Analysis of Quantum Linear Systems’ Response to Multi-photon States

# Analysis of Quantum Linear Systems’ Response to Multi-photon States

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###### Abstract

The purpose of this paper is to present a mathematical framework for analyzing the response of quantum linear systems driven by multi-photon states. Both the factorizable (namely, no correlation among the photons in the channel) and unfactorizable multi-photon states are treated. Pulse information of multi-photon input state is encoded in terms of tensor, and response of quantum linear systems to multi-photon input states is characterized by tensor operations. Analytic forms of output correlation functions and output states are derived. The proposed framework is applicable no matter whether the underlying quantum dynamic system is passive or active. The results presented here generalize those in the single-photon setting studied in ([Milburn, 2008]) and ([Zhang & James, 2013]). Moreover, interesting multi-photon interference phenomena studied in ([Sanaka, Resch & Zeilinger, 2006]), ([Ou, 2007]), and ([Bartley, et al., 2012]) can be reproduced in the proposed framework.

zhang]Guofeng Zhang

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong

Key words:  quantum linear systems, multi-photon states, tensors.

## 1 Introduction

Analysis of system response to various types of input signals is fundamental to control systems engineering. Step response enables a control engineer to visualize system transient behavior such as rise time, overshoot and settling time; frequency response design methods are among the most powerful methods in classical control theory; response analysis of linear systems initialized in Gaussian states driven by Gaussian input signals is the basis of Kalman filtering and linear quadratic Gaussian (LQG) control (see, e.g., [Anderson & Moore, 1971]; [Kwakernaak & Sivan, 1972]; [Anderson & Moore, 1979]; [Zhou, Doyle & Glover, 1996]; [Qiu & Zhou, 2009]).

Over the last two decades, there has been rapid advance in experimental demonstration and theoretical investigation of quantum (namely, non-classical) control systems due to their promising applications in a wide range of areas such as quantum communication, quantum computation, quantum metrology, laser-induced chemical reaction, and nano electronics ([Gardiner & Zoller, 2000]; [Loudon, 2000]; [Nielsen & Chuang, 2000]; [D’Alessandro, 2007]; [Walls & Milburn, 2008]; [Wiseman & Milburn, 2010]; [Belavkin, 1983]; [Huang, Tarn & Clark, 1983]; [Yurke & Denker, 1984]; [Gardiner, 1993]; [Doherty & Jacobs, 1999]; [Khaneja, Brockett & Glaser, 2001]; [Albertini & D’Alessandro, 2003]; [Yanagisawa & Kimura, 2003]; [Stockton, van Handel & Mabuchi, 2004]; [Mabuchi & Khaneja, 2005]; [van Handel, Stockton & Mabuchi, 2005]; [Altafini, 2007]; [Mirrahimi & van Handel, 2007]; [James, Nurdin & Petersen, 2008]; [Rouchon, 2008]; [Bonnard, Chyba & Sugny, 2009]; [Gough & James, 2009]; [Li & Khaneja, 2009]; [Mirrahimi & Rouchon, 2009]; [Nurdin, James & Doherty, 2009]; [Yamamoto & Bouten, 2009]; [Bloch, Brockett & Rangan, 2010]; [Bolognani & Ticozzi, 2010]; [Brif, Chakrabarti & Rabitz, 2010]; [Dong & Petersen, 2010]; [Gough, James & Nurdin, 2010]; [Munro, Nemoto & Milburn, 2010]; [Wang & Schirmer, 2010]; [Maalouf & Petersen, 2011]; [Zhang & James, 2011]; [Altafini & Ticozzi, 2012]; [Amini, Mirrahimi & Rouchon, 2012]; [Zhang, et al., 2012]; [Qi, 2013]). Within this program quantum linear systems play a prominent role. Quantum linear systems are characterized by linear quantum stochastic differential equations (linear QSDEs). In quantum optics, linear systems are widely used because they are easy to manipulate and, more importantly, linear dynamics often serve well as good approximation of more general dynamics ([Gardiner & Zoller, 2000]; [Loudon, 2000]; [Walls & Milburn, 2008]; [Wiseman & Milburn, 2010]). Besides their broad applications in quantum optics, linear systems have also found applications in many other quantum-mechanical systems such as opto-mechanical systems ([Massel, et al., 2011, Eqs. (15)-(18)]), circuit quantum electrodynamics (circuit QED) systems ([Matyas, et al., 2011, Eqs. (18)-(21)]), atomic ensembles ([Stockton, van Handel & Mabuchi, 2004, Eqs. (A1),(A4)]), quantum memory ([Hush, Carvalho, Hedges & James, 2013, Eqs. (12)-13]). From a signals and systems point of view, quantum linear systems driven by Gaussian input states have been studied extensively, and results like quantum filtering and measurement-based feedback control have been well established ([Wiseman & Milburn, 2010]).

