Quantum theory of light scattering in a one-dimensional channel: Interaction effect on photon statistics and entanglement entropy

# Quantum theory of light scattering in a one-dimensional channel: Interaction effect on photon statistics and entanglement entropy

Mikhail Pletyukhov and Vladimir Gritsev Institute for Theory of Statistical Physics and JARA – Fundamentals of Future Information Technology, RWTH Aachen, 52056 Aachen, Germany
Institute for Theoretical Physics, Universiteit van Amsterdam, Science Park 904, Postbus 94485, 1098 XH Amsterdam, The Netherlands
###### Abstract

We provide a complete and exact quantum description of coherent light scattering in a one-dimensional multi-mode transmission line coupled to a two-level emitter. Using recently developed scattering approach we discuss transmission properties, power spectrum, the full counting statistics and the entanglement entropy of transmitted and reflected states of light. Our approach takes into account spatial parameters of an incident coherent pulse as well as waiting and counting times of a detector. We describe time evolution of the power spectrum as well as observe deviations from the Poissonian statistics for reflected and transmitted fields. In particular, the statistics of reflected photons can change from sub-Poissonian to super-Poissonian for increasing values of the detuning, while the statistics of transmitted photons is strictly super-Poissonian in all parametric regimes. We study the entanglement entropy of some spatial part of the scattered pulse and observe that it obeys the area laws and that it is bounded by the maximal entropy of the effective four-level system.

## I Introduction

### i.1 Overview of studies of scattering in one-dimensional channel with emitter

With current advances in experimental nanooptics, the problem of light scattering in quasi-one dimensional waveguides becomes an important cornerstone for understanding physics behind the light-matter interaction in a confined geometry. A number of recent experimental studies have been devoted to photon scattering when a single emitter is coupled to a one-dimensional (1D) scattering channel Akimov (), Astafiev (), Dayan (), coherent1 (), coherent2 (), chal1 (), chal2 (), Eichler1 (), Eichler2 (). The focus of these studies is made on a possibility of making few-photon devices (transistors, mirrors, switchers, transducers, etc.) as building blocks for either all-photonic or hybrid quantum devices. While a number of few-photon emitters based on single molecules, diamond color centers and quantum dots are available nowadays Yamamoto (),1-photon-rev (), an understanding of the extreme quantum regime of a few-photon scattering in a 1D fiber or transmission line Claudon (),Imamoglu () should be supplemented by microscopic studies of scattering of a coherent light (e.g., generated by a laser driving) off an emitter in a confined 1D geometry. This is the main motivation of the present work. In addition, it is worth mentioning that the model studied here can be derived as an effective model in a 3D scattering geometry when scattering channels are restricted to the photonic states with the lowest angular momentum values (-wave scattering).

Theoretical studies of quantum models describing light propagation in 1D geometry have been pioneered in 1980’s by Rupasov and Yudson R1 (), R2 (), RY1 (), RY2 (), Y1 (), Y2 (), Y-entropy (). They introduced and solved a broad class of the Bethe ansatz integrable one-dimensional models, and even managed to determine exactly time evolution of the certain initial states Y1 (). In the next decade these studies have extended by the other authors Le1 (), KoLe (), LLLS (), Le2 (). The exactly solvable class of models includes linearly dispersed photons interacting with a single qubit, a Dicke cluster, and distributed emitters. However, the integrability imposes a rather strict constraint – it requires the absence of backscattering thus limiting this class to the chiral, or unidirectional, models. This constraint, however, is not restrictive if a scatterer is local: transforming left- and right-propagating states of photons to the basis of their even and odd combinations, one can observe that the odd modes decouple from the scatterer and thus a model with backscattering is mapped onto an effective chiral model for even modes. In turn, to realize the physical chiral model with distributed emitters it has been recently proposed RPG () to employ scattering of edge states in topological photonic insulators. Experimentally a quantum nondemolition measurement of a single unidirectionally propagating microwave photon has been achieved in Ref. QNDChalmers () using a chain of transmons cascaded through circulators which suppress photon backscattering.

