Quantum trajectories for environment in

Quantum trajectories for environment in superposition of coherent states Anita Magdalena D¡browska Nicolaus Copernicus University in Toru«, Collegium Medicum Bydgoszcz, ul.

Jagiello«ska 15, 85-067 Bydgoszcz, Poland

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

I.

INTRODUCTION

Quantum ltering theory [18] 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 eld interacting with the system. The ltering theory is formulated with the making use of input-output formalism [11] wherein the input eld is interpreted as the eld before interaction with the system and the output eld is interpreted as the eld after this interaction. Information about the quantum system is gained in an indirect way by performing the measurements on the output eld. In general, there are two types of the measurement considered in the ltering theory, namely, the photon counting and homodyne/heterodyne measurements which corresponds respectively to the counting and diusion stochastic processes [6]. The evolution of an open quantum system conditioned on the results of the continuous in time measurement of the output eld is given by the stochastic master equation called also in the literature the quantum ltering 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 measurement we get from the a posteriori evolution the a priori evolution given by the master equation. Clearly, the form of the ltering equation depends on the initial state of the environment and on the type of measurement performed on the output eld. There exist many derivations of the ltering equations (see, for example, [14, 8, 1215]). The rigorous derivations of the conditional evolution for the case when the Bose eld is prepared in the Gaussian state one can nd, for instance, in [1621]. The standard methods of determination of the ltering equation stop working when the Bose eld is prepared in non-classical state. The initial temporal correlations in the Bose eld makes 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 the ltering 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 [2225], for a Fock state in [2628], and for a superposition of coherent states in [22, 24]. Note, however that ancilla system serve 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 become thereby more dicult. In the paper we present derivation of the ltering 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 [2932], known also in the physical literature as a collision model [33]. We consider the environment modeled by an innite 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 as an approximation of the symmetrical Fock space we use the toy Fock space [15, 29, 34, 3841]. The idea of obtaining the dierential ltering equations from their dierence versions were 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 Section 2, we introduce the description of the environment and its interaction

2 with the quantum system. Section 3 is devoted to derivation of the conditional evolution of open system for the case when the environment prepared in a coherent state. In Section 4 the conditional evolution of open system for the bath in a superposition of coherent states is investigated. Our results are briey summarized in Section 5. II.

THE UNITARY SYSTEM AND ENVIRONMENT EVOLUTION

Let us consider a quantum system S of the Hilbert space HS interacting with the 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 +∞ O

HE =

(1)

HE,k ,

k=0

where HE,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 nally 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]

Uj = Vj−1 Vj−2 . . . V0 for j ≥ 1,

U0 = 1,

(2)

where Vk is the unitary operator acting non-trivially only in the Hilbert space HE,k ⊗ HS , that is, k−1 O

1 i ⊗ Vk ,

(3)

Vk = exp (−iτ Hk ) ,

(4)

Vk =

i=0

and

with

Hk =

+∞ O i=k

i 1 i ⊗ HS + √ τ

σk+

⊗

+∞ O

1i ⊗ L −

σk−

i=k+1

⊗

+∞ O

1i ⊗ L

! †

,

(5)

i=k+1

where HS is the Hamiltonian of S , L is a bounded operator of S , and σk+ = |1ik h0|, σk− = |0ik h1|, where by |0ik and |1ik we indicated respectively the ground and excited states of the k -th qubit. The Hamiltonian Hk is written in the interaction picture eliminating the free evolution of the bath. A detailed discussion on the physical assumptions leading to (5) one can nd, for instance, in [33, 35, 42]. For simplicity, we set the Planck constant ~ = 1. Note that N+∞ Uj describes the j -th rst interactions and it has trivial action on k=j HE,k . Let us dene in HE,k the vector |αk ik by the formula [35] √

|αk ik = e

+ − τ (αk σk −α∗ k σk )

|0ik ,

(6)

where αk ∈ C. One can check that

 |αk ik =

|αk |2 τ 1− 2



√ |0ik + αk τ |1ik + O(τ 3/2 )

(7)

and

hαk |σk− |αk i =

√

τ αk + O(τ 3/2 ), hαk |σk+ σk− |αk i = τ |αk |2 + O(τ 2 ).