In addition to Gaussian states there are other types of non-classical states, for example single-photon states and multi-photon states. Such states describe electromagnetic fields with a definite number of photons. Due to their highly non-classical nature and recent hardware advance, there is rapidly growing interest in the generation and engineering (e.g., pulse shaping) of photon states, and it is generally perceived that these photon states hold promising applications in quantum communication, quantum computing, quantum metrology and quantum simulations ([Cheung, Migdall & Rastello, 2009]; [Gheri, Ellinger, Pellizzari & Zoller, 1998]; [Sanaka, Resch & Zeilinger, 2006]; [Ou, 2007]; [Bartley, et al., 2012]; [Milburn, 2008]; [Gough, James & Nurdin, 2013]; [Hush, Carvalho, Hedges & James, 2013]). Thus, a new and important problem in the field of quantum control engineering is: How to characterize and engineer interaction between quantum linear systems and photon states? The interaction of quantum linear systems with continuous-mode photon states has recently been studied in the literature, primarily in the physics community. For example, interference phenomena of photons passing through beamsplitters have been studied, see, e.g., [Sanaka, Resch & Zeilinger, 2006]; [Ou, 2007]; [Bartley, et al., 2012]. Milburn discussed how to use an optical cavity to manipulate the pulse shape of a single-photon light field ([Milburn, 2008]). Quantum filtering for systems driven by single-photon fields has been investigated in [Gough, James & Nurdin, 2013], based on which nonlinear phase shift of coherent signal induced by single-photon field has been studied in [Carvalho, Hush & James, 2012]. Intensities of output fields of quantum systems driven by continuous-mode multi-photon light fields have been studied in [Baragiola, Cook, Brańczyk & Combes, 2012]. In [Zhang & James, 2013] the response of quantum linear systems to single-photon states has been studied. Formulas for intensities of output fields have been derived. In particular, a new class of optical states, photon-Gaussian states, has been proposed.

In the analysis of the response of quantum linear systems to single-photon states, matrix presentation is sufficient because two indices are adequate: one for input channels, and the other for output channels. However, this is not the case in the multi-photon setting. In addition to indices for input and output channels, we need another index to count photon numbers in channels. As a result, tensor representation and operation are essential in the multi-photon setting. To be specific, multi-photon state processing by quantum linear systems can be mathematically represented in terms of tensor processing by transfer functions. The key ingredient for such an operation is the following (for the passive case). Let be the transfer function of a quantum linear passive system with input channels. For each , let be an -way -dimensional tensor function that encodes the pulse information of the -th input channel containing photons. Denote the entries of by . For all given , define an -way -dimensional tensor with entries given by the following multiple convolution

 Wj,r1,…,rℓj(t1,…,tℓj) =m∑k1,…,kℓj=1 ∞∫−∞⋯∞∫−∞Er1k1(t1−ι1)⋯Erℓjkℓj(tℓj−ιℓj)Vj,k1,…,kℓj(ι1,…,ιℓj)dι1…dℓj.

It turns out that the tensors () encode the pulse information of the output field. That is, an -way input tensor is mapped to an -way output tensor by the quantum linear passive system.