A revival of interest to problems of photonic transport in 1D geometries has been triggered in 2000’s by the progress in quantum information science, which resulted in series of publications from various groups Kojima1 (), Kojima2 (), Kojima3 (), Shen-Fan1 (), Shen-Fan2 (), Shen-Fan3 (), Shen-Fan4 (), Chang (), Nori1 (), YR (), Chang2 (), Busch (), Sorensen (), Nori2 (), Baranger1 (), Roy (), Shi-Fan-Sun (), Hafezi1 (), Baranger2 (), Hafezi2 (), Liao1 (), Liao2 (), PGnjp (), Oehri (), sanchez (), werra (), martens (), auf1 (), auf2 (). In these works, a variety of different setups have been carefully analyzed, comprising three- and four-level emitters, the nonlinear photon dispersion, effects of driving and dissipation. In addition, the recent experimental achievements Wallraff-2 () motivate a theoretical consideration of models containing both distributed emitters and the backscattering Baranger3 (), LP (), auf3 (), 2qubit-plasmon ().

### i.2 Scattering approach, role of detector

In this paper we focus on a basic model consisting of a two-level qubit coupled to a 1D channel and driven by a coherent field. We develop a complete and exact quantum description of all physical properties of this system using the scattering formalism.

A characterization of different scattering regimes in this model can be obtained by introducing parameters quantifying (i) an initial state, (ii) a qubit (iii), a waveguide, and (iv) a detector. Throughout the paper we assume that the initial state is a pulse of the spatial length . In the units , where is the group velocity of linearly dispersed photons in a waveguide, the parameter defines one of the important energy scales – the wavepacket width in the -space. Another energy scale is given by the qubit relaxation rate , where is a photon-qubit interaction strength. In addition, we have a dimensionless parameter characterizing the mean number of photons in the initial pulse. In terms of these parameters, one can distinguish three different regimes in this problem: (a) ; (b) ; (c) . Note that in all cases we assume meaning the long- (or narrow bandwidth) pulse’s limit.

The regime (a) was studied back in 1970’s in connection with the resonance fluorescence phenomenon Mollow () (see also the review KM-rev ()). In this regime, a semiclassical description of the laser beam is sufficient.

In contrast to the regime (a), in the regime (c) single-photon (elastic) processes dominate, while a contribution of many-photon (inelastic) processes to the scattering outcome is weak; the most remarkable inelastic effect in this regime is, perhaps, a formation of the two-photon bound state. Various aspects of the regime (c) have been recently studied in numerous publications cited above.

To complement the previous studies, we wish to achieve a comprehensive understanding of the crossover regime (b), where the mean number of scattered photons is already large, but the Rabi frequency is still much smaller than . To this end, we apply the quantum scattering approach developed in our earlier paper PGnjp (). It will be shown in the following that it is eventually capable to cover all three regimes, thereby establishing a theoretical platform for studying the classical-to-quantum crossover in this model.

In addition to the system related parameters, our approach can accommodate information contained in the detection protocol as shown in Fig. 1. A pulse of the spatial length (shown in pink) is injected into the waveguide at time and the coordinate (recall that ). Due to the linear dispersion, it moves without changing its shape toward the qubit coupled to the waveguide at the point . From the time instant the photons in the pulse start interacting with the qubit, and this interaction lasts approximately for a time . Subsequently the scattered pulse leaves the interaction region around (only the transmitted part is shown in blue in the figure), its shape is being modified, and at its front reaches the detector (the “eye”) at the position . To make the scattering formalism applicable we must assume that , meaning that the detector is far away from the scattering region. After the front of the scattered pulse reaches the detector at , the latter remains switched off during the waiting time , and at it switches on and starts to count transmitted photons during the subsequent time interval (counting time). Thus, the setup in Fig. 1 is characterized by the two important time scales and . In the following we will assume (which, in particular, allows us to take the limit before the limit ). This assumption is based on an instrumental possibility to create an initial wavepacket of a sufficiently narrow width in the -space; from the conceptual viewpoint we impose this condition to enable an open-system description of the scattered pulse, for which the aforementioned order of limits is essential.

The detector-related time scales together with the system-related energy scales exhaust the most relevant parameters of this problem.

