattered in the time interval τ) is then found to be

U(ω) =−2iγ2

(ω + iγ/2)(ω − 2ω+)(ω − 2ω−), (7.3)

while the function H(ω) (defined in Appendix B) is givenby

H(ω) = − 2iγ2ρ0

(ω + i0)(ω − Ω+)(ω − Ω−), (7.4)

with the complex frequencies defined by

Ω± = −i34γ ±

√Ω2

R − (γ/4)2, (7.5)

and where ΩR = (4γρ0)1/2 is the Rabi frequency. In

free space, we also obtain a simple expression for thecoefficient F−1

0

1

F0= − 2γ2ρ0

Ω+Ω−=

1

2

γΩ2R

Ω2R + γ2/2

. (7.6)

Since the correlation function G(τ) ≡ G(τ − τ ′, 0) isan even function, we restrict our further computations tothe case τ > 0. Then, we obtain [48]

G(τ) = G(1)(τ) +G(2)(τ), (7.7a)

G(1)(τ) =ρ20

F0

∫ ∞

−∞

dω

2πH(ω)D(ω) e−iωτ , (7.7b)

G(2)(τ) =ρ20

F0

∫ ∞

−∞

dω

2πΓ(2)(ω)e−iωτ , (7.7c)

where we introduced

D(ω) =Γ(1)(ω)Γ(1)∗(−ω)

U(ω). (7.7d)

At the first glance, the behavior of the correlator ismuch richer than in the Mollow theory, because, apartfrom the contributions of three poles of the functionH(ω)corresponding to the elastic component, ω = 0, and twosideband components, ω = Ω±, Eqs. (7.7) contain alsocontributions from singular points of the functions D(ω)and Γ(2)(ω). By direct computation we obtain

Γ(1)(ω) = −ω + iγ

2iγρ0U(ω), (7.8a)

Γ(2)(ω) =1

2ρ20

i

ω + iγ/2− (ω + iγ)2

4γ2ρ20

U(ω) + c.c.,

(7.8b)

and hence

Γ(2)(ω) =1

2ρ20

i

ω + iγ/2+D(ω) + c.c. . (7.8c)

By substituting these expressions in Eqs. (7.7), we finallyobtain in the ordinary vacuum limit

G(τ) =1

2

Ω2R

Ω2R + γ2/2

Re

[∫ ∞

−∞

dω

2π

−2(ω + iγ)2

Ω2R

H(ω)

+iγ

ω + iγ/2

e−iωτ

]. (7.9)

This expression for the spectral density of the scatteredfield reproduces the Mollow results for the spectrum ofthe scattered radiation. We note that the central broad-ened line of the Mollow spectrum is represented by thesecond term in Eq. (7.9).

As we argue below, the value of the Rabi frequencyused in this paper differs by a

√2 factor from the value

used in Mollow’s work. This difference is caused bythe different normalization of the atom-field interactionHamiltonian. We start by writing the Rabi frequencyΩR =

√4γρ0 in terms of the amplitude of the electric

field of the incident laser pulse. Inserting the explicitexpressions for γ and ρ0, we obtain

Ω2R = 4

(4

3ω3

Ad2

)(3π

ω2A

N

SL

)= 16πd2ωAN

SL, (7.10)

where we used the expression for the impact area ofdipole photons, σ1 = 3π/ω2

A, and set ωL = ωA. The lastfactor in this expression represents the density of energyin the incident pulse. If the electric field of the incidentpulse is given by

E(x) =1

2E0

(eikLx + e−ikLx

), (7.11)

the density of energy is found to be

ωAN

SL=E2

0

8π, (7.12)

and hence

ΩR =√

2dE0. (7.13)

Therefore, in our approach the Rabi frequency is√

2times bigger that the expression for the Rabi frequencyusually used in the optical Bloch equations [2]. To ex-plain this, we compute the coupling of the atom to theincident coherent field. This requires averaging the fieldoperators involved in the operator of atom-field coupling,

HAF = −√γ

∫dω

2π[c(ω)σ+ + σ−c

†(ω)], (7.14a)

over the state of the incident field. We find

〈In|HAF |In〉 = −√γρ0(σ+ + σ−) = − 1√

2dE0(σ+ + σ−),

(7.14b)

in which the coupling magnitude 1√2dE0 is again

√2

times bigger than 12dE0 used in OBE. Clearly, the dif-

ference in the values of the Rabi frequency is caused bythe different normalization of the atom-field interactionHamiltonian.

14

B. Numerical Results

The resonance fluorescence theory developed in thispaper is generally valid, regardless the magnitude of thequantum driving field and the specifics of the photonicreservoir considered. In this section, we consider a spe-cific model for the photonic reservoir DOS and evaluatethe resonance fluorescence spectrum associated with thescattered radiation from a two-level atomic system drivenby an external laser field. Generally, the spectral densityof the scattered field cannot be obtained in an analyticalform and the integrals involved in Eq. (6.16) have to beevaluated numerically. The elastic and inelastic [2] partsof the scattered spectrum are separated in the usual way:

G(ω) = Gel(ω) +Gin(ω) (7.15a)

Gel(ω) = 2Re

[1

zlimz→0

z · G(z)

]

z=ω

, (7.15b)

Gin(ω) = 2Re

[G(z) − 1

zlimz→0

z · G(z)

]

z=ω

, (7.15c)

where

G(z) =ρ20

2F0Γ(2)(z) +

ρ30

F0

Γ(1)(z)Γ(1)∗(z)

1 − ρ0U(z). (7.15d)

Further, using the expansion of the function U(ω) in theneighborhood of the origin, Eq. (C7b), (the frequencyis measured from ωL), the elastic part of the scatteredspectrum gives the usual delta component:

Gel(ω) = 2πF0

∣∣∣Γ(1)(0)∣∣∣2

δ(ω). (7.16)

1. Ordinary Vacuum

We now extend the results obtained in the previous sec-tion to arbitrary intensities and detunings of the drivinglaser field. The ordinary vacuum results are well known[2], [3] and are used here as a consistency check of ourapproach.