(8)

The coherent state in HE we dene as

|αi =

+∞ O k=0

|αk ik

(9)

3 with the condition

+∞ X

|αk |2 τ < ∞.

k=0

Note that the vector state |αi is a discrete analogue of coherent state dened 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. III.

QUANTUM TRAJECTORIES FOR THE COHERENT STATE

In this section we consider the case when the composed E + S system is prepared initially in the pure product state (10)

|αi ⊗ |ψi, where |αi is the coherent state of the environment. A.

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

σk− σk+ = |1ik h1|,

(11)

k = 0, 1, 2, . . . .

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 ˜ j i = p |Ψj i , |Ψ hΨj |Ψj i

(12)

where |Ψj i =

+∞ O

(13)

|αk ik ⊗ |ψj i

k=j

and the conditional vector |ψj i from HS satises the recurrence formula (14)

|ψj+1 i = Mηjj+1 |ψj i,

where ηj+1 stands for a random variable describing the (j + 1)-th output of (11), and Mηjj+1 has the form M0j M1j

1 |αj |2 = 1S − iHS + L† L + L† αj + 2 2 √ 3/2 = (L + αj ) τ + O(τ ). 



τ + O(τ 2 ),

(15) (16)

˜ j=0 i = |αi ⊗ |ψi. Initially |ψj=0 i = |ψi such that |Ψ ˜ j i is the product state vector belonging to the Hilbert space It is clear that |Ψ

+∞ O HE,k ⊗ HS . Note also that the k=j

conditional vector |ψj i 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     +∞ O |αj |2 1 τ + O(τ 2 ) |ψj i Vj |Ψj i = |0ij ⊗ |αk ik ⊗ 1S − iHS + L† L + L† αj + 2 2 k=j+1

+|1ij ⊗

+∞ O k=j+1

h i √ |αk ik ⊗ (L + αj ) τ + O(τ 3/2 ) |ψj i.

(17)

4 Now using the fact that the conditional vector |Ψj+1 i from the Hilbert space

+∞ O

HE,k ⊗ HS is dened by

k=j+1

 Πjη ⊗ j+1

+∞ O



1k ⊗ 1S  Vj |Ψj i = |ηj+1 ij ⊗ |Ψj+1 i,

(18)

k=j+1

where

Πj0 = |0ij h0|,

Πj1 = |1ij h1|,

(19)

|αk ik ⊗ |ψj+1 i

(20)

we readily nd that |Ψj+1 i has the form +∞ O

|Ψj+1 i =

k=j+1

with |ψj+1 i given by (14), which ends the proof. B.

Homodyne detection

Now we describe the evolution conditioned on the results of the measurements of the observables

σkx = σk+ + σk− = |+ik h+| − |−ik h−|,

k = 0, 1, 2, . . . ,

(21)

where

1 |+ik = √ (|0ik + |1ik ) , 2 1 |−ik = √ (|0ik − |1ik ) , 2

(22) (23)

are vectors from the Hilbert space HE,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 ˜ j i = p |Ψj i , |Ψ hΨj |Ψj i

(24)

where |Ψj i =

+∞ O

|αk ik ⊗ |ψj i

(25)

k=j

and the conditional vector |ψj i from HS satises the recurrence formula |ψj+1 i = Rζjj+1 |ψj i,

(26)

where ζj+1 = ±1 stands for a random variable describing (j + 1)-th output of (21), and Rζjj+1

1 =√ 2



1S

   √ 1 † |αj |2 † 3/2 − iHS + L L + L αj + τ + (L + αj )ζj+1 τ + O(τ ) . 2 2

˜ j=0 i = |αi ⊗ |ψi. Initially |ψj=0 i = |ψi such that |Ψ

(27)

5

Proof. Assuming that (25) holds we get    +∞ O 1 |αj |2 1 † † √ Vj |Ψj i = |+ij ⊗ τ |αk ik ⊗ 1S − iHS + L L + L αj + 2 2 2 k=j+1 o √ + (L + αj ) τ + O(τ 3/2 ) |ψj i    +∞ O |αj |2 1 1 τ + √ |−ij ⊗ |αk ik ⊗ 1S − iHS + L† L + L† αj + 2 2 2 k=j+1 o √ − (L + αj ) τ + O(τ 3/2 ) |ψj i. +∞ O

The conditional vector |Ψj+1 i from the Hilbert space

(28)

HE,k ⊗ HS satises for the measurement of (21) the

k=j+1

equation

 Πj ζj+1

⊗



+∞ O

1k ⊗ 1S  Vj |Ψj i = |ζj+1 ij ⊗ |Ψj+1 i,

(29)

k=j+1

where ζj+1 has to possible values ±1, and

Πj+1 = |+ij h+|,

Πj−1 = |−ij h−|.

