The position non Markovian master equation

Chaos, Solitons and Fractals 95 (2017) 57–64

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

The position non Markovian master equation A. Pourdarvish a, J. Sadeghi b, N.J. Hassan a,∗ a b

Department of Statistics, Faculty of Mathematical Sciences, University of Mazandran, Babolsar, Iran Department of Physics, Faculty of Basic Science, University of Mazandaran, Babolsar, Iran

a r t i c l e

i n f o

Article history: Received 3 May 2016 Accepted 14 December 2016

Keywords: Post Markovian perturbation Position Non Markovian master equation Statistical operator Functional operator

a b s t r a c t In this paper, we derive the non Markovian master equation (NMME) that correspond to position non Markovian stochastic Schrödinger equation (PNMSSE) in linear and non linear cases. In this case, using Nivokov property we derive four formulas of (NMME) for linear and non linear cases respectively. The functional derivative operator may depend on time and independent with respect to noise. Here, we determine the functional derivative of statistical operator. When the functional derivative operator depends on time and noise, one can calculate the perturbation and post Markovian perturbation for the functional operator, which exists in position non Markovian equation of motion (PNMEM). In order to explain our theory, we present a simple non Markovian example. Finally, we give the conclusion and the plan for future works. © 2016 Published by Elsevier Ltd.

1. Introduction

such that hˆ (t, s, z∗ ) = δδz∗ is the functional derivative operator. Also

An open quantum system is a system interacting with external bath [1–3]. It can be described in different ways including: Local and non local master equation for statistical operator [4–6] and stochastic Schrödinger equations (SSEs) [7–11]. The exact convolutionless master equation for quantum Brownian model is calculated in different contexts [12–25]. The exact non Markovian master equation for the two level system is derived in Ref. [26]. The relation between system states |ψ z  and reduced density operator (statistical operator) ρ (t) [27] is defined in linear case as,

the second functional operator is,

s

ρ (t ) = E˜[|ψ˜ z (t )ψ˜ z (t )|],

(1)

where ρ (t ) = E˜ [P˜(t )] and

P˜(t ) = |ψ˜ z (t )ψ˜ z (t )|,



 i ∂t |ψ˜ z (t ) = − Hˆint + Lˆz∗ (t , t ) − LˆDˆ (z, t ) − Lˆ† Mˆ (z, t ) |ψ˜ z (t ), η

(3)

Dˆ (z, t ) =

0

γ ∗ (t + s )hˆ (t, s, z∗ )ds

∗

Corresponding author. E-mail addresses: [email protected] (A. Pourdarvish), [email protected] (J. Sadeghi), [email protected], [email protected] (N.J. Hassan). http://dx.doi.org/10.1016/j.chaos.2016.12.016 0960-0779/© 2016 Published by Elsevier Ltd.

0

t

α (t − s )hˆ (t, s, z∗ )ds,

(5)

where the conjugate correlation function is defined,

γ ∗ (t + s ) = E˜[z∗ (t , t )z∗ (t , s )] =



(g∗ )2k expi k (t+s) .

(6)

k

And

α (t − s ) = E˜[z(t , t )z∗ (t , s )] =



|gk |2 exp−i k (t−s) ,

(7)

k

z(t, s ) =



√

|gk | 2xk (t ) exp−i k (t−s) ,

(4)

(8)

k

where xk represent the actual results of the measurement. In non linear case we have the following expression,

ρ (t ) = E[|ψz (t )ψz (t )|],

(9)

where ρ (t ) = E[P (t )] and

P (t ) = |ψz (t )ψz (t )|,

where the first functional operator is t



such that the noise function is

(2)

where |ψ˜ z (t ) is solution of the linear (PNMSSE) which is defined by the following equation,



ˆ (z, t ) = M

(10)

where |ψ z (t) is the solution of non linear PNMSSE which is defined as,

 i ∂t |ψz (t ) = − Hˆint + Lˆz∗ (t ) − LˆDˆ (z, t ) +  LˆDˆ (z, t ) η  ˆ (z, t ) +  Lˆ† M ˆ (z, t ) |ψz (t ), − Lˆ† M (11)

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A. Pourdarvish et al. / Chaos, Solitons and Fractals 95 (2017) 57–64

where Lˆ = Lˆ − Lˆt . The convolutionless non Markovian SSE may serve as a powerful tool for the derivation of convolutionless master equation for non Markovian quantum systems [28]. The post Matkovian perturbation to the non Markovian SSE, leads us to approximate results for the statistical operator, when the system is closely Markovian [29]. A perturbative approach is presented for non Markovian coherent and quadrature SSE [30], which is based on the expansion of the functional operators and the consistency condition that it has the following expression

∂t

δ ˜ δ |ψ (t ) = ∗ ∂t |ψ˜ z (t ), δ zt∗ z δ zt

(12)

In this paper, we derive non Markovian master equation (NMME), that correspond to PNMSSE in linear and non linear cases. In this side, using Nivokov property we derive four formulas of NMME in linear and non linear cases respectively. When the functional derivative operator depend on time and independent with noise, also here we study the convolutionless PNMMEs by determining functional derivative operator without using SSE. When functional operator in PNMEM depend on time and noise, we derive the perturbation and post Markovian perturbation to the corresponding operator (in order to have definite operator). Also, without using SSE here we note that the PNMEM can be solved numerically. In this case, we take the combined functional derivative for probability operator to derive stochastic differential equation (SDE) for the functional derivative operator. In this work, can be divided into six sections. In Section 2, we derive NMME that correspond to PNMSSE in linear and non linear cases. Also we derive four formulas for linear and non linear NMME respectively. We discuss the convolutionless PNMMEs without using PNMSSE. In Section 3, we derive zero and first order stochastic perturbation equations for the functional operator which exist in PNMEM to calculate this operator. In Section 4, we achive the post Markovian perturbation to PNMEM and obtain position post Markovian equation of motion (PPMEM) for probability operator. In Section 5, we present simple example to explain our theory. Finally, we give our conclusion and future work.