In Fig. 9 we plot the exact resonance inelastic part ofthe scattered spectrum, evaluated by numerical integra-tion of Eq. (6.16), for some arbitrarily chosen values ofthe laser field intensity. In the case of strong driving field,ΩR ≫ γ, the delta-function component gives only a verysmall contribution to the central component, while inthe opposite regime, ΩR ≪ γ, it can dominate the spec-trum [2]. In the case of weak driving field, illustratedin Fig. 9(curve (a)), the inelastic spectrum consists ofonly one central component. As the driving field inten-sity becomes larger than the critical value ΩC

R = γ/2, thespectrum develops two additional sidebands at frequen-cies Ω± = ±

√Ω2

R − (γ/4)2, as shown by curves (b) and(c). In the limit of strong driving laser field, the classicalMollow spectrum is regained in Fig. 9 (curve (c)). Thepeak heights of the center and sidebands are in ratio 3:1,

the integrated intensities are in the ratio 2:1, the centralcomponent width is γ/2, and the sideband widths are3γ/4, respectively.

FIG. 9: Inelastic scattered spectrum in the free space case.The atomic frequency is resonant with the laser field fre-quency. The three curves correspond to different values ofthe driving field intensity: (a) ΩR/γ = 0.2, (b) ΩR/γ = 3.0,(c) ΩR/γ = 8.0, respectively.

In Fig. 10, we plot the peak height of the central com-ponent as function of the Rabi frequency. As expected,we obtain that the weight of the inelastic scattered ra-diation increases with the increase of the strength of thelaser field. For higher intensities, the atomic system be-comes saturated, and the maximum value of the centralcomponent becomes insensitive to further changes in thedriving field intensity.

FIG. 10: The peak height of the central component as a func-tion of the scaled Rabi frequency, ΩR/γ. The atomic fre-quency is resonant with the laser field frequency.

This picture is drastically changed for a non-resonantdriving laser field. As the laser field is further de-tuned from the atomic resonant frequency, the center-to-sideband peak ratio decreases, and can become much

15

smaller than 1 (Fig. 11). Also, the larger the detun-

FIG. 11: Inelastic scattered spectrum in the free space case.The driving laser field intensity is ΩR/γ = 5.0. The curvescorrespond to different values of the detuning of the laser fieldfrequency from the resonant atomic frequency, ∆AL = eωA −ωL: (a) ∆AL/γ = 3.0, (b) ∆AL/γ = 5.0, (c)∆AL/γ = 10.0,(d)∆AL/γ = 15.0, respectively.

ing of the laser field from the resonant atomic frequency,the less effective the coupling between the atomic sys-tem and the driving field. Consequently, the amount ofscattered radiation diminishes considerably, as exempli-fied in Fig. 12, where we plot the maximum value ofthe central component as a function of the driving laserfield strength. Our results, (Fig. 9, Fig. 10, Fig. 11 and

FIG. 12: The maximum value of the central component as afunction of the scaled detuning of the atomic frequency withrespect to the laser field frequency, ∆AL/γ = (eωA − ωL)/γ.The driving field Rabi frequency is fixed to ΩR/γ = 5.

Fig. 12) are clearly in agreement with those obtained byother approaches [2], [3] for ordinary vacuum.

2. Colored Vacuum

The analytical properties of the functions u(λ) are ofcentral importance in our study. As argued before, thefunction u(λ) is analytical in the upper half-pane (UHP),provided that a full expression for the self-energy Σ(λ)is used. As seen from its definition (Eq. (4.7)), the self-energy is analytical in the upper half-plane, since theintegration contour lies below the pole ω = λ+ i0. Thisproperty is the mathematical expression of the CausalityPrinciple. The formal divergence of the self-energy can beremoved through a regularization procedure and then in-corporated into the definition of the physical atomic tran-sition frequency, ωA, as is done in the Wigner-Weisskopftheory. We note that if, for a given frequency dependentphotonic reservoir, the scattering amplitude acquires asingular point in the UHP (as a result of the chosenregularization procedure), this inconvenience can be ad-dressed by slightly modifying the definition of the func-tion u in the time domain (Eq. (B5a)). In all such cases,the corresponding Fourier transform is defined assum-ing that the integration contour in the Fourier integrallies upper to all singular points of the scattering ampli-tude in the frequency domain. This assumptions mustbe treated as a part of the regularization procedure,which, as a result, preserves causality. This renormal-ization procedure adds more complexity to the numeri-cal methods used to evaluate the spectral density of thescattered field. The parameter space range that can beexplored becomes strongly limited, since, for particularchoices of the atomic form factor z(ω), its complex planeextension may cause loss of convergence of the numericalalgorithms used to evaluate the spectral density of thescattered field.

We restrict our study for colored vacua to the simplecase of the laser field resonant with both the atomic tran-sition and the “band edge” frequency: ωL = ωA = ωr.We choose the atomic form factor corresponding to amodel density of states given by Eq. (5.1) with γc =γ/100, h = 0.99. In this case we find that the spectralfeatures are considerably narrowed (at low intensities ofthe driving field, the central component width is aboutfive times narrower than its free space correspondent, asshown in Fig. 13). Moreover, the shift of the sidebandcomponents relative to the central component is also less.

These effects indicate a dynamical decoupling of theatomic system from the radiation reservoir of the pho-tonic crystal, confirming earlier calculations made in thecontext of driven atoms coupled to frequency-dependentphotonic reservoirs [23], [24], [25], [26]. The suppres-sion of vacuum fluctuations and atomic decay rate overa appreciable spectral range (corresponding to the re-gion ω ∈ (ωl, ωr)) explains the narrowing of the spectraldensity components of the scattered field. Meanwhile,the displacement of the sidebands components can be ex-plained by the less effective coupling between the atomicsystem and the driving field that produces a smaller effec-tive Rabi frequency and, implicitly, modifies the position

16

FIG. 13: Inelastic scattered spectrum for a photonic crys-tal reservoir. The atomic frequency is resonant with thelaser field frequency. The three curves correspond to differ-ent values of the driving field intensity: (a) ΩR/γ = 0.2, (b)ΩR/γ = 3.0, (c) ΩR/γ = 8.0, respectively. The inset shows acomparison between the central component in the case of thephotonic crystal for γc = γ/100, h = 0.99, (the thin curve)and the same component in the case of ordinary vacuum (thebroad curve), which corresponds to setting h = 0.

of the sidebands components of the spectrum.Also, we note that the maximum value of the central

peak exhibits a resonance-like behavior (see Fig. 14), ata “threshold” value of the intensity of the driving field(for the specific choice of parameters here, Ωth

R ≈ 0.3 γ).Below the “threshold” intensity, the maximum value of

FIG. 14: The maximum value of the central component, asa function of the scaled Rabi frequency, ΩR/γ. The atomicfrequency is resonant with the laser field frequency.

the central peak increases, if the driving field intensityis increased, whereas above the “threshold” value, thecentral maximum decreases. This dependence of centralpeak maximum on the driving field intensity is reminis-cent of the switching behavior of atomic systems coupledto frequency dependent radiation reservoirs described in

Ref. [27], [16], [28]. This connection is further analyzedin Sec. VIII.