(30)

It is seen that |Ψj+1 i has the form of (25) and the vector |ψj i from HS satises the recurrence equation (27). IV.

QUANTUM TRAJECTORIES FOR A SUPERPOSITION OF COHERENT STATES

Let us assume that the initial state of the compound E + S system is given by (31)

(cα |αi + cβ |βi) ⊗ |ψi, where |αi and |βi are coherent states of HE , and

|cα |2 + c∗α cβ hα|βi + cα c∗β hβ|αi + |cβ |2 = 1.

(32)

Note that in this case the bath qubits are prepared in the entangled state. A.

Photon counting

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 ˜ j i = p |Ψj i , |Ψ hΨj |Ψj i

(33)

where |Ψj i = cα

+∞ O k=j

|αk ik ⊗ |ψj i + cβ

+∞ O

|βk ik ⊗ |ϕj i.

(34)

k=j

The conditional vectors |ψj i, |ϕj i from HS in (34) are given by the recurrence formulas j |ψj+1 i = Mηαj+1 |ψj i,

(35)

j |ϕj+1 i = Mηβj+1 |ϕj i,

(36)

6

where ηj+1 = 0, 1 stands for a random variable describing the (j + 1)-th output of (11), and α M0 j

= 1S

|αj |2 1 − iHS + L† L + L† αj + 2 2 



τ + O(τ 2 ),

(37)

  1 |βj |2 β M0 j = 1S − iHS + L† L + L† βj + τ + O(τ 2 ), 2 2

(38)

√ α M1 j = (L + αj ) τ + O(τ 3/2 ),

(39)

√ β M1 j = (L + βj ) τ + O(τ 3/2 ),

(40)

and initially we have |ψ0 i = |ϕ0 i = |ψi. 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 |Ψj i 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 |Ψj ihΨj | over S we get the unnnormalized state of the environment of the form

ρfj ield = |cα |2

+∞ O

|αk ik hαk |hψj |ψj i + cα c∗β

k=j

+cβ c∗α

+∞ O

|αk ik hβk |hϕj |ψj i

k=j +∞ O

|βk ik hαk |hψj |ϕj i+|cβ |2 |

+∞ O

|βk ik hβk |hϕj |ϕj i.

(41)

k=j

k=j

The operator ρfj ield 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 changes our knowledge about the state of the future part of the environment. ˜ j ihΨ ˜ j | over the environment. In order to obtain the conditional state of S one has to take the partial trace of |Ψ One can check that the a posteriori state of S at the time jτ has the form

ρ˜j =

ρj , Trρj

(42)

where

ρj = |cα |2 |ψj ihψj | + cα c∗β

+∞ Y

hβk |αk i|ψj ihϕj | + cβ c∗α

k=j

+∞ Y

hαk |βk i|ϕj ihψj |

k=j

(43)

+|cβ |2 |ϕj ihϕj |

and Trρj 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 ∗ αβ 2 ββ ρ˜j = |cα |2 ρ˜αα ˜j + cβ c∗α ρ˜βα ˜j , j + cα cβ ρ j + |cβ | ρ

(44)

where

ρ˜αα j = +∞ Y

ρ˜αβ j

=

1 |ψj ihψj |, Trρj

(45)

k hβk |αk ik

k=j

Trρj

|ψj ihϕj |,

(46)

7 +∞ Y

ρ˜βα j

=

k hαk |βk ik

k=j

ρ˜ββ j =

Trρj

|ϕj ihψj |,

1 |ϕj ihϕj |. Trρj

In our derivation we will use several times the formula   +∞ +∞ Y Y
1 2 2 ∗ 2 |αj | + |βj | − 2αj βj τ + O(τ ) k hβk |αk ik = k hβk |αk ik 1 − 2 k=j