The contributions of this paper are three-fold. First, the analytic form of the steady-state output state of a quantum linear system driven by a multi-photon input state is derived. When the quantum linear system is a beamsplitter (a static passive device), interesting multi-photon interference phenomena studied in ([Sanaka, Resch & Zeilinger, 2006]), ([Ou, 2007]), and ([Bartley, et al., 2012]) are re-produced by means of our approach, see Examples 1,2,3. Second, when the underlying quantum linear system is not passive (e.g., a degenerate parametric amplifier), the steady-state output state with respect to a multi-photon input state is not a multi-photon state. In terms of tensor representation, a more general class of states is defined. Such rigorous mathematical description paves the way for multi-photon state engineering. Third, both the factorizable and unfactorizable multi-photon states are treated in this paper. Here a factorizable multi-photon state is a state for which the photons in a given channel are not correlated, while for an unfactorizable multi-photon state there exists correlation among the photons. This difference cannot occur in the single-photon state case. Thus, the mathematical framework presented here is more general.

The rest of the paper is organized as follows. Preliminary results are presented in Section 2. Specifically, quantum linear systems are briefly reviewed in Subsection 2.1 with focus on stable inversion and covariance function transfer, in Subsection 2.2 several types of tensors and their associated operations are introduced. The multi-photon state processing when input states are factorizable in terms of pulse shapes is studied in Section 3. (Here the word “factorizable” means there is no correlation among photons in each specific channel.) Specifically, single-channel and multi-channel multi-photon states are presented in Subsections 3.1 and 3.2 respectively, covariance functions and intensities of output fields are studied in Subsection 3.3, while an analytic form of steady-state output states is derived in Subsection 3.4. The unfactorizable case is investigated in Section 4. Specifically, unfactorizable multi-channel multi-photon states are defined in Sebsection 4.1, the analytic form of the steady-state output state is presented in Subsection 4.2 where the underlying system is passive, the active case is studied in Subsection 4.3. Some concluding remarks are given in Section 5.

Notations. is the number of input channels, and is the number of degrees of freedom of a given quantum linear stochastic system. denotes the initial state of the system which is always assumed to be vacuum, denotes the vacuum state of free fields. Given a column vector of complex numbers or operators where is a positive integer, define , where the asterisk indicates complex conjugation or Hilbert space adjoint. Denote . Furthermore, define the doubled-up column vector to be . Let be an identity matrix and a zero square matrix, both of dimension . Define and (The subscript “” is often omitted.) Then for a matrix , define . denotes the Kronecker product. Given a function in the time domain, define its two-sided Laplace transform ([Sogo, 2010, (13)]) to be . Given two constant matrices , , define . Similarly, given time-domain matrix functions and of compatible dimensions, define . Given two operators and , their commutator is defined to be . For any integer , we write for integration in the space . We also write for . Finally, given a column vector , we use to denote its entries. Given a matrix , we use to denote its entries. Given a 3-way tensor (also called a tensor of order 3), we use to denote its entries; we do the similar thing for higher order tensors.

## 2 Quantum linear systems and tensors

This section records preliminary results necessary for the development of the paper. Quantum linear systems are briefly discussed is Subsection 2.1. Tensors and their associated operations, the appropriate mathematical language to describe the interaction of a quantum linear system with multi-photon channels, is introduced in Subsection 2.2.

### 2.1 Quantum linear systems

In this subsection quantum linear systems are described in the input-output language, which makes it natural to present transfer of covariance function of input fields. Moreover, the input-output framework also enables the definition of the stable inversion of quantum linear systems.

#### 2.1.1 Fields and systems

In this part we set up the model which is a quantum linear system driven by boson fields, ([Gardiner & Zoller, 2000]; [Walls & Milburn, 2008]; [Wiseman & Milburn, 2010]).

The triple provides a compact way for the description of open quantum systems ([Gough & James, 2009]; [Gough, James & Nurdin, 2010]; [Zhang & James, 2012]). Here the self-adjoint operator is the initial system Hamiltonian, is a unitary scattering operator, and is a coupling operator that describes how the system is coupled to its environment. The environment is an -channel electromagnetic field in free space, represented by a column vector of annihilation operators . Let be the initial time, namely, the time when the quantum system starts interacting with its environment. Define a gauge process by with operator entries on the Fock space for the free field ([Gardiner & Zoller, 2000, Walls & Milburn, 2008]). In this paper it is assumed that these quantum stochastic processes are canonical, that is, they have the following non-zero Ito products