VIII. SEMI-CLASSICAL PICTURE

In this section, we review the semi-classical picture ofthe resonance fluorescence in colored vacua. The purposeof this section is to justify the assertion made at the endof the previous section that the resonance-like featurepresent in Fig. 14 is associated with mechanisms similarto those responsible for the atomic population inversion.The analysis is based on the dressed-atom picture [29]used in [16], [17] and assumes that the atomic systemis driven by an classical external single-mode laser fieldand that this “dressed” atomic system is coupled to thefrequency-dependent photonic reservoir. We adapt theformalism and the notations in order to make connec-tion to the multi-photon scattering approach developedin [17].

In the dipole and RWA approximations and in a rotat-ing frame of reference (rotating with the laser frequency,ωL), the effective Hamiltonian of the system is given by[16], [28]

H = H0 +H1, (8.1a)

where

H0 =1

2∆AL σ3 +

∫ ∞

0

dω

2π∆ω c

†(ω)c(ω) + ǫ [σ+ + σ−] ,

(8.1b)represents the Hamiltonian of the atomic system and theexternal driving field, and

H1 = i√γ

∫ ∞

0

dω

2π

√z(ω)

[c†(ω)σ− − c(ω)σ+

], (8.1c)

is the interaction Hamiltonian between the atomic systemand the frequency-dependent photonic bath. Here, ǫ =~d21 · ~E is the Rabi frequency, ~E is the laser electrical fieldat the atom position, ∆ω = ω−ωL and ∆AL = ωA −ωL,are the detuning of the ω−mode with respect to the laserfield frequency, and the detuning of the atomic frequency,respectively.

The Hamiltonian H0 can be diagonalized by trans-forming to the dressed atom basis defined in [16]. Thistransformation leads to the non-interacting dressed stateHamiltonian:

H0 = hΩR3 +

∫ ∞

0

dω

2π∆ω c(ω)†c(ω), (8.2)

with Ω =[ǫ2 + ∆2

AL/4]1/2

the generalized Rabi fre-quency.

We define the time-dependent interaction-picture

Hamiltonian, H1 = U†(t)H1U(t), with the unitary trans-formation operator U(t) = exp(−iH0t/h). In this pic-

17

ture, the interaction Hamiltonian H1 takes the form:

H1 = i√γ

∫ ∞

0

dω

2π

√z(ω)

[c(ω)†

(csR3 e

i∆ωt

+c2R12 ei(∆ω−2Ω)t

−s2R21 ei(∆ω+2Ω)t

)]+ h.c. (8.3)

The dressed atomic operators in the interaction pic-

ture exhibit the time dependence given by: R12(t) =

R12(0) exp(−2iΩt), R21(t) = R21(0) exp(2iΩt), and

R3(t) = R3(0). Hereafter, we drop the tilde on the inter-action picture operators.

In the case of frequency-dependent photonic reservoir,we assume that the “atomic” form factor (and, implic-itly, the photonic density of modes) exhibits a step-likediscontinuity, so that the Mollow spectral componentsexperience very different photonic mode densities. For asimplified model of a photonic crystal, in which the pho-tonic DOS, while rapidly varying at a given frequency,is constant over the spectral regions surrounding thedressed-state frequencies ωL, ωL−2Ω and ωL+2Ω. Thenone can make the Markov approximation, and associatespontaneous emission rates γ0 and γ± with each of thefrequencies ωL and ωL ± 2Ω, where

γ0 = γ · z(ωL), (8.4)

γ± = γ · z(ωL ± Ω). (8.5)

Moreover, we further simplify our analysis by neglectingthe energy shifts caused by atom-reservoir interaction,and discard the fast oscillating terms with frequencies±2Ω, ±4Ω.

The connection between the resonance-like behaviorexhibited by peak height of the central line obtained inSec. VIIB 2 and the atomic switching behavior analyzedin [16], [17], can be understood using the intuitive dressedatom picture. In the secular approximation, the dressedatom approach leads to a simple interpretation of thedynamics of the atomic system and the characteristicsof the scattered radiation [29], [30]. The bare (dressed)picture diagram is presented on the left (right) side ofFig. 15. The equations of evolution for the dressed atomicpopulations are given by [29], [28]

π1 = −π1Γ1→2 + π2Γ2→1, (8.6a)

π2 = −π2Γ2→1 + π1Γ1→2, (8.6b)

where π1, π2 represent the populations on the groundand excited dressed atomic states, respectively, Γ1→2 =∣∣∣〈1, n|H1, |2, n− 1〉

∣∣∣2

= γ− s4 is the transition rate for

∣∣∣1, n〉 →∣∣∣2, n− 1〉 and Γ2→1 =

∣∣∣〈2, n|H1, |1, n− 1〉∣∣∣2

=

γ+ c4 is the transition rate for

∣∣∣2, n〉 →∣∣∣1, n− 1〉 . The

steady state solution of these equations is

πst1 =

Γ2→1

Γ1→2 + Γ2→1=

γ+c4

γ−s4 + γ+c4, (8.7a)

πst2 =

Γ1→2

Γ1→2 + Γ2→1=

γ−s4

γ−s4 + γ+c4. (8.7b)

As shown in [17], in the case of negative detunings of

FIG. 15: Bare Atom and Dressed Atom States Diagram.

the atomic frequency with respect to the frequency ofthe laser field, the condition to preferentially populatethe ground state |1〉 is

Γ1→2 < Γ2→1 =⇒ γ+ c4 > γ− s

4 . (8.8)

and the threshold Rabi frequency is given by

ǫthr

|∆AL|=

4√γ+γ−√

γ+ −√γ−

. (8.9)

In the secular approximation, the dressed state picturerenders the analysis of the spectral characteristics of thescattered radiation highly intuitive and simple. Here, wefollow the analysis presented in Ref. [29] and extend it tothe case of a frequency dependent photonic reservoir. Byusing the quantum regression theorem, it can be shownthat the spectrum of the scattered radiation is symmetri-cal about the laser field frequency ωL and exhibits threelines centered on the Mollow triplet frequencies ωL − 2Ω,ωL, ωL + 2Ω. The central line appears as a result ofthe transitions |1, n〉 → |1, n − 1〉 and |2, n〉 → |2, n − 1〉(which occur at a rate Γ1→1 = Γ2→2 = γ0 c

2s2) and itsstructure is given the sum of an elastic component,

Gel(ω) = γB δ(ω − ωL), (8.10)

with a weight of γB and an inelastic component, centeredat ωL, with a weight γA and a FWHM of 2Γpop.