−[P (t )Bˆ† (t ) Lˆ† + P (t )Aˆ † (t ) Lˆ] +  LˆBˆ(t )P (t ) +P (t )Bˆ† (t ) Lˆ†  +  Lˆ† Aˆ (t )P (t ) + P (t )Aˆ † (t ) Lˆ i = − [Hˆ , P (t )] + ( Lˆz∗ (t , t )P (t ) + P (t )z(t , t ) Lˆ† )

η

− 2 LˆBˆ(t )P (t ) − 2 Lˆ† Aˆ (t )P (t ) − P (t )Bˆ† (t )Lˆ† −P (t )Aˆ † (t ) 2 Lˆ

i

ρ˙ (t ) = − [Hˆ , ρ (t )] + LˆE (z∗ (t , t )P˜(t )) η +E (P˜(t )z(t, t ))Lˆ† − LˆLˆBˆ(t )ρ (t )ds −Lˆ† Aˆ (t )ρ (t )ds − ρ (t )Bˆ† (t )Lˆ† ds − ρ (t )Aˆ † (t )Lˆds, i

ρ˙ (t ) = − [Hˆ , ρ (t )] + LˆE (z∗ (t )P (t )) η +E (P (t )z(t )) Lˆ† − 2 LˆBˆ(t )ρ (t ) − 2 Lˆ† Aˆ (t )ρ (t ) − ρ (t )Bˆ† (t )Lˆ† − ρ (t )Aˆ † (t ) 2 Lˆ,

∂t

= [∂t |ψ˜ z (t )ψ˜ t | + |ψ˜ z (t )ψ˜ z (t )|∂t ],

E[P (t )z(t )] = E



t

0

dsα (t − s )

δ P (t ) δ z∗ (s )



E (z (t , t )P˜(t )) = ∗



t 0 t 0



 δ ˜ E[z (t , t )z (t , S )]E P (t ) ds δ z∗ (s ) ∗

∗

γ ∗ (t + s )hˆ (t, s, z∗ )ρ (t )ds

and also find E (P˜(t )z(t, t )),

E (z(t , t )P˜(t )) =



t

0

 =

t

0



E[z(t , t )z(t , s )]E

γ (t + s )ρ (t )hˆ † (t, s, z∗ )ds

(14)

and substituting Eqs. (3) and (11) in Eqs. (13) and (14), one can obtain respectively as,

E (z (t )P (t )) =

η

−[LˆBˆ(z, t )P˜(t ) + Lˆ† Aˆ (z, t )P˜(t )] −[P˜(t )Bˆ† (z, t )Lˆ† + P˜(t )Aˆ † (z, t )Lˆ].

η −[ LˆBˆ(t )P (t ) + Lˆ† Aˆ (t )P (t )]

t

0

=

t

0



 δ E[z (t )z (s )]E P (t ) ds δ z∗ (s ) ∗

∗

γ ∗ (t + s )hˆ (t, s, z∗ )ρ (t )ds

(23)

and also use E(P(t)z(t)) for the non linear state,

and

i P˙ (t ) = − [Hˆ , P (t )] + ( Lˆz∗ (t )P (t ) + P (t )z(t ) Lˆ† )

 

(15)

(22)

Eq. (22) is first formula of the linear position non Markovian master equation (LPNMME). Same as before, we back to first case of non linear state and use E(z∗ (t)P(t)), so we have ∗

i P˜˙ (t ) = − [Hˆ , P˜(t )] + (Lˆz∗ (t , t )P˜(t ) + P˜(t )z(t , t )Lˆ† )

(21)

Substituting Eqs. (20) and (21) in Eq. (17), one can obtain,

i

∂ Pt = [∂t ψz (t )ψt | + ψz (t )ψz (t )|∂t ], ∂t

(20)

 δ ˜ P (t ) ds δ z (s )

ρ˙ (t ) = − [Hˆ , ρ (t )] − Lˆ† Aˆ (t )ρ (t ) − ρ (t )Aˆ † (t )Lˆ η

and

(19)

If we define the terms E (z∗ (t , t )P˜(t )) and E (P˜(t )z(t, t )) in linear and E(z∗ (t)P(t)) and E(P(t)z(t)) in non linear cases, so we have four cases. Here, we note that linear and non linear are based on the correlation functions γ ∗ (t + s ) and γ (t + s ). Now we are going to investigate the first case of linear state and find E (z∗ (t , t )P˜(t )), which is given by,

=

(13)

(18)

Eqs. (17) and (18) are still far from being closed evolution equation for ρ (t). We introduce the Novikov’s theorem [31] which is given by,

2. The master equations corresponding to position non Markovian stochastic Schrödinger equation

∂ P˜t

(17)

and



In this section, we derive the master equation (ME) that correspond to PNMSSE in linear and non linear cases. In this side, we assume that the functional derivative operator hˆ (t, s, z∗ ) is independent with noise. We take the time derivative for the probability operator, one can obtain

(16)

where Eqs. (15) and (16) linear and non-linear. Taking the mathematical expectation of equation with respect to (15) and (16), we have following equations,

