Quantum trajectories for environment in superposition of coherent states

We derive stochastic master equations for a quantum system interacting with a Bose field prepared in a superposition of continuous-mode coherent states. To determine a conditional evolution of the quantum system, we use a collision model with an environment given as an infinite chain of not interacting between themselves qubits prepared initially in an entangled state being a discrete analogue of a superposition of coherent states of the Bose field. The elements of the environment chain interact with the quantum system in turn one by one, and they are subsequently measured. We determine a conditional evolution of the quantum system for continuous in time observations of the output field as a limit of discrete recurrence equations. We consider the stochastic master equations for a counting as well as for a diffusive stochastic process.


Introduction
Quantum filtering theory [1][2][3][4][5][6][7][8] formulated within the framework of quantum stochastic Itô calculus (QSC) [9,10] gives the best state estimation of an open quantum system on the basis of a continuous in time measurement preformed on the Bose field interacting with the system. The filtering theory is formulated with making use of input-output formalism [11] wherein the input field is interpreted as the field before interaction with the system and the output field is interpreted as the field after this interaction. Information about the quantum system is gained in an indirect way by performing the measurements on the output field. In general, there are two types of B Anita Magdalena Dąbrowska amdabrowska@gmail.com 1 Collegium Medicum Bydgoszcz, Nicolaus Copernicus University in Toruń, ul. Jagiellońska 15, 85-067 Bydgoszcz, Poland the measurement considered in the filtering theory, namely the photon counting and homodyne/heterodyne measurements which corresponds, respectively, to the counting and diffusion stochastic processes [6]. Evolution of an open quantum system conditioned on the results of the continuous in time measurement of the output field is given by the stochastic master equation called also in the literature the quantum filtering equation. The conditional state, depending on all past results of the measurement, creates quantum trajectory. By taking the average over all possible outcomes of the measurements, we get from the a posteriori evolution the a priori evolution given by the master equation. Clearly, the form of the filtering equation depends on the initial state of the environment and on the type of measurement performed on the output field. There exist many derivations of the filtering equations (see, for example, [1][2][3][4]8,[12][13][14][15]). One can find the rigorous derivations of the conditional evolution for the case when the Bose field is prepared in the Gaussian state, for instance, in [16][17][18][19][20][21]. The standard methods of determination of the filtering equation stop working when the Bose field is prepared in non-classical state. The initial temporal correlations in the Bose field make then the evolution of open system non-Markovian. The system becomes entangled with the environment, and its evolution is no longer given by one equation but by a set of equations. In this case to determine the conditional evolution of the system, one can apply a cascaded approach [7] with an ancilla system being a source of non-classical signal. The methods of determination of the filtering equation based on the idea of enlarging the Hilbert space of the compound system by the Hilbert space of ancilla were used for single-photon state in [22][23][24][25], for a Fock state in [26][27][28], and for a superposition of coherent states in [22,24]. Note, however, that ancilla system serves here only as a convenient theoretical mathematical device allowing to solve the problem of determination of the conditional evolution. Unfortunately, by introducing such auxiliary system we loose some physical intuition and the interpretation of quantum trajectories becomes thereby more difficult.
In the paper, we present derivation of the filtering equations for the environment prepared in a superposition of coherent states. Instead of the methods based on the concept of ancilla and QSC, we use quantum repeating interactions and measurements model [29][30][31][32], known also in the physical literature as a collision model [33]. We consider the environment modeled by an infinite chain of qubits which interact in turn one by one with a quantum system. After each interaction, the measurement is preformed on the last qubit interacted with the system. The essential properties of our model are that each qubit interacts with the system only once and that the environment qubits do not interact between themselves. So in the paper we use the toy Fock space as an approximation of the symmetrical Fock space [15,29,34,[38][39][40][41]. The idea of obtaining the differential filtering equations from difference equations was implemented for the Markovian case in [14,15,34,35]. As shown in [36,37], it can be successfully applied also for the non-Markovian case.
The paper is organized as follows. In Sect. 2, we introduce a description of the environment and its interaction with the quantum system. Section 3 is devoted to derivation of the conditional evolution of open system for the case when the environment is prepared in a coherent state. In Sect. 4 the conditional evolution of open system for the bath in a superposition of coherent states is investigated. As an example, we present the a priori and the a posteriori dynamics of a single mode cavity in Sect. 5. Our results are briefly summarized in Sect. 6.