(47)

(48)

(49)

k=j+1

following from k hβk |αk ik

=1−


1 |αk |2 + |βk |2 − 2αk βk∗ τ + O(τ 2 ). 2

(50)

Let us notice rst that the conditional operator ρj+1 is given by the recurrence formula j j† ρj+1 = |cα |2 Mηαj+1 |ψj ihψj |Mηαj+1 + cα c∗β

+∞ Y

j j† hβk |αk iMηαj+1 |ψj ihϕj |Mηβj+1

k=j+1

+cβ c∗α

+∞ Y

j j† j j† |ϕj ihψj |Mηαj+1 |ϕj ihϕj |Mηβj+1 hαk |βk iMηβj+1 + |cβ |2 Mηβj+1 ,

(51)

k=j+1

where ηj+1 stands for the random variable having two possible values: 0, 1. Let us note that in order to determine ρ˜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

1 † L L, |ψj ihψj | τ 2 −|ψj ihψj |Lαj∗ τ − L† |ψj ihψj |αj τ − |ψj ihψj ||αj |2 τ + O(τ 2 ),

|ψj+1 ihψj+1 | = |ψj ihψj | − i[HS , |ψj ihψj |]τ −

 
1 |ψj+1 ihϕj+1 | = |ψj ihϕj | 1 − |αj |2 + |βj |2 τ 2 1 † −i[HS , |ψj ihϕj |]τ − L L, |ψj ihϕj | τ 2 −|ψj ihϕj |Lβj∗ τ − L† |ψj ihϕj |αj τ + O(τ 2 ), 1 † L L, |ϕj ihϕj | τ 2 −|ϕj ihϕj |Lβj∗ τ − L† |ϕj ihϕj |βj τ − |ϕj ihϕj ||βj |2 τ + O(τ 2 ).

(52)

(53)

|ϕj+1 ihϕj+1 | = |ϕj ihϕj | − i[HS , |ϕj ihϕj |]τ −

(54)

The conditional probability of the outcome 0 at the moment (j + 1)τ when the a posteriori state of S at jτ was ρ˜j is dened as

pj+1 (0|˜ ρj ) =

Trρj+1 , Trρj

(55)

where ρj+1 is given by (51) for ηj = 0. Hence, we obtain the formula

pj+1 (0|˜ ρj ) = 1 − νj τ + O(τ 2 ),

(56)

νj = |cα |2 νjαα + cα c∗β νjαβ + c∗α cβ νjβα + |cβ |2 νjββ ,

(57)

where

8

νjαα = Tr




 L† L + Lαj∗ + L† αj + |αj |2 ρ˜αα , j

(58)

νjαβ = Tr

h

i
, L† L + Lβj∗ + L† αj + αj∗ βj ρ˜αβ j

(59)

νjβα = Tr

h

i
L† L + Lαj∗ + L† βj + αj βj∗ ρ˜βα , j

(60)

νjββ = Tr

h

i
. L† L + Lβj∗ + L† βj + |βj |2 ρ˜ββ j

(61)

Now, making use of the fact that


1 1 = 1 + νj τ + O(τ 2 ) , Trρj+1 Trρj

(62)

we obtain the set of dierence equations

1 † ∗ L L, ρ˜αα τ − ρ˜αα j j Lαj τ 2 2 2 −L† ρ˜αα ˜αα j αj τ − ρ j |αj | τ + O(τ ),

(63)

o 1n † ∗ L L, ρ˜αβ τ − ρ˜αβ j j Lβj τ 2 ∗ 2 −L† ρ˜αβ ˜αβ j αj τ − ρ j βj αj τ + O(τ ),

(64)

o 1n † ∗ L L, ρ˜ββ τ − ρ˜ββ j j Lβj τ 2 2 2 −L† ρ˜ββ ˜ββ j βj τ − ρ j |βj | τ + O(τ ).