### i.3 Physical observables and their generating function

Major statistical properties of a scattered light field are characterized by a collection of -point () correlation functions Glauber (). Another basic observable quantity is a transmission (reflection) spectrum. It quantifies the amplitude of the transmitted (reflected) field, and it has been measured in various experiments with microwave transmission lines Astafiev (), Dayan (), coherent1 (), coherent2 (). A more involved quantity of interest is the resonance fluorescence power spectrum, which is a Fourier transform of the field-field () correlation function. This (Mollow) spectrum is know to feature a three-peak structure Mollow (). Further experimental practice is to collect information about the density-density () correlation function using the Hanbury-Brown-Twiss setup HBT (). Higher order correlators can also (at least, in principle) be accessed in an experiment, and this motivates us to study the generating function of all moments of the density operator, being related to the -point functions. In mesoscopic physics, such a generating function is known as the Full Counting Statistics (FCS), though in the historical retrospective this concept has been originally introduced in the quantum-optical context Glauber (), MaWo () in order to characterize statistical properties of a non-interacting quantized electromagnetic field. Its fully quantum derivation has been presented in Ref. KK (). For a coherent light field driving a two-level system (the resonance fluorescence model) the FCS has been studied in Mandel (), Cook (), Lenstra (), Smirnov-Troshin (). It has been observed Mandel () that the FCS distribution of the fluorescent photons is narrower than the Poissonian distribution. An important measure of this effect is the Mandel’s -parameter, which is obtained from the first and the second factorial moments of the FCS.

Later on, the concept of the FCS has been borrowed and actively developed in the field of mesoscopic physics LL (), LLL () for studying statistical properties of electronic currents in meso- and nanoscopic devices for both non-interacting and interacting electrons NB (), Belzig (), BN (), BUGS (), GK (), Schonh (), KS (), BSS (), BBSS (), CBS (). It has been recently shown that the FCS can be very useful for characterizing classical-quantum crossover SB (), quantum entanglement TF (), KL (), LeHur (), and phase transitions IA (), FG (). The FCS of nonlocal observables can be used to quantify correlations GADP (), IGD (), Hoffer (), LF (), and prethermalization in many-body systems Kitagawa (), prethermalization (), as well as to define some kind of a topological order parameter IA2 ().

The studies of the FCS in mesoscopic physics have generated a back flow of ideas to the quantum optics community. Inspired by the recent experiments in the context of the 1D resonance fluorescence, the subject of the FCS has received the renewed attention, see deflection (), BB (), Vogl (), LJ ().

### i.4 This paper: content and results

Motivated by previous developments we revisit the original problem of computing the FCS for photons interacting with an emitter. This is the first goal of the present work. We give it a detailed quantum consideration, treating the interaction nonperturbatively. We present an exact calculation of the FCS in the basic model of light-matter interaction (see Fig. 1): a multimode propagating photonic field in 1D interacting with a two-level emitter. Since one of the objectives in nanophotonics research is to obtain strong photon nonlinearities as well as a strong photon-emitter interaction for the purpose of an efficient control over individual atoms and photons, knowledge of statistical properties of an interacting photon-emitter device becomes essential.

The geometry of our system (see Fig. 1) suggests to define the three types of the counting statistics: the FCS of transmitted photons, the FCS of reflected photons, and the FCS of the chiral model. Below we discuss this classification in more detail. The multimode nature of the waveguide implies an emergence of many-body correlations. As such, interactions modify the statistics of photons in forward and backward scattering channels in comparison with the Poissonian one of the incident coherent beam. As a by-product of our FCS computation, we revisit the transmission properties and evaluate the Mandel’s -factor exhibiting super-Poissonian, sub-Poissonian, and Poissonian statistics for the three counting models in question.

We also provide a new derivation of the Mollow spectrum based on knowledge of the exact scattering wavefunction that avoids the usage of the quantum regression theorem, and reproduces the original expression Mollow () derived for the 3D scattering geometry. It helps to understand how the resonance fluorescence can be decomposed into elementary scattering processes.