The unitary system and environment evolution
Let us consider a quantum system S of the Hilbert space H S interacting with an environment consisting of a sequence of qubits. We assume that the environment qubits do not interact between themselves, but they interact in a successive way with the system S each during the time interval of the length τ . At a given moment S interacts with only one of the environment qubits. The Hilbert space of the environment is where H E,k stands for the Hilbert space of the k-th qubit interacting with S in the time interval [kτ, (k + 1)τ ). We start from a discrete in time model of repeated interactions (collisions) to show finally its limit with time treated as a continuous variable. We will treat τ as a small time and work to linear order in τ (we neglect all higher order terms in τ ). We assume that the unitary evolution of the compound E + S system is governed by [29,31] where V k is the unitary operator acting non-trivially only in the Hilbert space H E,k ⊗ H S , that is, and with where H S is the Hamiltonian of S, L is a bounded operator of S, and σ + k = |1 k 0|, σ − k = |0 k 1|, where by |0 k and |1 k we denoted, respectively, the ground and excited states of the k-th qubit. The Hamiltonian H k is written in the interaction picture eliminating the free evolution of the bath. One can find a detailed discussion on the physical assumptions leading to (5), for instance, in [33,35,42]. For simplicity, we set the Planck constant = 1. Note that U j describes the j-th first interactions and it has trivial action on +∞ k= j H E,k . Let us define in H E,k the vector |α k k by the formula [35] where α k ∈ C. One can check that and The coherent state in H E is defined as with the condition +∞ k=0 |α k | 2 τ < ∞. Note that the vector state |α is a discrete analogue of coherent state defined in the symmetric Fock space considered in QSC. We will show that it allows in the continuous time limit to reproduce all results for the coherent state received within QSC.

Quantum trajectories for a coherent state
In this section we consider the case when the composed E + S system is prepared initially in the pure product state where |α is the coherent state of the environment.

Photon counting
We assume that after each interaction the measurement is performed on the last element of the environment chain just after its interaction with S. A goal of this subsection is providing a description of the state of S conditioned on the results of the measurements of the observables

Theorem 1
The conditional state of S and the part of the environment which has not interacted with S up to jτ for the initial state (10) and the measurement of (11) at the moment jτ is given by where and the conditional vector |ψ j from H S satisfies the recurrence formula where η j+1 stands for a random variable describing the ( j + 1)-th output of (11), and M j η j+1 has the form Initially |ψ j=0 = |ψ such that |˜ j=0 = |α ⊗ |ψ .
It is clear that |˜ j is the product state vector belonging to the Hilbert space +∞ k= j H E,k ⊗ H S . Note also that the conditional vector |ψ j depends on all results of the measurements performed on the bath qubits up to time jτ .
Proof We prove the above theorem by an induction technique. So we start from the assumption that (13) holds and then check that Now using the fact that the conditional vector | j+1 from the Hilbert space we readily find that | j+1 has the form with |ψ j+1 given by (14), which ends the proof.

Homodyne detection
Now we describe the evolution conditioned on the results of the measurements of the observables where are vectors from the Hilbert space H E,k .

Theorem 2
The conditional state of S and the part of the environment which has not interacted with S up to jτ for the initial state (10) and the measurement of (21) at the moment jτ is given by where and the conditional vector |ψ j from H S satisfies the recurrence formula where ζ j+1 = ±1 stands for a random variable describing ( j + 1)-th output of (21), and Initially |ψ j=0 = |ψ such that |˜ j=0 = |α ⊗ |ψ .
Proof Assuming that (25) holds we get The conditional vector | j+1 from the Hilbert space where ζ j+1 has two possible values ±1, and j +1 = |+ j +|, It is seen that | j+1 has the form of (25) and the vector |ψ j from H S satisfies the recurrence equation (27).

Quantum trajectories for a superposition of coherent states
Let us assume that the initial state of the compound E + S system is given by where |α and |β are coherent states of H E , and Note that in this case the bath qubits are prepared in an entangled state.

Theorem 3
The conditional state of S and the part of the environment which has not interacted with S up to jτ for the initial state (31) and the measurement of (11) at the moment jτ is given by where The conditional vectors |ψ j , |ϕ j from H S in (34) are given by the recurrence formulas where η j+1 = 0, 1 stands for a random variable describing the ( j + 1)-th output of (11), and and initially we have |ψ 0 = |ϕ 0 = |ψ .