Gin(ω) = γA1

π

Γpop

[ω − ωL]2

+ Γ2pop

. (8.11)

18

Here, 1/Γpop is the time constant that governs the tran-sient regime of the dressed atomic populations, and isgiven by

Γpop = Γ1→2 + Γ2→1 = γ−s4 + γ+c

4. (8.12)

The total weight of the central line is defined as numberof photons emitted per second in this line [29] and iscalculated using the dressed state diagram in Fig. 15:

γ(A+B) = πst1 · Γ1→1 + πst

2 · Γ2→2 (8.13)

Further calculations [29], [28] show that the weight ofelastic line takes the form

γB = Γ1→1

(πst

1 − πst2

)2= Γ2→2

(πst

2 − πst1

)2, (8.14)

while the maximum value of the inelastic component ofthe central line is given by

Gin(ωL) =γ A

π

1

Γpop=

2Γ1→1

π

πst1 · πst

2

Γ1→2 + Γ2→1(8.15)

It is thus clear that the dependence of the atomic pop-ulation on the Rabi frequency will leave their mark onthe dependence of the characteristics of the scattered ra-diation spectrum on the laser field intensity (the weightsand widths of the spectral components are determined byatomic population and dressed-states transition rates).

In the case of positive atomic detunings, we obtain amonotonic dependence of the peak height of the inelas-tic component of the central line on the Rabi frequency,shown in Fig. 17, whereas in the case of the negative de-tunings, presented in Fig. 16, the peak height of the in-elastic component of the central line has a resonance-likeshape, associated with the ability of the atomic systemto reach positively inverted states.

Due to the non-Markovian interaction between theatomic system and the frequency dependent reservoir,even in the case of exact resonance between the atomicfrequency in the medium ωA and the band edge frequencyωr, a small part of the radiation spontaneously emittedby the atomic system (about 2.5% for a frequency depen-dent reservoir characterized by γc = γ/100 and h = 0.01)falls in the spectral range of the pseudo-gap (see Fig. 7),which is negatively detuned from the band edge fre-quency ωr. We suggest that this contribution gives riseto the characteristic behavior of maximum value of theinelastic component of the central line Gin(ωL) as a func-tion of Rabi frequency shown in Fig. 14. We infer fromFig. 16 and Fig. 17, that the magnitude of this maxi-mum value for negative detunings, is much larger thanin the opposite regime of positively detuned atomic fre-quencies (in a ratio of about 1000). Thus, even if only asmall percentage of the characteristic atomic radiation isnegatively detuned, its contribution to the total emittedscattered radiation is still important.

FIG. 16: The peak height H0max of the central component of

the scattered spectrum (in arbitrary units) as a function ofthe Rabi frequency ǫ/|∆AL| for |∆AL|/γ+ = 3, sign(∆AL) =−1, and the magnitude of the jump in the photonic DOS,γ−/γ+, in the absence of the dipolar dephasing, γp/γ+ = 0.The atomic resonant frequency is detuned negatively from thelaser field frequency ∆AL = ωA − ωL < 0.

FIG. 17: The peak height of the inelastic component of thecentral line Gin(ωL) as a function of ǫ/|∆AL| for positive de-tuning of the atomic frequency ωA with respect to the laserfield frequency ωL, sign(∆AL) = 1 and the ratio of the decayconstants γ−/γ+ for γ0/γ+.

IX. QUANTUM OPTICAL BLOCH EQUATIONS

As it has been shown in the previous sections, theMollow spectrum for resonance fluorescence in vacuumis derived directly from the dipole-approximation Hamil-tonian, HD (Eq. (3.16)), with no additional assumptions,such as the Born-Markov approximation. Therefore, theOptical Bloch equations should also be derived directlyfrom the Hamiltonian HD, without any additional ap-proximations. In this section, we derive general form ofthe quantum optical Bloch equations. In order to sim-plify the calculations, we limit ourselves to the free space

19

case.In the ordinary vacuum, the Hamiltonian HD can be

cast in the form

HD =1

2ωA (σ3 + 1) − i

∫ ∞

−∞dx c†(x)

∂

∂xc(x)

+√γ

∫ ∞

−∞dx δ(x) [c(x)σ+ + σ−c

†(x)].(9.1a)

Since in ordinary vacuum ω = k, we introduce inEq. (9.1a) the radiation operators

c(x) =

∫ ∞

−∞

dk

2πc(k) exp (ikx). (9.1b)

The Heisenberg equations of motion are then found tobe

i

(∂

∂t+

∂

∂x

)c(x, t) = −√

γ δ(x)σ−(t), (9.2a)

id

dtσ−(t) = ωA σ−(t) +

√γ σz(t) c(0, t), (9.2b)

id

dtσ3(t) = 2

√γ

[c†(0, t)σ−(t) − σ+(t) c(0, t)

].(9.2c)

These equations describe particles which propagate tothe positive direction of the x axis and are scattered by atwo-level atom located at the point x = 0. The negative xcorrespond to incoming dipole photon, while the positivex - to outgoing one.

From the Maxwell equation (9.2a), we can infer thatthe radiation field operator is a discontinuous function ofthe coordinate x, and hence this operator at the atom(assumed a point-like system) position,

c(0, t) =

∫ ∞

−∞dx δ(x) c(x, t), (9.3)

is not well defined. To correct this unphysical behavior,we replace the δ-function by a smooth function, δǫ(x),and set δǫ(x) → δ(x) only after Eqs. (9.2) have beensolved. This physical redefinition results in

c(0, t) =1

2[c(0+, t) + c(0−, t)], (9.4a)

i.e., the discontinuous field at the atom position is foundto be the semi-sum of the fields at the points x → ±0.Thus, the atom interacts with both incoming and outgo-ing components of the field. The jump of the field at thepoint x = 0 is found by integrating Eq. (9.2a) over aninfinitely small interval around the point x = 0,

i[c(0+, t) − c(0−, t)] = −√γσ−(t). (9.4b)