Yet another quantity that has recently received a great deal of attention due to developments in quantum information science is the entanglement entropy, see, e.g., the recent review Eisert (). While several measures of entanglement exist, the entropy of entanglement has several nice properties like additivity and convexity. In quantum information theory, the entanglement entropy gives the efficiency of conversion of partially entangled to maximally entangled states by local operations Bennett1 (), Bennett2 (). In other terms, it gives the amount of classical information required to specify the reduced density matrix. A large degree of entanglement is what makes quantum information exponentially more powerful than classical information, so states with lower entanglement entropy are less complex. For extended systems of condensed matter physics it is customary to distinguish between area and volume law Eisert () behavior of the entanglement entropy. Here the notions of area or volume refer to a typical geometric measure of a region bounded by a subsystem with respect to the rest of a system. Thus in 1D case, relevant for our discussion here, the area of an interval consists of just two end points, while the volume is a length of the interval of the subsystem . Systems with volume law behavior entanglement possess much higher potential for applications in quantum simulations and computing. It was shown Eisert () that in most cases a quantum ground state wave function of gapped systems exhibits the area law, while typical excited states mostly follow the volume law. An intermediate logarithmic behavior of the entanglement entropy is related to gapless systems. These features should be understood as asymptotic properties of a system, when the area and the volume of a subsystem entangled with the rest part of a system become large. Our complete knowledge of the scattering state allows us to calculate explicitly the entanglement entropy of the scattered pulse’s interval of the length (see Fig. 1) for different values of , , and system parameters. The (dimensionless) duration of the observation interval plays the role of the volume of the subsystem in this context. One of our central results in that section is a demonstration of the existence of the absolute limit for the entanglement entropy in our system: it is bounded by , the entropy of four-level system. Another important observation is that while the entanglement entropy at large asymptotically approaches the area law bounded by , it can behave very differently for small and intermediate values of – we even observe its nonmonotonous oscillatory behavior for some intermediate regime of parameters.

## Ii Definitions and approximations

We start our analysis defining the model and approximations involved in its derivation, the bosonic operators creating the initial pulse, and the FCS.

### ii.1 Theoretical model.

#### ii.1.1 Approximations and the effective Hamiltonian.

Our model is described by the Hamiltonian , where is the Hamiltonian of the free propagating photonic field, is the Hamiltonian of an emitter, and describes the field-emitter coupling. We involve approximations which are customary in quantum optics: (i) the dipole approximation for the interacting Hamiltonian; (ii) the two-level approximation for the emitter Hamiltonian; (iii) the rotating wave approximation (RWA); and (iv) the Born-Markov approximation (energy independence) for the coupling constant. In addition, we linearize the photonic spectrum around some appropriately chosen frequency which is commensurate with the emitter’s transition frequency , and extend the linearized dispersion to infinities. With these assumptions [except for (iii)] we obtain an effective low-energy Hamiltonian

 H = ∑ξ=r,l∫dk(Ω0+ξk)a†ξkaξk+Ω2σz (1) + g0∑ξ=r,l∫dk(a†ξk+aξk)(σ++σ−),

featuring the two-branch linear dispersion with right- () and left- () propagating modes. Here are expressed in terms of the Pauli matrices , . The states of the emitter are separated by the transition frequency . To implement the RWA in a systematic way, we first perform the gauge transformation with

 U=exp⎡⎣−iΩ0t⎛⎝∑ξ=r,l∫dka†ξkaξk+σz2⎞⎠⎤⎦, (2)

which leads us to the Hamiltonian

 H = ∑ξ=r,l∫dkξka†ξkaξk+Δ2σz (3) + g0∑ξ=r,l∫dk(a†ξkσ−+aξkσ+) + g0∑ξ=r,l∫dk(a†ξkσ+e2iΩ0t+aξkσ−e−2iΩ0t),

where . As soon as , the time-oscillating terms in (3) can be treated as a time-dependent perturbation. In zeroth order it is simply neglected, which is equivalent to the RWA. Note that this approximation is consistent with the assumption about the absence of lower and upper bounds in the linearized dispersion.

#### ii.1.2 Transformation to the “even-odd” basis.

Due to energy independence of the coupling constant , one can decouple the Hilbert space of the model defined in (3) into two sectors. To this end, one introduces even (symmetric) and odd (antisymmetric) combinations of fields corresponding to the same energy

 aek=ark+al,−k√2,aok=ark−al,−k√2. (4)

By virtue of this canonical transformation the Hamiltonian (3) turns into a sum of the two terms, , defined by

 He = ∫dk[ka†ekaek+g(a†ekσ−+aekσ+)]+Δ2σz, Ho = ∫dkka†okaok, (5)

where . Note that the odd Hamiltonian is noninteracting, and therefore odd modes do not scatter off a local emitter (). The even Hamiltonian can be interpreted in terms of a chiral model with a single branch of the linear dispersion.

A similar decomposition can be applied to an initial state. Both even and odd photons are labeled by a momentum value lying on a single branch of the linear dispersion.