E (z(t )P (t )) =

 

=

t 0 t 0



 δ E[z(t )z(s )]E P (t ) ds δ z (s ) γ (t + s )ρ (t )hˆ † (t, s, z∗ )ds

(24)

A. Pourdarvish et al. / Chaos, Solitons and Fractals 95 (2017) 57–64

−ρ (t )Bˆ† (t )Lˆ†

Substituting Eqs. (23) and (24) in Eq. (18), one can obtain,

i

ρ˙ (t ) = − [Hˆ , ρ (t )] − ( 2 Lˆ − Lˆ )Bˆ(t )ρ (t ) η −ρ (t )Bˆ† (t )( 2 Lˆ† − Lˆ† ) − 2 Lˆ† Aˆ (t )ρ (t ) − ρ (t )Aˆ † (t ) 2 Lˆ



t

0

 = and

E (z(t , t )P˜(t )) =

t

0

 

=

t 0 t 0



E[z∗ (t , t )z(t , s )]E

(25)

 δ ˜ P (t ) ds δ z (s )

α ∗ (t − s )ρ (t )hˆ † (t, s, z∗ )ds 

E[z(t , t )z∗ (t , s )]E

 δ ˜ P ( t ) ds δ z∗ (s )

α (t − s )hˆ (t, s, z∗ )ρ (t )ds

 = and

E (z(t )P (t )) =

t

(27)

 0

t

(29)

E (z (t , t )P˜(t )) = ∗

0

 = and

E (z(t , t )P˜(t )) = =

0

 

t

t 0 t 0



E[z(t , t )z∗ (t , s )]E

=

E (z(t , t )P˜(t )) =

0



t 0 t 0

α ∗ (t − s )ρ (t )hˆ † (t, s, z∗ )ds

(38)



 δ ˜ E[z(t , t )z(t , S )]E P (t ) ds δ z (s ) γ (t + s )ρ (t )hˆ † (t, s, z∗ )ds

(39)

ρ˙ (t ) = − [Hˆ , ρ (t )] + [Lˆ, ρ (t )Aˆ † (t )] − Lˆ† Aˆ (t )ρ (t ) η ˆ −LBˆ(t )ρ (t )

(40)

where the Eq. (40) is fourth formula of LPNMME. Again we continue the above process for the non linear state, which is given by,

(31)

 δ ˜ P ( t ) ds δ z∗ (s )

E (z (t )P (t )) = ∗

= and

E (z(t )P (t )) =

δ ˜ P (t ) ds δ z∗ (s )

α (t − s )hˆ (t, s, z∗ )ρ (t )ds

t 0 t 0



t

0

 =



 

(32)

ρ˙ (t ) = − [Hˆ , ρ (t )] + [Aˆ (t )ρ (t ), Lˆ† ] − ρ (t )Aˆ † (t )Lˆ η

t

 δ ˜ E[z (t , t )z(t , s )]E P (t ) ds δ z (s )

i

(30)

Substituting Eqs. (32) and (33) in Eq. (17), one can obtain

i

0



∗

Putting Eqs. (38) and (39) in Eq. (17), then we have

γ ∗ (t + s )hˆ (t, s, z∗ )ρ (t )ds 

t



where Eq. (31) is second formula of NLPNMME. As we pointed before, we are going to investigate the third case of linear state, so we have following,

E[z∗ (t , t )z∗ (t , s )]E



=

i

t

(36)

ρ˙ (t ) = − [Hˆ , ρ (t )] − ( 2 Lˆ − Lˆ )Bˆ(t )ρ (t ) − ρ (t )Bˆ† (t ) 2 Lˆ† η +[Aˆ (t )ρ (t ) Lˆ† − 2 Lˆ† Aˆ (t )ρ (t )] − ρ (t )Aˆ † (t ) 2 Lˆ (37)



ρ˙ (t ) = − [Hˆ , ρ (t )] − ( 2 Lˆ − Lˆ )ρ (t )Aˆ † (t ) η −Aˆ (t )ρ (t )( 2 Lˆ† − Lˆ† ) − 2 LˆBˆ(t )ρ (t ) − ρ (t )Bˆ† (t ) 2 Lˆ†



(35)

 δ P ( t ) ds δ z∗ (s )

α (t − s )hˆ (t, s, z∗ )ρ (t )ds



Putting Eqs. (29) and (30) in Eq. (18), we get

E (z∗ (t , t )P˜(t )) =

t

0

and

α (t − s )hˆ (t, s, z∗ )ρ (t )ds

0

E[z(t )z∗ (s )]E

0

 =

γ ∗ (t + s )hˆ (t, s, z∗ )ρ (t )ds 

t

(28)

δ E[z(t )z∗ (s )]E P (t ) ds δ z∗ (s )

t

t

 δ P ( t ) ds δ z∗ (s )

where Eq. (37) is third formula of the NLPNMME. Finally, we concentrate the final case for the linear and non linear states. First, we investigate linear state for the corresponding case. It has following form,

δ P (t ) ds δ z (s )



0

0





α ∗ (t − s )ρ (t )hˆ † (t, s, z∗ )ds

0

 =

E[z∗ (t )z(s )]E

0

and

E[z∗ (t )z∗ (s )]E

i

Eq. (28) is second formula of lineae position non Markovian master equation (LPNMME). Again, we back the second case of non linear state, which is given by following expression,

E (z∗ (t )P (t )) =

=



t

Substituting Eqs. (35) and (36) in Eq. (18), one can obtain

i



 