In the scattering problem of a given incident field, theoperator of the incoming photon, c(0−, t), is associatedonly with the incident field at the point x = 0, i.e.,c(0−, t) = cin(t). Its time evolution does not dependon the state of the atom, and is completely determined

by the state of the incident field. Similarly, the opera-tor of outgoing dipole photon, c(0+, t), is associated withthe out-field of the scattering problem, c(0+, t) = cout(t).Correspondingly, the difference of these operators definesthe scattered field,

cscatt(t) = cout(t) − cin(t) = i√γσ−(t), (9.5a)

while the total field at the point x = 0 involved in theBloch equations (9.2b) and (9.2c) is found to be

c(0, t) =1

2[cout(t) + cin(t)] = c(t) +

i

2

√γσ−(t). (9.5b)

We note that the operators of the out- and scatteredfields at a given moment of time t are determined by timeevolution of atomic operators for all previous moments oftime. Therefore, in the Bloch equations determining timeevolution of the atomic operators, the atomic and fieldoperators on the r. h. s. must be ordered as in Eqs.(9.2b)and (9.2c). Inserting then Eq. (9.5b) into Eqs.(9.2b) and(9.2c) we obtain

id

dtσ−(t) = (ωA − iγ/2)σ−(t) +

√γσ3(t)cin(t), (9.6a)

d

dtσ3(t) = −γ[1 + σ3(t)] − 2i

√γ

[cin(t)†σ−(t) − σ+(t)cin(t)

], (9.6b)

where we used σ3(t)σ−(t) = −σ−(t) and σ+(t)σ−(t) =12 [1 + σ3(t)].

Equations (9.6) are quantum optical Bloch equations,because the external field is represented here by the oper-ator cin(t) rather then its expectation value. These equa-tions are valid for an arbitrary given quantum state of theincident field. Similar to the classical optical Bloch equa-tions [2], they allow us to derive a closed set of equationsfor the correlation functions of atomic operators, and toobtain the Mollow spectrum of fluorescence. However, inthis case, the Mollow expression for the spectrum is in anoperator form and is valid for an arbitrary statistics of anincident laser field. To obtain the fluorescence spectrumfor a given quantum state of the incident field, one hasto average the operator Mollow spectrum over this quan-tum state. If and only if the incident light is in a coherentstate, the averaging simply results in the replacement ofthe operator cin(t) by its expectation value, Ein(t), andthis way we derive the standard Mollow expression forfluorescence spectrum. In all other cases, the averagingcan dramatically change the analytical structure of theMollow expression, and may lead to a strong dependenceof the fluorescence spectrum on quantum-statistical prop-erties of incident light.

X. CONCLUSIONS

In this paper, we have extended the multi-photon the-ory of resonance fluorescence to colored vacua. Our

20

approach is valid for an arbitrary driving field inten-sity, and an arbitrary photonic reservoir bath. This ap-proach is not restricted to weak coupling between atomand reservoir required by the perturbational methods[31], and can describe a general non-Markovian system.The only restriction of our theoretical framework is that,strictly speaking, it requires either an isotropic or an one-dimensional dispersion relation for the photonic reservoir.

For a strongly colored vacuum, the scattering crosssection becomes strongly non-Lorentzian, reflecting thenon-exponential character of the atomic dipole decay. Asthe intensity of the driving field increases, the scatteredspectrum suffers a global narrowing (the position of thesidebands components of spectrum is shifted towards thecentral component) and a local narrowing (the width ofthe individual components considerably decreases). Us-ing a qualitative analysis (see Sec. VIII), we argue thatsome features present in the spectral characteristics of thescattered radiation can also be associated with switchingto a positive inversion of the atomic system described inRefs. [16], [28].

Our formalism also enables derivation of a quantum

version of the optical Bloch equations (for ordinary vac-uum case, see Sec. IX), without recourse to the Born-Markov approximation. They represent an extension oftheir classical counterparts (here, the driving field is rep-resented by its creation and annihilation operators ratherthan its expectation value), and provide a relationshipbetween the quantum statistical properties of incidentand scattered light. To obtain the fluorescence spectrumfor a given quantum state of the incident field, one hasto average the operator Mollow spectrum over this quan-tum state. If and only if the incident light is in a coherentstate, the averaging simply results in the replacement ofthe incident field operators by their expectation values,and this results in the standard Mollow expression for thefluorescence spectrum. In all other cases, the averagingcan dramatically change the analytical structure of theMollow expression, and may lead to a strong dependenceof the fluorescence spectrum on quantum-statistical prop-erties of incident light.

The strength of our approach is given by the simplicityof its results (the scattered density of the scattered field isobtained by simple integration over analytical functionsdependent on the “atomic form-factor” characterizingthe photonic reservoir) and by its wide range of validitygenerated by its non-perturbational character.

APPENDIX A: MULTI-PARTICLE SCATTERING

PROBLEM SOLUTION

In the case of multi-photon scattering, the n−particle

eigenstate of the Hamiltonian (3.16), |~λ〉 ≡ |λ1, . . . , λn〉,parameterized by the set of “rapidities”λj , j = 1, n;λi 6=

λj for i 6= j is defined by

HD |~λ〉 =

n∑

j=1

λj

|~λ〉, (A1a)

N |~λ〉 = n |~λ〉. (A1b)

The “rapidities”are, in turn, defined by the Bethe ansatz[32] for the multi-particle wave function [4, 6]:

|~λ〉 =

∫ +∞

−∞dτ1 . . . dτn

×∏

k<l

[1 +

iγ

h(λi) − h(λj)sign (τk − τl)

]

×n∏

j=1

[Φ(τj |λj) e

iλjτj

×(c†(τj) −

√γ

h(λj)δ(τj)σ+

])|0〉. (A2)

Here, the Bethe factor,

∏

k<l

[1 +

iγ

h(λi) − h(λj)sign (τk − τl)

], (A3)

is induced by the photon-photon correlations in the pro-cess of the scattering of photons on the atomic system[12].

Consider now an initial state of the system, |Ψin〉,which, in the ~λ representation, has the expansion:

|Ψin〉 =

∫d~λ |~λ〉 〈~λ|Ψin〉. (A4)

This initial state evolves according to |Ψ(τ)〉 =exp (−iHτ)|Ψin〉, and its asymptotic behavior for τ → ∞is given by

|Ψout〉 ≡ limτ→∞

|Ψ(τ)〉 =

limτ→∞

∫d~λ |λ〉 〈~λ|Ψin〉 exp (−iτ

n∑

j=1

λj). (A5)

In the following, we consider the asymptotic behavior(τ → ∞) of a specific multi-particle state that initially(τ → −∞) has the form

|In, N〉 =1√N !