### ii.2 Definitions of wave packet field operators and the initial state.

In order to define the incident coherent state we need field operators annihilating/creating states which are normalized by unity. The field operators and , which are the Fourier transforms of and , do not fulfill this requirement, as they obey the commutation relation (in other words, they annihilate and create unnormalizable states). To circumvent this difficulty, we construct wavepacket field operators

 b†k = 1√L∫L/2−L/2dxa†(x)eikx, (6)

which do satisfy the desired commutation relation . For example, the operators and create wavepackets which are centered around of the right (left) branch of the spectrum and broadened over the width . In the coordinate representation, they create states which are spatially localized on a finite interval of the length . We note the identity . In the following, denotes the laser driving frequency (measured from the linearization point).

Having introduced , we define the initial (incoming) state , where the incident right-moving photons are prepared in the coherent state , and the two-level emitter is initially in the ground state . Here  denotes the photonic vacuum, and is the coherent state displacement operator.

The mean number of photons in the state is given by . We also quote a useful relation

 D†r(α0)ar(x)Dr(α0)=ar(x)+α0eik0x√LΘ(L/2−|x|), (7)

which follows from the commutation relation between and operators.

The initial coherent state defined in the original right-left basis admits a decomposition into the product state in the even-odd basis

 |α0⟩r = eα0b†r,k0−|α0|2/2|0⟩ = e(α0/√2)b†e,k0−|α0|2/4e(α0/√2)b†o,k0−|α0|2/4|0⟩ ≡ De(α)Do(α)|0⟩≡|α⟩e⊗|α⟩o,

where , and the displacement operators are defined using the mutually commuting operators and , respectively. Importantly, , i.e. the incoming state is properly normalized.

The major consequence of the even-odd decoupling is a factorization of the scattering operator into the product , where is the identity operator, and can be studied in the context of the effective one-channel chiral model described by . Scattering in chiral models has been studied by us in Ref. PGnjp () for arbitrary initial states, including the coherent state. In particular, we have established the explicit form of the operator in the latter case, which provides the full information about the scattering wavefunction. Here we take over this result and use it for calculation of observables announced in the Introduction. All expressions necessary for this purpose are quoted below for readers’ convenience.

### ii.3 Definition of the full counting statistics

The statistics of the initial field, defined by the probability to find photons in the mode , is given by the Poissonian distribution for the coherent field, with the mean value . Due to the photonic dispersion and inelastic scattering processes photons can leak from the right-moving mode to other modes on both branches of the spectrum by virtue of scattering processes, thus modifying the photon statistics. A fraction of photons is reflected, and their statistics is also of great interest. We propose a calculation of the FCS in both forward and backward scattering channels, which is exact and nonperturbative in both and .

Generally speaking, the FCS can be defined as a generating function associated with a probability distribution to detect photons in some given state. The function generates -th order moments of the distribution which are determined by evaluating the -th derivative with respect to at . The Fourier expansion of the -periodic function yields power series in terms of the “fugacity” , . The normalization of a probability distribution implies . The expansion around gives the factorial moments of a distribution .

To define the photon FCS we need the two main constituents: (i) the scattering (outgoing) state necessary to perform the average, and (ii) a meaningful and experimentally measurable counting operator . In particular, as such we can choose the number of transmitted photons which pass through the detector during the time (see the Fig. 1). Because of the linear dispersion, the same operator characterizes the number of photons in the spatial interval of the length viewed in the frame co-moving in the right direction with the velocity . Introducing the coordinate system in this co-moving frame such that the pulse’s front has the coordinate value , we define the photon number operator

 Nr,τ=∫z2z1dxa†r(x)ar(x) (8)

of transferred photons which appear in the spatial interval . Here (we note that the tail of the scattered pulse extends to , see below; however, we will focus on counting intervals ). The corresponding FCS reads

 Fr,τ(χ0)=⟨eiχ0Nr,τ⟩. (9)

In the chosen coordinate system, the waiting time is expressed by .

In the same setting one can consider the FCS of the chiral model.

To define the FCS for reflected photons, one can put the second detector at and consider the co-moving frame with the velocity , defining in it the counting operator via .

## Iii Method of computing observables based on exact scattering matrix

### iii.1 Scattering of the coherent state in the chiral model

As we discussed above, we need to know an expression for the scattering state in the effective chiral model for the even modes. In this subsection we quote the result of Section 5 in Ref. PGnjp () for the coherent light scattering in the chiral model. In particular, we copy the Eq. (134) from this reference, adapting notations therein to the present paper. The subscript is also omitted in the following.