 dBj(t)dB∗k(t)= δjkdt, dΛjkdB∗l(t)=δkldB∗j(t), dBj(t)dΛkl(t)= δjkdBl(t), dΛjk(t)dΛlm(t)=δkldΛjm(t), j,k,l=1,…,m,

where is a column vector of the integrated field operators defined via . In the interaction picture the stochastic Schrodinger’s equation for the open quantum system driven by the free field is, in Ito form ([Gardiner & Zoller, 2000, Chapter 11]),

 dU(t,t0)={Tr[(S−Im)dΛ(t)T]+dB†(t)L−L†SdB(t)−(12L†L+iH)dt}U(t,t0),  t≥t0, (1)

with being an identity operator for all .

Specific to the linear case, the open quantum linear system shown in Fig. 1 represents a collection of interacting quantum harmonic oscillators (defined on a Hilbert space ) coupled to boson fields described above ([Gardiner & Zoller, 2000, Wiseman & Milburn, 2010, Zhang & James, 2011, Zhang & James, 2012]). Here, () is the annihilation operator of the th oscillator satisfying the canonical commutation relations . Denote . The vector operator is defined as with . The initial Hamiltonian is with satisfying and . By (1), the dynamic model for the system is

 ˙˘a(t) = A˘a(t)+B˘b(t),  ˘a(t0)=˘a, (2) ˘bout(t) = C˘a(t)+D˘b(t), (3)

in which system matrices are given in terms of the physical parameters , specifically,

The transfer function (impulse response function) for the system is

 gG(t):={δ(t)D+CeAtB,t≥0,0,t<0. (4)

This, together with (2) and (3), yields

 ˘bout(t)=CeA(t−t0)˘a+t∫t0gG(t−r)˘b(r)dr. (5)

The system is said to be passive if both and . The system is said to be asymptotically stable if the matrix is Hurwitz ([Zhang & James, 2011, Section III-A]).

Define matrix functions

 gG−(t) :=⎧⎪ ⎪⎨⎪ ⎪⎩δ(t)S−[C−C+]eAt[C†−−C†+]S,t≥0,0,t<0, gG+(t)

(Note that when is passive, .) With these functions, the transfer function in (4) can be re-written as

 gG(t)=Δ(gG−(t),gG+(t)).

Finally, assume that the system (2)-(3) is asymptotically stable. Letting and noticing (4), equation (5) becomes

 ˘bout(t)=∞∫−∞gG(t−r)˘b(r)dr, (6)

which characterizes the steady-state relation between the input and the output.

#### 2.1.2 Stable inversion

In this par some results for the stable inversion of quantum linear systems are recorded, which are used in the derivation of output states of quantum linear systems driven by multi-photon states, cf. Sections 3 and 4.

For the transfer function defined in (4), let be its two-sided Laplace transform (see the Notations part and [Sogo, 2010, Eq. (13)]. Define a matrix function to be

 gG−1(t):=L−1b{ΞG[s]−1}(t). (7)

The following result is proved in [Zhang & James, 2013, Lemma 1].

###### Lemma 1

Assume that the system is asymptotically stable. Then

 gG−1(t) = Δ(gG−(−t)†,−gG+(−t)T). (8)

Remark 1. Because the system is asymptotically stable, it has no zeros on the imaginary axis, is well defined. It is worth pointing out that the matrix function turns out to be the transfer function of the inverse system defined in [Gough, James & Nurdin, 2010, (71)].

###### Lemma 2

Assume the quantum linear system is asymptotically stable. Define an operator

 ˘b−(t,t0):=U(t,t0)˘b(t)U(t,t0)∗,   t≥t0.

Then

 ˘b−(t,−∞)=∞∫−∞gG−1(t−r)˘b(r)dr. (9)

Proof.  Because , equation (5) gives

That is,

Letting and noticing (4) we have

 ˘b(t)=∞∫−∞gG(t−r)˘b−(r,−∞)dr. (10)

However, by Eq. (7), we have . Substituting it into (10) yields (9).

Remark 2. Operators and are formally defined mathematically, they may not correspond to physical variables. However, they do help in the derivation of the steady-state state of output fields.