[ak]N |0〉

=1√N !

N∑

n=0

CnN βn/2

[A†

k

]N−n [c†(ω)

]n |0〉,(A6)

(here, CnN = N !/n!(N − n)! are the binomial coefficients

and Ak, β are defined in (3.15)) and describes an incidentpulse containing N plane wave photons.

Due to the specific form of the interaction Hamiltonian(8.1c), only the dipole photons interact with the atomic

21

system. Therefore, we analyze the scattering of n dipolephotons in an initial state

|Ψin, n〉 =1√n!

∫d~τ ′Ψin(~τ ′)

n∏

j=1

c†(τ ′j)|0〉, (A7)

where ~τ ′ = (τ ′1, . . . , τ′n). In the case of steady-state res-

onance fluorescence, the initial multi-photon wave func-tion is given by

Ψin(~τ ′) =n∏

j=1

exp (iωLτ′j). (A8)

Since the integrand in Eq. (A7) is symmetric with respectto permutations of particle coordinates, the in-state canbe represented as

|Ψin, n〉 =√n!

∫d~τ ′θ(τ ′1 > . . . > τ ′n)Ψin(~τ ′)

n∏

j=1

c†(τ ′j)|0〉,

(A9)where the integration is carried out over the ordered par-ticle coordinates, τ ′1 > . . . > τ ′n.

After scattering, the out-state of the n scattered dipolephotons can be parameterized in an analogous way,

|Ψout, n〉 =√n!

∫d~τ θ(τ1 > . . . > τn)

×Ψout(~τ)

n∏

j=1

c†(τj)|0〉. (A10)

Similar to the one-particle scattering case, we use theexpansions

n∏

j=1

c†(τj)|0〉 τ→∞=

n∏

j=1

∫ ∞

−∞

dλj

2π

∫

Dj

dτ ′j eiλj(τ

′j−τj)

[1 + ϕ(λj)] c†(τ ′j)|0〉, (A11)

where the scattered amplitude ϕ(λj) is defined byEq. (4.20). The function Ψout(~τ) in Eq. (A10) becomes

Ψout(~τ) =

n∏

j=1

eiωLτj

∫ +∞

−∞

dλj

2π

∫

Dj

dτ ′j ei(ωL−λj)(τj−τ ′

j)

× [1 + ϕ(λj)] . (A12)

The impossibility of double excitation of a single atom(which eliminates two-photon scattering processes) im-plies that the integration over the coordinate of an in-cident particle τ ′j has to be carried out in the domainDj ≡ (τj−1, τj) [4], and Eq. (A12) simplifies to

Ψout(~τ) =

n∏

j=1

eiωLτjψout(τj−1, τj), (A13)

where

ψout(τj−1, τj) =

∫ +∞

−∞

dλj

2π

∫ τj

τj−1

dτ ′j ei(ωL−λj)(τj−τ ′

j)

× [1 + ϕ(λj)]

=

∫ +∞

−∞

dλj

2πi

1 + ϕ(λj)

λj − ωL + i0

×

1 − ei(ωL−λj)(τj−1−τj). (A14)

Consider now an initial state containing N plane wavephotons, |In, N〉. From these N plane wave photons, onlyn (the dipole photons) interact with the atomic system.Using the definitions (6.1), (A6) we obtain

|Out, N〉 =1√N !

∞∑

n=0

N !

(N − n)!n!βn/2

[a†k

]N−n

⊗|Φscatt, n〉, (A15)

with |Φscatt, n〉 the unsymmetrized state vector of thescattered dipole photons. We isolate the contribution ofthe N − n plane wave photons and the contribution ofthe n dipole photons, and symmetrize them with respectto permutations of the respective photons. We obtain

|Out, N〉 =

∞∑

n=0

√N !

(N − n)!n!βn/2 |Φscatt, n〉√

n!

⊗ 1

(N − n)!

[a†k

]N−n

=

∞∑

n=0

√Cn

Nβn/2|Ψscatt, n〉

× ⊗ 1

(N − n)!

[a†k

]N−n

, (A16)

where |Ψscatt, n〉 = 1/√n!|Φscatt, n〉 is now the sym-

metrized state vector of the scattered dipole photons.

If the incident electric field is in a coherent state con-taining an average number of photons N and given by

|In〉 = eN/2∞∑

N=0

NN/2

N !

[a†k

]N

|0〉 =

∞∑

N=0

AN |In, N〉,

(A17)where

AN =

√e−N NN

N !, (A18)

22

the out-state of the field becomes

|Out〉 =∞∑

N=0

AN |Out, N〉 = eN/2∞∑

N=0

NN/2

√N !

|Out, N〉

= eN/2∞∑

N=0

NN/2

√N !

N∑

n=0

√Cn

Nβn/2|Ψscatt, n〉

⊗ 1√(N − n)!

[a†k

]N−n

= eN/2∞∑

N=0

N∑

n=0

N (N−n)/2

(N − n)!

[a†k

]N−n

⊗[

1

n!Nn/2 βn/2 |Ψscatt, n〉

]

= eN/2∞∑

N=0

N∑

n=0

N (N−n)/2

(N − n)!

[a†k

]N−n

⊗[

1

n!ρ

n/20 |Ψscatt, n〉

], (A19)

with ρ0 = β N . Now, using the identity

∞∑

N=0

N∑

n=0

F (N − n) · g(n) =

∞∑

N=0

∞∑

n=0

F (N) · g(n), (A20)

we have

|Out〉 = eN/2∞∑

N=0

[N (N)/2

N !

[a†k

]N]

⊗N∑

n=0

[1

n!ρ

n/20 |Ψscatt, n〉

]

=

[N∑

n=0

1

n!ρ

n/20 |Ψscatt, n〉

]⊗ |In〉

and Eq. (6.11) is therefore proved.

APPENDIX B: RENORMALIZED OUT-STATE

The out-state Eq. (6.11) allows us to calculate anyphysical characteristics of the radiation field. Thestationary expectation value of an arbitrary operatorQ

c†(τ), c(τ)

is found by averaging over the out-state,

Q ≡ 〈Out|Q c†, c |Out〉 = e−N∞∑

n=0

∞∑

m=0

ρn/20 ρ

m/20√

m!n!