Thus, for the incoming coherent state in the mode , the outgoing scattering state amounts to , where

 S0=Sa0[L/2,−L/2]+Sb0[L/2,−L/2]λ√2ΓA†0. (10)

The operator

 A†0=√2Γ∫−L/2−∞dx0eik0x0e−i(δ+iΓ)(L/2+x0)a†(x0), (11)

describing the states in the tail of the scattered pulse, is normalized by .

In turn, the states within the initial pulse’s size are expressed via the operators

 Sa0[y,x] = 1+∞∑n=1λn∫Dxn × d0(y−xn)a†(xn)eik0xn × d0(xn−xn−1)a†(xn−1)eik0xn−1 × …d0(x2−x1)a†(x1)eik0x1, Sb0[y,x] = d0(y−x)+∞∑n=1λn∫Dxn × d0(y−xn)a†(xn)eik0xn × d0(xn−xn−1)a†(xn−1)eik0xn−1 × …d0(x2−x1)a†(x1)eik0x1d0(x1−x).

Here the parameter

 λ=−2iΓα(δ+iΓ)√L, (14)

is expressed in terms of the detuning and the relaxation rate ; is the bare single-photon propagator, and the short-hand notation has been used for the integration measure . We also consider .

For the later use we also define the following operators

 Sc0[y,x] = 1+∞∑n=1λn∫Dxn × a†(xn)eik0xnd0(xn−xn−1) × a†(xn−1)eik0xn−1d0(xn−1−xn−2)… × d0(x2−x1)a†(x1)eik0x1,
 S¯a0[y,x] = 1+∞∑n=1λn∫Dxn × a†(xn)eik0xnd0(xn−xn−1) × a†(xn−1)eik0xn−1d0(xn−1−xn−2)… × a†(x1)eik0x1d0(x1−x).

The set of operators is complete in that sense that they exhaust all possible arrangements of the bare propagators .

### iii.2 Algebra of scattering operators

The scattering state (134) of Ref. PGnjp () (equivalent of ) can be used for a computation of observable quantities. In particular, we will be interested in correlation functions , and therefore we need to know how the local annihilation operators commute with the many-body scattering operator defined on a finite spatial interval. For a systematic treatment, we observe the following algebraic properties of the scattering operators .

Let us choose an arbitrary point . Using the obvious identity

 Θ(y>xn>…>x1>x) = n∑j=0Θ(y>xn>…>xj+1>z>xj>…>x1>x),

where and , one can show by rewriting Eqs. (III.1), (III.1), (III.1), and (III.1), that the operators satisfy the following closed algebra with respect to the interval splitting operation

 Sa0[y,x]=Sa0[y,z]Sa0[z,x]+Sb0[y,z]{Sc0[z,x]−Sa0[z,x]}, Sb0[y,x]=Sa0[y,z]Sb0[z,x]+Sb0[y,z]{S¯a0[z,x]−Sb0[z,x]}, Sc0[y,x]=Sc0[y,z]Sa0[z,x]+S¯a0[y,z]{Sc0[z,x]−Sa0[z,x]}, S¯a0[y,x]=Sc0[y,z]Sb0[z,x]+S¯a0[y,z]{S¯a[z,x]−Sb0[z,x]}.

If one divides the interval into three parts by arbitrary points and , one can prove by a direct calculation that the algebra (LABEL:split_alg) is associative, as expected.

The algebra (LABEL:split_alg) also allows us to express the action of annihilation operators on the scattered state in a simple form

 a(z)Sa0[y,x]|0⟩=Sb0[y,z]λeik0zSa0[z,x]|0⟩, (19) a(z)Sb0[y,x]|0⟩=Sb0[y,z]λeik0zSb0[z,x]|0⟩, (20) a(z)Sc0[y,x]|0⟩=S¯a0[y,z]λeik0zSa0[z,x]|0⟩, (21) a(z)S¯a0[y,x]|0⟩=S¯a0[y,z]λeik0zSb0[z,x]|0⟩. (22)

For the proof of (19)-(22) we used the property .