#### 2.1.3 Steady-state covariance transfer

Here we record esults concerning covariance function transfer by the quantum linear system defined in (2)-(3).

Assume the quantum linear system is in the vacuum state . Assume further that the input field is in a zero-mean state . (specific types of will be studied in the sequel.) Denote the covariance functions of the input filed and the output field by and respectively, that is,

 R(t,r)=Tr[ρf˘b(t)˘b†(r)],  Rout(t,r)=Tr[|ϕ⟩⟨ϕ|⊗ρf˘bout(t)˘b†out(r)]. (11)

According to (6) and (11) we have

###### Lemma 3

Assume that the system (2)-(3) is asymptotically stable. Let the input field have covariance defined in (11). The steady-state (namely ) output covariance function is

 Rout(t,r)=∞∫−∞∞∫−∞gG(t−τ1)R(τ1,τ2)gG(r−τ2)†dτ1dτ2. (12)

In the frequency domain, we have

###### Theorem 4

Assume that the system (2)-(3) is asymptotically stable. If the input field is stationary with spectral density matrix (namely, the Fourier transform of ), the output spectral density matrix is given by

 Rout[iω]=ΞG[iω]R[iω]ΞG[iω]†. (13)

In particular, if the input field is in the vacuum state , that is, , then the output state is a Gaussian state with output spectral density matrix

 Rout[iω]=ΞG[iω][Im000m]ΞG[iω]†. (14)

In what follows we focus on the Gaussian input field states. Denote the initial joint system-field Gaussian state by where is a Gaussian field state. Define the steady-state joint state

 ρ∞g:=limt0→−∞t→∞U(t,t0)ρ0gU(t,t0)∗, (15)

and the steady-state output field state

 ρfield,g:=Trsys[ρ∞g],

where the subscript “sys” indicates that the trace operation is performed with respect to the system. Moreover, if the input state is stationary with spectral density , according to Theorem 4, is the steady-state output field density with covariance function given in (13). Finally, if , then is stationary zero-mean Gaussian with given in (14).

Remark 3. Because is obtained by tracing out the system, it is in general a mixed state. Moreover, is in general not the vacuum state even if . However, if the system is passive, then by (14),

 Rout[iω]=[Im000m]=Rin[iω]. (16)

That is, in the passive case the steady-state output state is again the vacuum state.

### 2.2 Tensors

In this subsection several types of tensors and their associated operations are introduced. Because different channels may have different numbers of photons, fibers of the tensors may thus have different lengths, see e.g., (17). Nonetheless, with slight abuse of notation, we still call these objects tensors.

Given positive integers and , let be a space of matrix-like objects, whose element is of the form

 ξ=⎡⎢ ⎢ ⎢⎣ξ11⋯ξ1ℓ1⋮⋱⋮ξm1⋯ξmℓm⎤⎥ ⎥ ⎥⎦.

In this paper is used to represent -channel multi-photon input states with denoting the photon number in the -th channel, . Because channels may have different numbers of photons, may not equal each other. Nonetheless in the paper we still call a matrix. Next we define a tensor space , whose elements are defined in the following way: For each , the model-3 fiber is

 Sij:=⎡⎢ ⎢⎣Sij1⋮Sijℓj⎤⎥ ⎥⎦∈Cℓj. (17)

That is, when the first two indices are fixed, we have a vector of dimension . looks like a 3-way tensor ([Kolda & Bader, 2009]), but its mode-3 fibers may have different dimensions for different . Nevertheless, in this paper we still call a 3-way tensor and a space of 3-way tensors over the field of complex numbers. Given a matrix , we may represent it as a 3-way tensor , by defining horizontal slices to be

 (18)

This update turns out to be very useful because the output state of a quantum passive linear system driven by an -channel multi-photon state encoded by a matrix can be characterized by a tensor in , see Sec. 3.4.

We adopt notations in [Kolda & Bader, 2009]. For each and ,

 S:jk=⎡⎢ ⎢⎣S1jk⋮Smjk⎤⎥ ⎥⎦∈Cm

is mode-1 (column) fiber. and are respectively horizontal and lateral slices (matrices) of the form

 Si::=⎡⎢ ⎢ ⎢⎣STi1:⋮STim:⎤⎥ ⎥ ⎥⎦∈Cm×(ℓ1,…,ℓm),  S:j:=[S:j1⋯S:jℓj]∈Cm×ℓj,  ∀i,j=1,…,m.