(26)

ρ˙ (t ) = − [Hˆ , ρ (t )] + [Lˆ, ρ (t )Aˆ † (t )] + [Aˆ (t )ρ (t ), Lˆ† ] η ˆ −LBˆ(t )ρ (t ) − ρ (t )Bˆ† (t )Lˆ†

t

E (z∗ (t )P (t )) =

E (z(t )P (t )) =

Putting Eqs. (26) and (27) in Eq. (17), we get



(34)

where Eq. (34) is third formula of LPNMME. In third case of non linear state also have,

So, the Eq. (25) is first formula of the non linear position non Markovian master equation (NLPNMME). Now, again we follow the second case linear state which is given by,

E (z∗ (t , t )P˜(t )) =

59

0

t



 δ E[z (t )z(s )]E P (t ) ds δ z (s ) ∗

α ∗ (t − s )ρ (t )hˆ † (t, s, z∗ )ds

(41)



 δ E[z(t )z(s )]E P (t ) ds δ z (s ) γ (t + s )ρ (t )hˆ † (t, s, z∗ )ds

(42)

Putting Eqs. (41) and (42) in Eq. (18), then we have

i

(33)

ρ˙ (t ) = − [Hˆ , ρ (t )] − ( Lˆρ (t )Aˆ † (t ) η −ρ (t )Aˆ † (t ) 2 Lˆ ) − ρ (t )Bˆ† (t )( 2 Lˆ† − Lˆ† ) − 2 Lˆ† Aˆ ρ (t ) − 2 LˆBˆρ (t ) where the Eq. (43) is fourth formula of NLPNMME.

(43)

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A. Pourdarvish et al. / Chaos, Solitons and Fractals 95 (2017) 57–64

3. Convolutionless linear position non Markovian equation of motion and master equation

where Eq. (53) is fourth formula of differential equation for operator hˆ (t, s, z∗ ).

In this section, we study the functional derivative operator hˆ (t, s, z∗ ) which is generally dependent on the noise z∗ . In that case we try to determine functional derivative operator, without using the stochastic Schrödinger equation (SSE). We take the combined functional derivative for the probability operator P˜(t ) (defined in Eq. (2)), so one can obtain,

4. The perturbative to linear position non Markovian equation of motion

 δ ˜ δ δ ˜ ˜ ˜ P (t ) = |ψ (t )ψz (t )| = |ψ (t ) ψ˜ z (t )| δ z∗ (s ) δ z∗ (s ) z δ z∗ (s ) z  δ ˜ ˜ +|ψz (t ) ψ (t )| . (44) δ z∗ (s ) z

In this section, in order to determine the functional operators we derive the perturbation to LPNMEM without using SSE. We take the combined functional derivative for the probability operator and obtain the zero and first orders stochastic perturbation equations for the functional operator. We rewrite the Eq. (15) under assump† tion as hˆ 0 (t, s, z∗ ) which is commutator with Lˆ and Lˆ† as

i P˜˙ (t ) = − [Hˆ , P˜(t )] + (Lˆz∗ (t , t )P˜(t ) + P˜(t )z(t , t )Lˆ† )

η

Taking time derivative of Eq. (44), we achieve the following expression,

∂t

  δ ˜ δ ˜ z (t ) ψ˜ z (t )| P ( t ) = | ψ ∂ t δ z∗ (s ) δ z∗ (s )   δ ˜ z (t ) ∂t ψ˜ z (t )| + | ψ δ z∗ (s )   δ ˜ z (t )| + ∂t |ψ˜ z (t )  ψ δ z∗ (s )   δ ˜ z (t )| . +|ψ˜ z (t )∂t  ψ δ z∗ (s )

where

ˆ 0 (t, z ) = W

 0

(45)

(56)

ˆ 0 (t, z ) = W



ˆ j (t, z ), W 0

(57)

j

and also the expansion for the memory is

λ(t, s ) =



λ(t , s ) j (t , s )

(58)

j

ˆ j (t, z ) is defined as, In that case the functional operator W 0

(46) (47)

ˆ j (t, z ) = W 0



t

0

λ(t , s ) j (t , s )hˆ 0 (t , s, z∗ )ds. 

 + (48)

(49)

(59)

The time derivative of Eq. (59) give us following expression,

∂t Wˆ 0j (t, z ) = λ(t, s ) j (t, t )Lˆ −

Eq. (48) is stochastic differential equation for operator hˆ (t, s, z∗ ) under initial condition hˆ (s, s, z∗ ) = Lˆ. Now, if the operator hˆ (t, s, z∗ ) is independent with the noise z∗ , we will take mathematical expectation for the Eq. (47), so we find following equation,

∂t hˆ (t, s, z∗ )ρ (t ) = ∂t ρ (t )hˆ (t , s, z∗ )

(55)

ˆ 0 (t, z ) is defined as and the expansion for W

Substituting Eq. (15) in Eq. (47), we will arrive at,

∂t hˆ (t, s, z∗ ) = [Lˆz∗ (t , t ) + z(t , t )Lˆ† ]hˆ (t , s, z∗ ) −[LˆBˆ(t, z ) + Bˆ† (t , z )Lˆ† ]hˆ (t , s, z∗ ) −[Lˆ† Aˆ (t, z ) + Aˆ † (t , z )Lˆ]hˆ (t , s, z∗ )

λ(t , s )hˆ 0 (t , s, z∗ )ds

λ(t, s ) = γ ∗ (t + s )Lˆ + α (t − s )Lˆ†

so

∂t hˆ (t, s, z∗ )P˜(t ) = ∂t P˜(t )hˆ (t , s, z∗ )

t

(54)

such that

By using Eq. (12) we get,

∂t hˆ (t, s, z∗ )P˜(t ) = [hˆ (t, s, z∗ )∂t |ψ˜ z (t )ψ˜ z (t )| +hˆ (t, s, z∗ )|ψ˜ z (t )∂t ψ˜ z (t )|] +[∂t |ψ˜ z (t )ψ˜ z (t )|hˆ (t , s, z∗ ) +|ψ˜ z (t )∂t ψ˜ z (t )|hˆ (t , s, z∗ )].