Similarly, for an action of the ordered product of two annihilation operators , with , we obtain

 a(z2)a(z1)Sa0[y,x]|0⟩ = Sb0[y,z2]λeik0z2Sb0[z2,z1]λeik0z1Sa0[z1,x]|0⟩, a(z2)a(z1)Sb0[y,x]|0⟩ = Sb0[y,z2]λeik0z2Sb0[z2,z1]λeik0z1Sb0[z1,x]|0⟩, a(z2)a(z1)Sc0[y,x]|0⟩ = S¯a0[y,z2]λeik0z2Sb0[z2,z1]λeik0z1Sa0[z1,x]|0⟩, a(z2)a(z1)S¯a0[y,x]|0⟩ = S¯a0[y,z2]λeik0z2Sb0[z2,z1]λeik0z1Sb0[z1,x]|0⟩.

Iterating this procedure, one can find an action of the ordered product of annihilation operators , . It produces the product of -operators, what can be symbolically written as

 amSa0 → Sb0…Sb0…Sa0, (27) amSb0 → Sb0…Sb0…Sb0, (28) amSc0 → S¯a0…Sb0…Sa0, (29) amS¯a0 → S¯a0…Sb0…Sb0. (30)

Note that in all intermediate positions appears only . To classify leftmost and rightmost operators in these expressions, we introduce the mappings and according to the table

 σ(a)= b,μ(a)= a, (31) σ(b)= b,μ(b)= b, (32) σ(c)= ¯a,μ(c)= a, (33) σ(¯a)= ¯a,μ(¯a)= b. (34)

In their terms, the relations (27)-(30) acquire the compact form

 (35)

### iii.3 Dressing S-operators

In the following we will also need shifted scattering operators

 Sβv[y,x] = D†(v)Sβ0[y,x]D(v) (36) = ev∗bk0Sβ[y,x]e−v∗bk0,

where , and is the displacement operator of the fields, such that . Our next goal is to establish explicit expressions for the operators for arbitrary complex-valued parameter .

Performing the displacement (36), we obtain the new series in field operators defining . Appropriately reorganizing (re-summing) them, we find the following expressions

 Sav[y,x] = ~dv(y−x)+∞∑n=1λn∫Dxn × dv(y−xn)a†(xn)eik0xn × dv(xn−xn−1)a†(xn−1)eik0xn−1 × …dv(x2−x1)a†(x1)eik0x1~dv(x1−x),
 Sbv[y,x] = dv(y−x)+∞∑n=1λn∫Dxn × dv(y−xn)a†(xn)eik0xn × dv(xn−xn−1)a†(xn−1)eik0xn−1 × …dv(x2−x1)a†(x1)eik0x1dv(x1−x),
 Scv[y,x] = ~~dv(y−x)+∞∑n=1λn∫Dxn × ~dv(y−xn)a†(xn)eik0xn × dv(xn−xn−1)a†(xn−1)eik0xn−1 × …dv(x2−x1)a†(x1)eik0x1~dv(x1−x),
 S¯av[y,x] = ~dv(y−x)+∞∑n=1λn∫Dxn × ~dv(y−xn)a†(xn)eik0xn × dv(xn−xn−1)a†(xn−1)eik0xn−1 × …dv(x2−x1)a†(x1)eik0x1dv(x1−x),

as well as

 Sv = D†(v)S0D(v) (41) = Sav[L/2,−L/2]+Sbv[L/2,−L/2]λ√2ΓA†0,

where

 dv(x) = −p++p−p+−p−[e−ip+x−e−ip−x], (42) ~dv(x) = −p−p+−p−e−ip+x+p+p+−p−e−ip−x, (43) ~~dv(x) = −p2−p2+−p2−e−ip+x+p2+p2+−p2−e−ip−x (44) ≡ ~dv(x)+iλv∗(δ+iΓ)√Ldv(x)

are the dressed single-photon propagators, and .

Additional details on evaluation of (III.3)-(III.3) are presented in the Appendix A.

### iii.4 Factorization property

To evaluate the FCS we prove the following key property of generalized -point correlation functions: their factorization into the -fold product of the two-point functions.

Generalized correlation functions are defined by

 G(m)β′β({zl};y,x) = ⟨0|Sβ′†u[y,x](:m∏l=1a†(zl)a(zl):)Sβv[y,x]|0⟩,

where the “time-forward” and the “time-backward” scattering operators depend on different and arbitrary displacement parameters and , respectively. We also assume here that