×∫d~τ

∫d~τ ′ Ψ∗

scatt(~τ′)Ψscatt(~τ)

×〈0|e√

Nak

m∏

l=1

c(τ ′l )Qc†, cn∏

j=1

c†(τj)e√

Na†

k |0〉,

where we have substituted the explicit form of the out-state Eq. (6.11) (and also Eq. (6.10)). In such an evalu-

ation, we need to estimate first the expression:

Q ∼ 〈0|e√

Nak

m∏

l=1

c(τ ′l )Qc†, cn∏

j=1

c†(τj)e√

Na†

k |0〉.

Here, the operators appearing in the exponents are gener-ated by the in-component of the out-state (see out-statedefinition (6.11)) and correspond to transmitted plane-wave photons.

As usual in the scattering theory [19], we assume thatthe incident beam has a finite cross-section S0 and thatmeasurements of physical observables (in our case, Q)are carried outside this section. Therefore, the measuredquantities do not include contributions from the trans-mitted components. Following the procedure outlined in[6], Eq. (B1) becomes

Q ∼ 〈0|m∏

j=l

[c(τ ′l ) +√ρ0e

iωLτ ′j ]Qc†, c

n∏

j=1

[c†(τj) +√ρ0e

−iωLτj ]|0〉, (B1)

and we note that the transmitted components result inan effective shift of the scattered photon operators bythe classical value proportional to the intensity of theincident light. For simplicity, we renormalize the out-state of the problem in the form[49]

|Out〉 = |0〉 +

∞∑

n=1

(ρ0)n/2

∫d~τ

n∏

j=1

ψ(τj−1 − τj)

×[eiωLτjc†(τj) +

√ρ0

]|0〉, (B2)

which verifies

〈Out|Qc†, c |Out〉 = 〈Out|Qc†, c |Out〉 (B3)

Since the renormalization procedure does not change theexpectation values of the physical observables, hereafterwe drop the tilde from the out-state.

To compute expectation values, it is necessary now toopen the brackets in Eq. (B2). In doing so, we assumethat the τ -coordinates of all scattered particles lie insidea finite interval between points −τ0 and +τ0, and we willset τ0 → ∞ only in expressions for expectation valuesof scattered field[50]. The out-state can be then repre-sented as a sum of terms containing different number ofscattered photons,

|Out〉 = v(2τ0)|0〉 +

∞∑

n=1

(ρ0)n/2

∫d~τv(τn + τo)

n∏

j=1

u(τj−1 − τj)eiωLτjc†(τj)|0〉, (B4)

where the function u(τ) is found as a solution of theintegral equation [6]

u(τ−τ ′) = ψ(τ−τ ′)+ρ0

∫ τ0

−τ0

dz ψ(τ−z)u(z−τ ′), (B5a)

23

while the function v(τ) is given by

v(τ) = 1 + ρ0

∫ τ

−τ0

dτ ′u(τ − τ ′). (B5b)

The Fourier transforms of these functions are found tobe

u(ω) =ψ(ω)

1 − ρ0ψ(ω)

=γ

ω[h(ω + ωL) + iγ/2] − γρ0, (B6a)

v(ω) =i

ω + i0[1 + ρ0u(ω)]

× i[h(ω + ωL) + iγ/2]

ω[h(ω + ωL) + iγ/2] − γρ0. (B6b)

As we will show in the next section, the wave functionv(τ) does not affect the expectation values of the scat-tered field, and one can set, for simplicity, v(2τ0) = 0.It is also convenient to omit hereafter the exponentialfactors exp (iωLτj), assuming that all the frequencies inthe problem are measured from the laser frequency ωL.Then, we finally obtain

|Out〉 =

∞∑

n=1

(ρ0)n/2

∫ ∞

−∞d~τv(τn + τ0)

×n∏

j=1

u(τj−1 − τj)c†(τj)|0〉. (B7)

The equations (B6) and (B7) solve completely the multi-particle scattering problem in frequency dependent pho-tonic reservoirs.

APPENDIX C: CALCULATION OF THE

SPECTRAL DENSITY OF THE SCATTERED

FIELD

The simplifications made concerning the function v(τ)do not affect expectation values in the case of steady-state resonance fluorescence (τ0 → ∞),

Q =〈Out|Qc†, c|Out〉

〈Out|Out〉 , (C1)

but they can change essentially the norm of the out-state.Therefore, we need first to evaluate

N = 〈Out|Out〉 =

∞∑

n=1

ρn0

∫d~τ V (τn + τ0)

×n∏

j=1

U(τj−1 − τj), (C2)

where we introduced U(τ) ≡ |u(τ)|2 and V (τ) ≡ |v(τ)|2.In deriving Eq. (C2) we took into account that, because

the wave function u(τ) vanishes for τ < 0, only directpairings of the photon operators, 〈0|c(τ ′j′)c†(τj)|0〉 with

j = j′, have a non-vanishing contribution (the contri-butions of indirect pairings with j 6= j′ containing thefunctions u(τ) with negative arguments vanish).

The Fourier images of the functions U(τ) and V (τ) aregiven by

U(ω) =

∫ ∞

0

dτ U(τ) eiωτ =

∫ ∞

−∞

dω′

2πu∗(ω′ − ω)u(ω′).

(C3)

V (ω) =

∫ ∞

0

dτ V (τ) eiωτ =

∫ ∞

−∞

dω′

2πv∗(ω′ − ω) v(ω′).

(C4)Equation (C2) yields

N =

∫dω

2πV (ω)H(ω) e−2iωτ0 , (C5a)

where

H(ω) =

∞∑

n=1

ρn0 [U(ω)]n =

ρ0U(ω)

1 − ρ0 U(ω). (C5b)

Since the norm of the out-state is canceled in the expres-sions for expectation values, we do not need to do anyfurther computations in Eqs. (C5). However, the spectraldensity of the scattered field will depend on the factor

H =

∫dω

2πH(ω) e−iωτ0 . (C6)

Making use of the explicit expression of the functionu(ω), we find

U(ω = 0) =

∫ ∞

−∞

dω

2πu∗(ω′)u(ω′)

=

∫ ∞

−∞

dω

2πi

1

ω − i0[u∗(ω′) − u(ω′)] =

1

ρ0, (C7a)

for an arbitrary function h(λ). This shows that the func-tion H(ω) has a pole at the point ω = 0 i.e., 1−ρ0U(ω =0) = 0. Therefore, for ω → 0 the denominator of thefunction H(ω) can be represented as

1 − ρ0U(ω) = −i(ω + i0)F (ω), (C7b)

where F (ω = 0) 6= 0. As τ0 → ∞, the value of H isdetermined only by the contribution of the pole at ω = 0,and we obtain

H =1

F0, (C8)

where F0 ≡ F (ω = 0). By expanding in series the func-tion U(ω) in the vicinity of ω = 0, the normalizationfactor F0 becomes

F0 = i ρ0

∫ ∞

−∞

dω′

2πU (1)(ω′), (C9)

24

with U (1)(ω′) given by

U (1)(ω′) = u(ω′)

[du∗(ω)

dω

]

ω=ω′

. (C10)

We now compute the correlator

G(τ, τ ′) =1

N 〈Out|c†(τ)c(τ ′)|Out〉, (C11)

whose Fourier image over the difference of coordinatesgives the spectral density of the scattered field per unitof time, and where the |Out〉 state is given by Eq. (B7).