ˆ 0 (t, z )P˜(t ) − P˜(t )W ˆ † (t, z ) −W 0

t 0



kj ˆ j (t, z ) + i j W 0 2

λ(t, s ) j (t, s )∂t hˆ 0 (t, s, z∗ )ds.

In order to find ∂t hˆ 0 (t, s, z∗ ), one can need to obtain the combined functional derivative for probability operator P˜(t ). So, we rewrite Eq. (47) as,

∂t hˆ 0 (t, s, z∗ )P˜(t ) = ∂t P˜(t )hˆ 0 (t , s, z∗ )

(61)

Substituting Eq. (54) into Eq. (61), we get

Putting first, second, third and fourth formulas of the linear position non Markovian master equations into Eq. (47) respectively, we get following expression,

∂t hˆ 0 (t, s, z∗ ) = [Lˆz∗ (t , t ) + z(t , t )Lˆ† ]hˆ 0 (t , s, z∗ ) ˆ 0 (t, z ) + W ˆ † (t , z )]hˆ 0 (t , s, z∗ ) −[W 0

∂t hˆ (t, s, z∗ ) = −[Lˆ† Aˆ (t ) + Aˆ † (t )Lˆ]hˆ (t , s, z∗ )

Putting Eq. (62) into Eq. (60), we have

(50)

where Eq. (50) is first formula of differential equation for operator hˆ (t, s, z∗ ),

∂t hˆ (t, s, z∗ ) = [Lˆ, Aˆ † (t )]hˆ (t , s, z∗ ) + [Aˆ (t ), Lˆ† ]hˆ (t , s, z∗ )

where Eq. (52) is third formula of differential equation for operator hˆ (t, s, z∗ ), and

∂t hˆ (t, s, z∗ ) = [[Lˆ, Aˆ † (t )] − LˆBˆ(t ) − Lˆ† Aˆ (t )]hˆ (t , s, z∗ )

(53)

(62)



kj ˆ j (t, z ) + i j W 0 2

ˆ j (t , z ) +[Lˆz∗ (t , t ) + z(t , t )Lˆ† ]W 0  ˆ † (t, z )W ˆ j (t, z ) − ˆ j,k (t, z ) −W W 0 0 1

(51)

(52)



2 ∂t Wˆ 0j (t, z ) = (g∗j )2 exp−k j t−2i j t+|g j | Lˆ −

where Eq. (51) is second formula of differential equation for operator hˆ (t, s, z∗ ),

∂t hˆ (t, s, z∗ ) = [[Aˆ (t ), Lˆ† ] − Aˆ † (t )Lˆ − Bˆ† (t )Lˆ† ]hˆ (t , s, z∗ )

(60)

(63)

k

Eq. (63) is zero order perturbation equation for functional operator ˆ 0 (t, z ), where operators Lˆ and Lˆ† are commutator. The first order W ˆ j,k (t, z ) is defined as, functional operator W 1

ˆ j,k (t, z ) = W 1



t 0

λ(t , s ) j (t , s )hˆ k1 (t , s, z∗ )ds,

(64)

A. Pourdarvish et al. / Chaos, Solitons and Fractals 95 (2017) 57–64

where

ˆ 0k (t, z ). hˆ k1 (t, s, z∗ ) = hˆ 0 (t, s, z∗ )W

61

+z(t , t )Lˆ† ]Lˆ† Lˆ

(65)



t

0

λ∗ (t, s )ds]

Taking time derivative of the Eq. (64), then we have following result,

+[−2Lˆ (Lˆz∗ (t , t ) + z(t , t )Lˆ† )Lˆ

∂t Wˆ 1j,k (t, z ) = (g∗j )2 exp−k j t−2i j t+|g j | hˆ k1 (t , t , z∗ )   kj ˆ j,k (t, z ) + W ˆ j (t, z )[Lˆz∗ (t , t ) − + i j W 1 0

+2Lˆ3

2

+Lˆ† Lˆ2

ˆ 0 (t, z ). Eq. (66) is first order perturbation equation for W

+Lˆ† Lˆ†

(67)

(69)

Substituting Eq. (69) into Eq. (68), we have the terms O(t − s )2 , O(t − s )3 and O(t − s )4 which are powers of (t − s )2 , (t − s )3 and (t − s )4 respectively, also all equal zero independently,

(70)

0

Putting Eq. (70) into Eq. (71), one can obtain

[∂t2 hˆ 0 (t, s, z∗ )|t=s ] = [[(Lˆz∗ (t , t ))2 + |z(t , t )|2 Lˆ, Lˆ† +(z(t , t )Lˆ† )2 ]Lˆ − [Lˆz∗ (t , t )  t +z(t , t )Lˆ† ]Lˆ2 λ(t , s )ds − [Lˆz∗ (t , t ) 0

t

0

λ∗ (t , s )ds

(71)