(65)

ρ˜αα ˜αα = ρ˜αα ˜αα j+1 − ρ j j νj τ − i[HS , ρ j ]τ −

ρ˜αβ ˜αβ = ρ˜αβ ˜αβ j+1 − ρ j j νj τ − i[HS , ρ j ]τ −

ρ˜ββ ˜ββ = ρ˜ββ ˜ββ j+1 − ρ j j νj τ − i[HS , ρ j ]τ −

 † βα αβ The equation for the operator ρ˜βα one can get using the fact that ρ ˜ = ρ ˜ . j j j When the result of the measurement at the moment (j + 1)τ is 1, we get the following recurrence formulas

|ψj+1 ihψj+1 | = L|ψj ihψj |L† + L|ψj ihψj |αj∗
+|ψj ihψj |L† αj + |ψj ihψj ||αj |2 τ + O(τ 2 ),

(66)

|ψj+1 ihϕj+1 | = L|ψj ihϕj |L† + L|ψj ihϕj |βj∗
+|ψj ihϕj |L† αj + |ψj ihϕj |αj βj∗ τ + O(τ 2 ),

(67)

|ϕj+1 ihϕj+1 | = L|ϕj ihϕj |L† + L|ϕj ihϕj |βj∗
+|ϕj ihϕj |L† βj + |ϕj ihϕj ||βj |2 τ + O(τ 2 ).

(68)

The conditional probability of the outcome 1 at the moment (j + 1)τ when the a posteriori state of S at the moment jτ was ρ˜j is dened by

pj+1 (1|˜ ρj ) =

Trρj+1 , Trρj

(69)

where ρj+1 is given by (51) with ηj = 1. One can check that

pj+1 (1|˜ ρj ) = νj τ + O(τ 2 ),

(70)

9 where the conditional intensity νj is dened by (57). So for the result 1 we nd that

ρ˜αα j+1 =


1 † † ∗ † † 2 L˜ ραα ραα ˜αα ˜αα + O(τ ), j L + L˜ j S αj + S ρ j L αj + S ρ j S |αj | νj

(71)

ρ˜αβ j+1 =

 1  αβ † † ∗ † † ∗ L˜ ρj |L + L˜ ραβ ˜αβ ˜αβ j S βj + S ρ j L αj + S ρ j S αj βj + O(τ ), νj

(72)

 1  ββ † † ∗ † † 2 + O(τ ). L˜ ρj L + L˜ ρββ ˜ββ ˜ββ j S βj + S ρ j L βj + S ρ j S |βj | νj

(73)

ρ˜ββ j+1 =

Let us introduce now the stochastic discrete process

nj =

j X

ηk ,

(74)

k=0

with the increment

∆nj = nj+1 − nj .

(75)

E[∆nj |˜ρj ] = νj τ + O(τ 2 ).

(76)

One can check that the conditional expectation

Finally, by combining Eqs. (63)(65) with Eqs. (71)(73), we obtain the set of stochastic dierence equations † ∗ ρ˜αα ˜αα = Lραα ραα ˜αα j+1 − ρ j j τ + [˜ j , L ]αj τ + [L, ρ j ]αj τ  1 † † ∗ + L˜ ραα ˜αα ραα j L +ρ j L αj + L˜ j αj νj 
2 αα +˜ ραα |α | − ρ ˜ (∆nj − νj τ ) , j j j

(77)

† ∗ ρ˜αβ ˜αβ = Lρββ ραβ ˜αβ j+1 − ρ j j τ + [˜ j , L ]αj τ + [L, ρ j ]βj τ   1 † † ∗ + L˜ ραβ ˜αβ ραβ j L +ρ j L αj + L˜ j βj νj   αβ ∗ +˜ ραβ β α − ρ ˜ (∆nj − νj τ ) j j j j

(78)

† ∗ ρ˜ββ ˜ββ = Lρβα ρββ ˜ββ j+1 − ρ j j τ + [˜ j , L ]βj τ + [L, ρ j ]βj τ   1 † † ∗ L˜ ρββ ˜ββ ρββ + j L +ρ j L βj + L˜ j βj νj   ββ ββ 2 +˜ ρj |βt | − ρ˜j (∆nj − νj τ ) ,

(79)

where

Lρ = −i[HS , ρ] −

1 † L L, ρ + LρL† 2

(80)

˜ββ ˜αβ and the initial condition ρ˜αα 0 = ρ 0 = |ψihψ|, ρ 0 = hβ|αi|ψihψ|. We dropped here all terms that do not contribute to the continuous time limit when τ → dt. Note that when ∆nj is equal to 0, then Eqs. (77)(79) reduce to Eqs. (63)(65), and when ∆nj is equal to 1, then all the terms proportional to τ in Eqs. (77)(79) are negligible and we obtain the formulas (71)(73).