Finally, let be a 3-way tensor. We say that is partially Hermitian in modes 2 and 3 if all the horizontal slices are Hermitian matrices. That is, for all , the horizontal slices satisfy . This is a natural extension of the concept partially symmetric discussed in ([Kolda & Bader, 2009]) to the complex domain.

In what follows we define operations associated to these tensors. Given 3-way tensors and partially Hermitian tensor , we define a matrix whose (i,k)-th entry is

 (S(t)⊛T(r))ik:=m∑j=11Nℓjℓj∑β=1ℓj∑α=1CjαβSijα(t)Tkjβ(r),   ∀i,k=1,…,m, (19)

where () are positive scalars. (The physical interpretation of will be given in Sec. 3.) It can be verified that

 (S(t)⊛T(r))†=T(r)#⊛S(t)#. (20)

In this paper, we call a “core tensor” for the operation . According to (19) and the definition of in (18), we have

 diagj=1,…,m(1Nℓjℓj∑i,k=1Cjikξji(r)∗ξjk(t)) = ξ↑(r)#⊛ξ↑(t), (21)

Given a matrix function and a -way tensor , define whose -th element is

In compact form we write

 T=S×1E,

where is called 1-mode matrix product ([Kolda & Bader, 2009, Sec. 2.5]).

Given two matrices and two tensors , define

 Δ(S,T)×1Δ(E,F):=Δ(S×1E+T#×1F,T×1E+S#×1F). (22)

That is, the operation is performed block-wise. This operation is useful in studying the output state of a quantum linear system driven by a multi-channel multi-photon input state.

Finally, we define another type of operations between matrices and tensors. Let be the transfer function of the underlying quantum linear system with input channels. For each , let be an -way -dimensional tensor function that encodes the pulse information of the th input field containing photons. Denote the entries of by . Define an -way -dimensional tensor with entries given by the following multiple convolution

 Wj,r1,…,rℓj(t1,…,tℓj) =m∑k1,…,kℓj=1 ∞∫−∞⋯∞∫−∞Er1k1(t1−ι1)⋯Erℓjkℓj(tℓj−ιℓj)Vj,k1,…,kℓj(ι1,…,ιℓj)dι1…dℓj

for all . In compact form we write

 Wj=Vj×1E×2⋯×ℓjE,   ∀j=1,…,m, (23)

cf. [Kolda & Bader, 2009, Sec. 2.5]. More discussions on tensors will be given in Section 4.3.

## 3 The factorizable case

In this section we study how a quantum linear system responds to a factorizable multi-photon input state, here the word “factorizable” means that photons in each input channel are not statistically correlated. The single-channel and multi-channel multi-photon input states are defined in Subsections 3.1 and 3.2 respectively, output covariance functions and intensities are presented in Subsection 3.3, while the output states are derived in Subsection 3.4.

### 3.1 Single-channel multi-photon states

In this subsection single-channel -photon states are defined and their statistical properties are discussed.

For any given positive integer and real numbers , let be a permutation of the numbers . Denote the set of all such permutations by . For arbitrary functions defined on the real line, define

 Nℓ:=∑P∈Sℓ∫ℓξℓ(tℓ)∗⋯ξ1(t1)∗ξ1(P(t1))⋯ξℓ(P(tℓ))dt1→ℓ, (24)

provided the above multiple integral converges (this is always assumed in the paper). The subscript “” in indicates the number of photons. It can be shown that . A single-channel continuous-mode -photon state is defined via

 |ψℓ⟩:=1√Nℓℓ∏k=1B∗(ξk)|0⟩, (25)

where , (.) Because is a product of single integrals, there is no correlation among the photons. This type of multi-photon states is therefore called factorizable photon states. It can be shown that

 ⟨0|ℓ∏i=1B(ξi)ℓ∏k=1B∗(ξk)|0⟩=∑P∈Sℓ∫ℓξℓ(tℓ)∗⋯ξ1(t1)∗ξ1(P(t1))⋯ξℓ(P(tℓ))dt1→ℓ=Nℓ.