0

0

0

 t

0

0 u 0

λ (t, s )λ(s, u )duds ∗

u

0 0 t u

 t 0

u

λ∗ (t, s )λ∗ (s, u )duds

λ(t, s )λ∗ (s, u )duds

λ(t, s )λ(s, u )duds



0

t

λ(t, s )hˆ 0 (t, s, z∗ )ds λ(t, s )[Lˆ + [∂t hˆ 0 (t, s, z∗ )|t=s ](t − s )

1 + [∂t2 hˆ 0 (t, s, z∗ )|t=s ](t − s )2 ]ds 2  t = Lˆ λ(t, s )ds + [Lˆz∗ (t , t )

−Lˆ2

λ∗ (t, s )[Lˆ† [∂t hˆ 0 (t, s, z∗ )|t=s ]

+[∂t hˆ †0 (t, s, z∗ )|t=s ]Lˆ]ds



+z(t , t )Lˆ† ]Lˆ

0

t

ˆ 0 (t, z ) = W

0

[∂t2 hˆ 0 (t, s, z∗ )|t=s ] = [Lˆz∗ (t , t ) + z(t , t )Lˆ† ][∂t hˆ 0 (t , s, z∗ )|t=s ]  t −2Lˆ λ(t, s )[∂t hˆ 0 (t, s, z∗ )|t=s ]ds 

0

t

(72)

ˆ 0 (t, z ) as Now, we can determine the corresponding operator W

=

and

−



λ∗ (t, s )λ∗ (s, u )duds

 t



+2Lˆ3

λ∗ (t, s )ds

λ∗ (t, s )λ∗ (s, u )duds]

λ∗ (t, s )λ(t, s )duds]Lˆ

0

+2LˆLˆ† Lˆ

(t, s, z∗ )|t=s ]Lˆ )(t − s )]

0

0

+2Lˆ† Lˆ† Lˆ

λ∗ (t, s )[Lˆ† Lˆ + (Lˆ† [∂t hˆ 0 (t, s, z∗ )|t=s ]

0

0 0 t u

+2Lˆ† Lˆ2

0

t

u

u

λ∗ (t , s )ds

λ∗ (t, s )λ(s, u )duds

0

 t

0

0

∂t hˆ 0 (t, s, z∗ ) = [Lˆz∗ (t , t ) + z(t , t )Lˆ† ]Lˆ + [Lˆz∗ (t , t ) +z(t , t )Lˆ† ][∂t hˆ 0 (t , s, z∗ )|t=s ](t − s ) + O(t − s )2  t − λ(t, s )[Lˆ2 + 2Lˆ[∂t hˆ 0 (t, s, z∗ )|t=s ](t − s )]ds

[∂t hˆ 0 (t, s, z∗ )|t=s ] = [Lˆz∗ (t , t ) + z(t , t )Lˆ† ]Lˆ  t  −Lˆ2 λ(t, s )ds − Lˆ† Lˆ

0

t

−[(LˆLˆ† Lˆ + 2Lˆ† Lˆ2 )z∗ (t , t )  t +3Lˆ† Lˆ† Lˆz(t , t )] λ∗ (t , s )ds

Putting Eq. (67) into Eq. (62), one can obtain

+O(t − s )2 + O(t − s )3 + O(t − s )4

0



= [(Lˆz∗ (t , t ))2 + |z(t , t )|2 Lˆ, Lˆ† + (z(t , t )Lˆ† )2 ]Lˆ −[(Lˆ3 + 2Lˆ4 )z∗ (t , t ) + (Lˆ† Lˆ2  t +2LˆLˆ† Lˆ )z(t , t )] λ(t , s )ds

∂t hˆ 0 (t, s, z∗ ) = [∂t hˆ 0 (t, s, z∗ )|t=s ] + [∂t2 hˆ 0 (t, s, z∗ )|t=s ](t − s ). (68)

+[∂

u

 t

 +Lˆ† Lˆ

Taking the time derivative of the Eq. (67), we get

ˆ† t h0

0

λ(t , s )ds

0

λ(t, s )λ∗ (s, u )duds]

+[−(Lˆ† Lˆz∗ (t , t ) + z(t , t )Lˆ† Lˆ† )

In this section, we derive post Markovian perturbation to LNˆ (t, z ) consequently. MEM in the limit of the functional operator W In that case the LNMEM become solvable numerically. The Taylor expansion of the operator hˆ 0 (t, s, z∗ ) in power of (t − s ), is defined as

0

0

 t

+Lˆ† Lˆ† Lˆ

5. The post Markovian perturbation to linear position non Markovian equation of motion

−

0

t

λ(t, s )λ(s, u )duds

+[−Lˆ† (Lˆz∗ (t , t ) + z(t , t )Lˆ† )Lˆ

l

t

u

0 t u



ˆ j (t , z )W ˆ 0k (t , z ) − W ˆ † (t , z )W ˆ 0k (t , z ) +z(t , t )Lˆ† ]W 0 0  j,k,l ˆ − W (t, z ) (66) 2



0

+2LˆLˆ† Lˆ

2

hˆ 0 (t, s, z∗ ) = Lˆ + [∂t hˆ 0 (t, s, z∗ )|t=s ](t − s ) 1 + [∂t2 hˆ 0 (t, s, z∗ )|t=s ](t − s )2 . 2

 t



 t

−Lˆ† Lˆ

0

u

0

 t 0

t 0

λ(t , s )(t − s )ds

λ∗ (t, s )λ(s, u )(t − s )duds

u 0



λ(t, s )λ∗ (t − s )(s, u )duds

1 + [(Lˆz∗ (t , t ))2 + |z(t , t )|2 Lˆ, Lˆ† 2  t +(z(t , t )Lˆ† )2 ]Lˆ λ(t , s )(t − s )2 ds 0