10 Let us notice that to get the continuous in time evolution of S we x time t = jτ such that when j → +∞ we have τ → 0. Of course, we take t xed but arbitrary. Thus in the continuous time limit we get from (77)(79) the set of the stochastic dierential equations of the form † ∗ d˜ ραα = Lραα dt + [˜ ραα ˜αα t t , L ]αt dt + [L, ρ t ]αt dt t 1 † † ∗ + L˜ ραα ˜αα ραα t L +ρ t L αt + L˜ t αt νj 
2 αα +˜ ραα |α | − ρ ˜ (dnt − νt dt) , t t t

(81)

† ∗ d˜ ραβ = Lραβ dt + [˜ ραβ ˜αβ t t , L ]αt dt + [L, ρ t ]βt dt t  1 † † ∗ L˜ ραβ ˜αβ ραβ + t L +ρ t L αt + L˜ t βt νt   αβ ∗ αβ +˜ ρt βt αt − ρ˜t (dnt − νt dt)

(82)

† ∗ d˜ ρββ = Lρββ dt + [˜ ρββ ˜ββ t t , L ]βt dt + [L, ρ t ]βt dt t  1 † † ∗ + L˜ ρββ ˜ββ ρββ t L +ρ t L βt + L˜ t βj νt   ββ 2 |β | − ρ ˜ (dnt − νt dt) +˜ ρββ t t t

(83)

and initially ρ˜αα ˜ββ ˜αβ 0 = ρ 0 = |ψihψ|, ρ 0 = hβ|αi|ψihψ|. The stochastic process nt is dened as the continuous limit of 2 the discrete process nj . The Itô table for dnt is (dnt ) = dnt and E [dnt |˜ ρt ] = νt dt, where

νt = |cα |2 νtαα + cα c∗β νtαβ + c∗α cβ νtβα + |cβ |2 νtββ ,

(84)

νtαα = Tr




 L† L + Lαt∗ + L† αt + |αt |2 ρ˜αα , t

(85)

νtαβ = Tr

h

i
L† L + Lβt∗ + L† αt + αt∗ βt ρ˜αβ , t

(86)

νtβα = Tr

h

i
L† L + Lαt∗ + L† βt + αt βt∗ ρ˜βα , t

(87)

νtββ = Tr

h

i
L† L + Lβt∗ + L† βt + |βt |2 ρ˜ββ . t

(88)

Moreover, the complex functions αt and βt satisfy the conditions Z +∞ Z +∞ 2 |αt | dt = 1, |βt |2 dt = 1, 0

(89)

0

and

  Z
1 +∞ |αt |2 + |βt |2 − 2αt βt∗ dt . hβ|αi = exp − 2 0

(90)

Thus, the a posteriori state of S is given as ∗ αβ 2 ββ ρ˜t = |cα |2 ρ˜αα ˜t + c∗α cβ ρ˜βα ˜t , t + |cβ | ρ t + cα cβ ρ

(91)

 † αβ ββ βα αβ where the conditional operators ρ˜αα , ρ ˜ , ρ ˜ satisfy Eqs. (81)-(83), and ρ ˜ = ρ ˜ . The equations (81)-(83) t t t t t agree with the stochastic master equations derived in [23] (see Section IV in [23]).

11 When we take an average of ρ˜t over all realizations of the stochastic process nt (all possible outcomes) then we get the a priori evolution of the system S . One can check that the a priori state of S is described by βα ∗ αβ ∗ 2 ββ %t = |cα |2 %αα t + cα cβ %j + cα cβ %t + |cβ | %t ,

(92)

αβ ββ where the operators %αα satisfy the dierential equations t , %t , %t αα † αα ∗ %˙ αα = L%αα t t + [%t , L ]αt + [L, %t ]αt ,

(93)

αβ αβ ∗ † %˙ αβ = L%αβ t t + [%t , L ]αt + [L, %t ]βt ,

(94)

ββ ββ ∗ † %˙ ββ = Lρββ t t + [ρt , L ]βt + [L, ρt ]βt ,

(95)

 † βα αβ αβ ββ . where L acts as (80). The initial condition is %αα 0 = %0 = |ψihψ|, %0 = hβ|αi|ψihψ|, and %t = %t

B.