Thus . That is, is normalized.

When , , is a single-photon state, ([Loudon, 2000, (6.3.4)]; [Milburn, 2008, (9)]). On the other hand, when and , the input light field contains indistinguishable photons; such states are called continuous-mode Fock states which have been intensely studied, in e.g., [Gheri, Ellinger, Pellizzari & Zoller, 1998, (3)]; [Baragiola, Cook, Brańczyk & Combes, 2012, (13)].

For convenience, define a matrix whose entries are

 Cik=⟨0|ℓ∏α=1,α≠iB(ξα)ℓ∏β=1,β≠kB∗(ξβ)|0⟩, ℓ≥2. (26)

Clearly, .

###### Lemma 5

defined in (26) satisfies

In what follows we study statistical properties of the -photon state . It is easy to show that for all , . That is, the field has zero average field amplitude. The following result summarizes the second-order statistical information of the -photon state .

###### Lemma 6

Let denote the field intensity with respect to the state , namely,

 ¯n(t)=⟨ψℓ|b∗(t)b(t)|ψℓ⟩.

(In quantum optics, the second-order moment is the count rate ([Gardiner & Zoller, 2000]).) Moreover, let the field covariance function be

 R(t,r)=⟨ψℓ|˘b(t)˘b†(r)|ψℓ⟩,

as given by (11). Then we have

 R(t,r) = δ(t−r)[1000]+1Nℓℓ∑i=1ℓ∑k=1Δ(Cikξk(t)ξi(r)∗,0), (27) ¯n(t) = 1Nℓℓ∑i=1ℓ∑k=1Cikξi(t)∗ξk(t). (28)

Proof.  Clearly,

 R(t,r)=δ(t−r)[1000]+Δ(⟨ψℓ|b∗(r)b(t)|ψℓ⟩,⟨ψℓ|b(t)b(r)|ψℓ⟩). (29)

Observing that

 b(t)|ψℓ⟩=1√Nℓℓ∑k=1ξk(t)ℓ∏r=1,r≠kB∗(ξr)|0⟩, (30)

we have

 ⟨ψℓ|b∗(r)b(t)|ψℓ⟩=1Nℓℓ∑i=1ℓ∑k=1Cikξi(r)∗ξk(t),  ⟨ψℓ|b(t)b(r)|ψℓ⟩=0. (31)

Substituting (31) into (29) establishes (27). Finally, because is the 2-by-2 entry of , (28) follows (27).

In particular, for the single-photon case, the field covariance function is

 R(t,r)=δ(t−r)[1000]+Δ(ξ1(t)ξ1(r)∗,0), (32)

which is the same as [Zhang & James, 2012, (35)].

Remark 4. According to Lemma 6, the -photon state is not Gaussian; it may not be stationary either. So, its first and second order moments cannot provide all statistical information of the input field.

### 3.2 Multi-channel multi-photon states

In this subsection multi-channel multi-photon states are defined.

Let there be input field channels. For the j-th field channel, let be the number of photons (). Similar to (25), define the -th channel -photon state by

 |Ψj⟩:=1√Nℓjℓj∏k=1B∗j(ξjk)|0⟩, (33)

where the subscript “” indicates the -th channel, and indicates that there are photons in this channel. In analog to (24), for each , the normalization coefficient is defined to be

 Nℓj:=∑P∈Sℓj∫ℓjξjℓj(tℓj)∗⋯ξj1(t1)∗ξj1(P(t1))⋯ξjℓj(P(tℓj))dt1→ℓj.

We define an -channel multi-photon state as

 |Ψ⟩:=|Ψ1⟩⊗|Ψ2⟩⊗⋯⊗|Ψm⟩=m∏j=11√Nℓjℓj∏k=1B∗j(ξjk)|0⊗m⟩. (34)

In particular, for each , if , then (34) defines a multi-channel continuous-mode Fock state, see eg., [Baragiola, Cook, Brańczyk & Combes, 2012, (D1)].

### 3.3 Output covariance functions and intensities

In this subsection analytical forms of output covariance functions