1 − [(Lˆ3 + 2Lˆ4 )z∗ (t , t ) + (Lˆ† Lˆ2 + 2LˆLˆ† Lˆ )z(t , t )] 2

62

A. Pourdarvish et al. / Chaos, Solitons and Fractals 95 (2017) 57–64

 t

×

0

u 0

In linear case, the first formula of LPNMME for two level atom model is

λ(t, s )λ(s, u )(t − s )2 duds

ρ˙ (t ) = −σˆ † Aˆ (t )ρ (t ) − ρ (t )Aˆ † (t )σˆ

1 − [(LˆLˆ† Lˆ + 2Lˆ† Lˆ2 )z∗ (t , t ) + 3Lˆ† Lˆ† Lˆz(t , t )] 2  t u × λ(t, s )λ∗ (s, u )(t − s )2 duds 0

0

+Lˆ† Lˆ2

 t 0

ˆ† ˆ† ˆ

+L L L

+Lˆ3

0

 t

0

u

0 0 t u

 +LˆLˆ† Lˆ

u v

0

0

 t 0

0

 v

u 0

 v

0

0

Putting Eq. (85) into Eq. (86), we get

ρ˙ (t ) = −σˆ † A(t )σˆ ρ (t ) − ρ (t )σˆ † A∗ (t )σˆ

λ(t, s )λ∗ (s, u )λ(u, v )(t − s )2 dvduds

 v

λ(t, s )λ (s, u )λ (u, v )(t − s ) dvduds ∗

∗

2

λ(t, s )λ(s, u )λ∗ (u, v )(t − s )2 dvduds

λ(t, s )λ(s, u )λ(u, v )(t − s )2 dvduds (73)

where

f0 (t ) =



t

0 t

0

 = f2 (t ) = f3 (t ) = f4 (t ) = f5 (t ) = f6 (t ) = f7 (t ) = f8 (t ) = f9 (t ) =

t

0

(γ (t + s )Lˆ + α (t − s )L )ds, ˆ†

(74)

Aˆ (t ) =

u 0

 t 0



u 0

t 0

0

u 0

 t 0

u 0

 t 0

u

0

(77)

u

(79)

λ(t, s )λ∗ (s, u )(t − s )2 duds,

(80)

 0

t

α (t − s )q(t, s )σˆ ds = A(t )σˆ

By using Eq. (50), we find the first differential equation (DE) for q(t, s), which is given by,

∂t q(t, s ) = −[σˆ † A(t ) + σˆ † A† (t )]q(t , s )

(89)

The second formula of LPNMME for the corresponding model is,

ρ˙ (t ) = [σˆ , ρ (t )σˆ † A∗ (t )] + [A(t )σˆ ρ (t ), σˆ † ] − σˆ B(t )σˆ ρ (t ) −ρ (t )σˆ † B∗ (t )σˆ † 

=

t 0 t 0

(90)

γ ∗ (t + s )hˆ 0 (t, s, z∗ )ds γ ∗ (t + s )q(t, s )σˆ ds = B(t )σˆ

(91)

∂t q(t, s ) = [σˆ , σˆ † A∗ (t )]q(t, s ) + [A(t )σˆ , σˆ † ]q(t, s )

(92)

The third formula of LPNMME is,

ρ˙ (t ) = [A∗ (t )σˆ ρ (t ), σˆ † ] − ρ (t )σˆ † A∗ (t )σˆ −ρ (t )σˆ † B∗ (t )σˆ †

(93)

the third DE for q(t, s) can be found by using Eq. (52), so we have

∂t q(t, s ) = [[A(t )σˆ , σˆ † ] − σˆ † A† (t )σˆ −σˆ † B† (t )σˆ † ]q(t, s )

(94)

The final formula of LPNMME is

 v  v  v

0

0

u 0

(78)

λ(t, s )λ(s, u )(t − s )2 duds,

0

 t 0

λ(t, s )λ∗ (s, u )(t − s )duds,

0

 t 0

(76)

0 u

0

α (t − s )hˆ 0 (t, s, z∗ )ds =

By using Eq. (51), the second DE for q(t, s) will be as,

λ(t, s )λ(s, u )(t − s )duds,

0

 t

t

 (75)

λ(t, s )(t − s )2 ds,

 t



(88)

Bˆ(t ) =

(γ ∗ (t + s )Lˆ + α (t − s )Lˆ† )(t − s )ds,

0

f10 (t ) =

0

∗

λ(t, s )(t − s )ds

 t

and

t

(87)

where

where



f1 (t ) =

λ(t, s )ds =



(86)

(95)

λ(t, s )λ (s, u )λ(u, v )(t − s ) dvduds,

(81)

λ(t, s )λ∗ (s, u )λ∗ (u, v )(t − s )2 dvduds,

(82)

∂t q(t, s ) = [[σˆ , σˆ † A∗ (t )] − σˆ B(t )σˆ − σˆ † A(t )σˆ ]q(t , s )

(83)

As we told before we applied four linear formula for two level atom as a example. So, this help us to consider the four non linear formula for the corresponding example. Now, we study with non linear case, the first formula of NLPNMME for this model which is given by,