Homodyne detection

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 ˜ j i = p |Ψj i , |Ψ hΨj |Ψj i

(96)

where |Ψj i = cα

+∞ O

|αk ik ⊗ |ψj i + cβ

k=j

+∞ O

(97)

|βk ik ⊗ |ϕj i.

k=j

The conditional vectors |ψj i, |ϕj i from HS in (34) are given by the recurrence formulas α

(98)

β

(99)

j |ψj+1 i = Rζj+1 |ψj i,

j |ϕj+1 i = Rζj+1 |ϕj i,

where ζj+1 stands for a random variable describing the (j + 1)-th output of (21), and αj Rζj+1

βj Rζj+1

1 |α |2 1S − iHS + L† L + L† αj + j 2 2    √ +(L + αj )ζj+1 τ + O τ 3/2 ,



1 |β |2 1S − iHS + L† L + L† βj + j 2 2   √ +(L + βj )ζj+1 τ + O τ 3/2 ,



1 = √ 2

1 = √ 2









τ (100)

τ (101)

and initially we have |ψ0 i = |ϕ0 i = |ψi. Proof. To prove Theorem (4) we use the result of Section 3.2 and the linearity of the evolution equation for the total system.

12 Clearly, the conditional state of S at the moment jτ has the form (44). We start derivation of the ltering equations for the stochastic operators (45)(48) from writing down the recursive formulas

2|ψj+1 ihψj+1 | = |ψj ihψj | + L|ψj ihψj |τ +[|ψj ihψj |, L† ]αj τ + [L, |ψj ihψj |]αj∗ τ 
 √ + (L + αj ) |ψj ihψj | + |ψj ihψj | L† + αj∗ ζj+1 τ ,

(102)

 
1 2|ψj+1 ihϕj+1 | = |ψj ihϕj | 1 − |αj |2 + |βj |2 − 2βj∗ αj τ 2 +L|ψj+1 ihϕj+1 |τ +[|ψj ihϕj |, L† ]αj∗ τ + [L, |ψj ihϕj |]βj∗ τ 
 √ + (L + αj ) |ψj ihϕj | + |ψj ihϕj | L† + βj∗ ζj+1 τ ,

(103)

2|ϕj+1 ihϕj+1 | = |ϕj ihϕj | + L|ϕj ihϕj |τ +[|ϕj ihϕj |, L]βj dt + [L, |ϕj ihϕj |]βj∗ τ 
 √ + (L + βj ) |ϕj ihϕj | + |ϕj ihϕj | L† + βj∗ ζj+1 τ ,

(104)

We can readily deduce that the conditional probability of the result ζj+1 at the moment (j+1)τ when the conditional state of S is ρ˜j at the time jτ is given by

pj+1 (ζj+1 |˜ ρj ) =

√
1 1 + µj ζj+1 τ + O(τ 3/2 ), 2

(105)

where βα ∗ αβ ∗ 2 ββ µj = |cα |2 µαα j + cα cβ µj + cα cβ µj + |cβ | µj

(106)

and

µαα j = Tr




 L + L† + αj + αj∗ ρ˜αα , j

(107)

µαβ j = Tr

h

i
L + L† + αj + βj∗ ρ˜αβ , j

(108)

µβα j = Tr

h

i
L + L† + βj + αj∗ ρ˜βα , j

(109)

µββ j = Tr

h

i
L + L† + βj + βj∗ ρ˜ββ . j

(110)

Thus for the discrete stochastic process ζj we obtain the conditional mean values

√

E[ζj+1 |˜ρj ] = µj τ + O(τ 3/2 ),

(111)

2 E[ζj+1 |˜ ρj ] = 1 + O(τ 3/2 ).

(112)

Let us introduce now the stochastic process

qj =

j √ X τ ζk k=1

One can easily check that

E[∆qj = qj+1 − qj |˜ρj ] = µj τ + O(τ 3/2 ).