Apart from pairings of the “internal” operators in-volved into the out-state, Eq. (C11) contains also twopairings between the “external” operators c(τ ′) andc†(τ), and internal operators, 〈0| c(τ ′l ) c†(τ) |0〉 = δ(τ−τ ′l )and 〈0| c(τ ′) c†(τj) |0〉 = δ(τ − τj). There are two differ-ent types of contributions corresponding to the externalpairings. The first type contains only one of two externalpairings, l 6= j, while in the second one both externalpairings are involved, l = j. The correlator G(τ, τ ′) isthen represented as G(τ, τ ′) = G(1)(τ, τ ′) + G(2)(τ, τ ′).We demonstrate this fact for the case of a small numberof scattered particles. For example, in the case n = 4, wehave:

G(1)n=4(τ, τ

′) =ρ40

N

∫dτ1 dτ2 dτ3V (τ3 + τ0)

×Γ(1)(τ ′|τ3, τ2) Γ(1)(τ |τ2, τ1)U(τ0 − τ1),

(C12)

where

Γ(1)(τ ′|τ3, τ2) = u(τ ′ − τ3)u(τ2 − τ ′)u∗(τ2 − τ3),

(C13a)

Γ(1)(τ |τ2, τ1) = u∗(τ − τ2)u∗(τ1 − τ)u(τ1 − τ2).

(C13b)

Adding the contributions from all diagrams containingthe contribution of the first type, we find in the Fourierrepresentation

G(1)(τ − τ ′) =ρ30

F0

∫dω

2π

Γ(1)(ω) Γ(1)∗(−ω)

1 − ρ0 U(ω)e−iω(τ−τ ′),

(C14a)where

Γ(1)(ω) =

∫dω′

2πu∗(ω′)u(ω′)u(ω′ + ω). (C14b)

Here, the term F−10 is generated by the summation of

all diagrams between the points τ and τ0, whereas thesum of diagrams between the points −τ0 and τ ′ givesthe norm of the out-state N , which cancels the normin the denominator of Eq. (C12). The summation ofcontributions between the points τ ′ and τ results in thedenominator 1 − ρ0U(ω) in Eq. (C14a).

Equation (C14a) is valid only for τ > τ ′. By repeatingthe calculations for the case τ < τ ′, we find the totalexpression for G(1)(τ−τ ′) valid for an arbitrary differenceof the coordinates,

G(1)(τ − τ ′) =ρ30

F0

∫dω

2π

[Γ(1)(ω)Γ(1)∗(−ω)

1 − ρ0U(ω)

+Γ(1)(−ω)Γ(1)∗(ω)

1 − ρ0U(−ω)

]e−iω(τ−τ ′). (C15)

Let us consider now the contributions of the secondtype, which contain both of the external pairings. In thecase of n = 3, we obtain

G(2)n=3(τ, τ

′) =ρ30

N

∫dτ1dτ2V (τ2 + τ0)

×Γ(2)(τ, τ ′, |τ2, τ1)U(τ0 − τ1), (C16a)

where

Γ(2)(τ, τ ′, |τ2, τ1) = u(τ ′ − τ2)u∗(τ − τ2)

×u(τ1 − τ ′)u∗(τ1 − τ). (C16b)

Summarizing all second type contributions that containΓ(2), we find

G(2)(τ − τ ′) =ρ20

F0

∫dω

2πΓ(2)(ω)e−iω(τ−τ ′), (C17a)

where

Γ(2)(ω) =

∫ ∞

−∞

dω′

2πu∗(ω′)u(ω′)u∗(ω′ + ω)u(ω′ + ω).

(C17b)The spectral density of scattered field is then found to be

G(ω) =

∫ ∞

−∞dτG(τ)eiωτ =

ρ30

F0

[ρ−10 Γ(2)(ω)

+Γ(1)(ω)Γ(1)∗(−ω)

1 − ρ0U(ω)+

Γ(1)(−ω)Γ(1)∗(ω)

1 − ρ0U(−ω)

]. (C18)

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[45] R. Wang and S. John, Phys. Rev. A 70, 043805 (2004).[46] For example, the photonic DOS can change by a factor of

100 on a frequency interval ∆ω = 10−n · ωC . Here ωC isthe band-edge frequency and n is a coefficient determinedby the structure length (empirically[9], n = L/5 + 1,where L is the number of unit cells in the sample).

[47] The time interval is chosen to be ∆t = 0.95∆/(c√

3).These parameters are within the stability bound [42]. Forlight propagating in vacuum, more than 20 grid pointsper wavelength, i.e., ω ≤ 0.5(2πc/a) if ∆ = 0.1a, willlead to an intrinsic grid velocity anisotropy to be lessthan 0.2%. The numerical wave propagating in vacuumwill grow exponentially when ∆t > ∆/(c

√3).

[48] As shown in Appendix C, the correlation function G(τ)can be represented as a sum of two contributions, corre-sponding to different pairings of the operators appearingin the evaluation of the spectral density of the scatteredfield (see Sec. C).

[49] The renormalization procedure presented here is neces-sary, since one of our goals is to compare the resultspredicted by the multi-photon scattering theory to theresults obtained using the more traditional optical Blochequations approach (OBE). In the OBE approach (see[33] for a detailed review of OBE picture of the reso-nance fluorescence) one eliminates the transmitted field(this elimination proceeds in a straightforward manner,because the incident field is assumed to be a classicalvariable), and evaluates only the fluorescent spectrumscattered by the atom.

[50] This approach allows us to avoid computational artifactsusually encountered in multi-particle scattering theories[11], caused by the non-commutation of limit and inte-gration operations in Eq. (B1).