Proof
The proof is straightforward. We simply refer to the results of the previous Section and the linearity of the evolution of the total system. Let us notice that the form of | j indicates that the system S becomes entangled with this part of the environment which has not interacted with S yet. Taking the partial trace of the operator | j j | over S, we get the unnormalized state of the environment of the form The operator ρ field j describes the conditional state of this part of the environment which has not interacted with S yet. It depends on all results of the measurements performed on the bath qubits up to jτ . Therefore, we can say that the results of the measurements change our knowledge about the state of the future part of the environment.
In order to obtain the conditional state of S, one has to take the partial trace of |˜ j ˜ j | over the environment. One can check that the a posteriori state of S at the time jτ has the formρ where and Trρ j is the probability of a particular trajectory.
To derive the set of recurrence equations describing the stochastic evolution of S, it is convenient to write down the conditional state of S at jτ in the form In our derivation, we will use several times the formula +∞ k= j k β k |α k k = +∞ k= j+1 following from Let us notice first that the conditional operator ρ j+1 is given by the recurrence formula where η j+1 stands for the random variable having two possible values: 0, 1. Let us note that in order to determineρ j+1 we need to know the operators (45)-(48) at the moment jτ and the result of the next measurement. When the result of the measurement is 0, then we obtain from Eqs. (35) and (36) the following set of discrete equations The conditional probability of the outcome 0 at the moment ( j + 1)τ when the a posteriori state of S at jτ wasρ j is defined as where ρ j+1 is given by (51) for η j = 0. Hence, we obtain the formula where Now, making use of the fact that we obtain the set of difference equations One can get the equation for the operatorρ βα j using the fact thatρ βα j = ρ αβ j † . When the result of the measurement at the moment ( j + 1)τ is 1, we get the following recurrence formulas The conditional probability of the outcome 1 at the moment ( j + 1)τ when the a posteriori state of S at the moment jτ wasρ j is defined by where ρ j+1 is given by (51) with η j = 1. One can check that where the conditional intensity ν j is defined by (57). So for the result 1 we find that Let us introduce now the stochastic discrete process with the increment One can check that the conditional expectation Finally, by combining Eqs. (63)-(65) with Eqs. (71)-(73), we obtain the set of stochastic difference equations where and the initial conditionρ αα 0 =ρ ββ 0 = |ψ ψ|,ρ αβ 0 = β|α |ψ ψ|. We dropped here all terms that do not contribute to the continuous time limit when τ → dt. Note that when n j is equal to 0, then Eqs. Let us notice that to get the continuous in time evolution of S, we fix time t = jτ such that when j → +∞, we have τ → 0. Of course, we take t fixed but arbitrary. Thus, in the continuous time limit we get from (77)-(79) the set of the stochastic differential equations of the form and initiallyρ αα 0 =ρ ββ 0 = |ψ ψ|,ρ αβ 0 = β|α |ψ ψ|. The stochastic process n t is defined as the continuous limit of the discrete process n j . The Itô table for dn t is (dn t ) 2 = dn t (we can measure at most one photon in the interval of length dt) and E dn t |ρ t = ν t dt, where Moreover, the complex functions α t and β t satisfy the conditions and Thus, the a posteriori state of S is given as where the conditional operatorsρ αα t ,ρ . One can check that Trρ t = 1 for any t ≥ 0. The equations (81)-(83) agree with the stochastic master equations derived in [23] (see Sec. IV in [23]).
When we take an average ofρ t over all realizations of the stochastic process n t (all possible outcomes), we get the a priori evolution of the system S. One can check that the a priori state of S is described by

Theorem 4
The conditional state of S and the part of the environment which has not interacted with S up to jτ for the initial state (31) and the measurement of (21) at the moment jτ is given by where The conditional vectors |ψ j , |ϕ j from H S in (34) are given by the recurrence formulas where ζ j+1 stands for a random variable describing the ( j + 1)-th output of (21), and and initially we have |ψ 0 = |ϕ 0 = |ψ .
Proof To prove Theorem (4), we use the result of Sect. 3.2 and the linearity of the evolution equation for the total system. Clearly, the conditional state of S at the moment jτ has the form (44). We start derivation of the filtering equations for the stochastic operators (45)-(48) from writing down the recursive formulas 2|ψ j+1 ψ j+1 | = |ψ j ψ j | + L|ψ j ψ j |τ We can readily deduce that the conditional probability of the result ζ j+1 at the moment ( j + 1)τ when the conditional state of S isρ j at the time jτ is given by where and Thus for the discrete stochastic process ζ j we obtain the conditional mean values Let us introduce now the stochastic process One can easily check that E[ q j = q j+1 − q j |ρ j ] = μ j τ + O(τ 3/2 ). Now, taking into account that after some algebra we find the set of the stochastic difference equations with the initial conditionsρ αα 0 = |ψ ψ|,ρ The process q j in the limit τ → 0 converges to the stochastic process q t with the conditional probability E[dq t = q t+dt − q t |ρ t ] = μ t dt.

An example: a cavity mode
One can read about the emergence of collision model in quantum optics , for instance, in [33,42]. To derive the discrete model of repeated interactions and measurements one starts from description of interaction of a quantum system with a Bose field propagating in only one direction, making the rotating wave approximation and taking the flat spectrum of the field. Then one passes to the interaction picture with respect to the free dynamics of the field and takes the Hamiltonian of the field in the frequency domain with the lower limit of integration extended to −∞. The time coarse-graining model arises from division of the field into some probe segments. Lack of an interaction between the system and the output field means that the photons emitted by the system leave immediately the interaction region and cannot be reabsorbed. We describe here briefly the a priori and the a posteriori evolution of a cavity mode coupled to a propagating one-dimensional Bose field in a superposition of two coherent states. Thus, we have where a stands for the annihilation operator, ω 0 > 0, and is a positive coupling constant. We consider here the case when the harmonic oscillator is initially in the coherent state Then, the solution to the set of the master equations can be written then in the form where and | f t , |g t are coherent states of the harmonic oscillator with the amplitudes satisfying the equationsḟ