(113)

13 Now, taking into account that


√ 1 2 1 − µj ζj+1 τ + µ2j τ = Trρj+1 Trρj

(114)

after some algebra we nd the set of the stochastic dierence equations † ∗ ρ˜αα ˜αα = L˜ ραα ραα ˜αα j+1 − ρ j j τ + [˜ j , L ]αj τ + [L, ρ j ]αj τ   † ∗ + (L + αj ) ρ˜αα ˜αα ˜αα (∆qj − µj τ ) , j +ρ j (L + αj ) − µj ρ j

(115)

† ∗ ραβ ˜αβ = L˜ ραβ ˜αβ ρ˜αβ j , L ]αj τ + [L, ρ j ]βj τ j τ + [˜ j j+1 − ρ i h † ∗ (∆qj − µj τ ) , ˜αβ ˜αβ + (L + αj ) ρ˜αβ j j (L + βj ) − µj ρ j +ρ

(116)

† ∗ ρββ ˜ββ = L˜ ρββ ˜ββ ρ˜ββ j , L ]βj τ + [L, ρ j τ + [˜ j ]βj τ j j+1 − ρ i h
˜ββ L† + βj∗ − µj ρ˜ββ (∆qj − µj τ ) + (L + βj ) ρ˜ββ j j +ρ j

(117)

˜αβ with the initial conditions ρ˜αα ˜ββ 0 = |ψihψ|, ρ 0 = hβ|αi|ψihψ|. Then in the continuous time limit we have 0 = |ψihψ|, ρ † ∗ d˜ ραα = Lραα dt + [˜ ραα ˜αα t t , L ]αt dt + [L, ρ t ]αt dt t
 + (L + αt ) ρ˜αα ˜αα L† + αt∗ − µt ρ˜αα (dqt − µt dt) t +ρ t t

(118)

† ∗ d˜ ραβ = Lραβ dt + [˜ ραβ ˜αβ t t , L ]αt dt + [L, ρ t ]βt dt ht i
+ (L + αt ) ρ˜αβ ˜αβ L† + βt∗ − µt ρ˜αβ (dqt − µt dt) , t +ρ t t

(119)

† ∗ d˜ ρββ = Lρββ dt + [˜ ρββ ˜ββ t t , L ]βt dt + [L, ρ t ]βt dt ht i
+ (L + βt ) ρ˜ββ ˜ββ L† + βt∗ − µt ρ˜ββ (dqt − µt dt) t +ρ t t

(120)

βα ∗ αβ ∗ 2 ββ µt = |cα |2 µαα t + cα cβ µt + cα cβ µt + |cβ | µt

(121)

where

and

µαα = Tr t




 L + L† + αt + αt∗ ρ˜αα , t

(122)

µαβ t = Tr

h

i
L + L† + αt + βt∗ ρ˜αβ , t

(123)

µβα t = Tr

h

i
L + L† + βt + αt∗ ρ˜βα , t

(124)

µββ t = Tr

h

i
L + L† + βt + βt∗ ρ˜ββ . t

(125)

The process qj in the limit τ → 0 converges to the stochastic process with qt with the conditional probability E[dqt = qt+dt − qt |˜ρt ] = µt dt.

14 V.

CONCLUSION

We derived the stochastic equation describing the conditional evolution of an open quantum system interacting with the Bose eld prepared in a superposition of coherent states. We consider two schemes of measurement of the output eld: photon counting and homodyne detection. Instead of methods based on the quantum stochastic calculus and the cascades system model [23], we used the collision model with the environment given by an innite chain of qubits. We assumed that te bath qubits do not interact between themselves and they are initially prepared in the entangled state being a discrete analogue of a superposition of continuous-mode coherent states. The initial state of the compound system was factorisable. Because of the temporal correlations present in the environment, the evolution of open quantum system becomes non-Markovian. We started from the discrete in time description of the problem and obtained in the continuous time limit dierential ltering equations consistent with the results published in [22, 24]. We would like to stress that the presented method is more straight and intuitive than the methods described in [22, 24]. It not only allows to derive the equations describing the conditional evolution of the system but also enables to nd the general structure of quantum trajectories. Acknowledgements

This paper was partially supported by the National Science Center project 2015/17/B/ST2/02026. VI.

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