∗

2

λ(t, s )λ(s, u )λ∗ (u, v )(t − s )2 dvduds

 v 0

ρ˙ (t ) = [σˆ , ρ (t )σˆ † A∗ (t )] − σˆ B(t )σˆ ρ (t ) −σˆ † A(t )σˆ ρ (t )

λ(t, s )λ(s, u )λ(u, v )(t − s )2 dvduds

(84)

6. The corresponding two level atom example

1 ) 0

and σˆ y = (0i

−i ) 0

are called

by Pauli matrices. The functional derivative operator is defined as,

hˆ 0 (t, s, z∗ ) = q(t, s )σˆ

(96)

ρ˙ (t ) = −( 2 σˆ − σˆ )B(t )σˆ ρ (t ) − ρ (t )σˆ † B∗ (t )( 2 σˆ † − σˆ † ) − 2 σˆ † A(t )σˆ ρ (t ) − ρ (t )σˆ † A∗ (t ) 2 σˆ (97) in this case the second formula of NLPNMME is

The model is a two level atom (TLA) [27] which is coupled linearly by a single mode bath with no deturning. It means that Hˆ = 0, k = 0 and k = 1. Let Lˆ = σˆ = σˆ x + iσˆ y is the lowering operator for the TLA where σˆ x = (10

the final DE for q(t, s) can be found by using Eq. (53). In that case we will arrive at following expression,

(85)

ρ˙ (t ) = −( 2 σˆ − σˆ )ρ (t )σˆ † A∗ (t ) − A(t )σˆ ρ (t )( 2 σˆ † − σˆ † ) − 2 σˆ B(t )σˆ ρ (t ) − ρ (t )σˆ † B∗ (t ) 2 σˆ † (98) and the third formula of NLPNMME is

ρ˙ (t ) = −( 2 σˆ − σˆ )B(t )σˆ ρ (t ) − ρ (t )σˆ † B∗ (t ) 2 σˆ † +A(t )σˆ ρ (t ) σˆ † − 2 σˆ † A(t )σˆ ρ (t ) −ρ (t )σˆ † A∗ (t ) 2 σˆ

(99)

A. Pourdarvish et al. / Chaos, Solitons and Fractals 95 (2017) 57–64

So, the final formula of NLPNMME for two level atom example will be following,

ρ˙ (t ) = − σˆ ρ (t )σˆ † A∗ (t ) − ρ (t )σˆ † A∗ (t ) 2 σˆ −ρ (t )σˆ † B∗ (t )( 2 σˆ † − σˆ † ) − 2 σˆ † A(t )σˆ ρ (t ) − 2 σˆ B(t )σˆ ρ (t )

(100)

0

λ(t , s )q(t , s )σˆ ds = [(g∗ )2 σˆ + |g|2 σˆ † ]

f9 (t ) =



t 0

q(t, s )σˆ ds

such that, γ ∗ (t + s ) = (g∗ )2 and α (t − s ) = |g|2 We calculate the ˆ (t ) under assumption J = 1, so, zero for the functional operator W by using Eq. (63), we get

(103)

By using Eq. (66), we can calculate the first order perturbation equation as

∂t Wˆ 1 (t ) = [(g∗ )2 σˆ + |g|2 σˆ † ]q1 (t , t )σˆ + Wˆ 0 (t )[(g∗ )2 σˆ ˆ 0 (t )[σˆ z∗ (t , t ) + z(t , t )σˆ † ]W ˆ 0 (t ) +|g|2 σˆ † ]q0 (t , t )σˆ + W † ˆ 0 (t )W ˆ 03 (t ) ˆ (t )W ˆ 0 (t ) − W −W (104) 0 ˆ (t ) we solve Eq. (103) with To determine the functional operator W ˆ 0 (t ) and substitute it in Eq. (104), we get the value of respect W ˆ 1 (t ). The first order position non Markovian master equation can W ˆ (t ). So now be obtained by computing the functional operator W we need to calculate the coefficients f0 (t ), f1 (t ), f2 (t ), . . . , f10 (t ) as

f0 (t ) = f1 (t ) =

f2 (t ) =

f3 (t ) =

f4 (t ) =

f5 (t ) =

f6 (t ) =

0



t 0

λ(t, s )ds = [(g∗ )2 σˆ + |g|2 σˆ † ]t,

0

t 2

u

 t



u

0

t 0

t3 3

0

u

0

 t 0

0

u

† 2t

2

2

† 2t

2

u, 2 (108) (109)

3

t u, 3 (110)

λ(t, s )λ∗ (s, u )(t − s )2 duds t3 u 3

 t 0

u

 v

0

0

 t 0

u

 v

0

0

and

f10 (t ) =

 t 0

u 0

(112)

λ(t, s )λ∗ (s, u )λ∗ (u, v )(t − s )2 dvduds t3 vu, 3

(113)

λ(t, s )λ(s, u )λ∗ (u, v )(t − s )2 dvduds

 v 0

t3 vu, 3

t3 vu, 3

(114)

λ(t, s )λ(s, u )λ(u, v )(t − s )2 dvduds t3 vu 3

(115)

here we note that the f2 (t ) = f3 (t ), f5 (t ) = f6 (t ) and f7 (t ) = f8 (t ) = f9 (t ) = f10 (t ) Substituting the value of coefficients in Eq. (73), and by using Pauli matrices, we get following expression,

ˆ 0 (t, z ) = [|g|2 σˆ σˆ † ]t + [|g|2 z(t , t )σˆ † σˆ σˆ † ] W +2(g∗ )4 |g|2 σˆ σˆ † σˆ

t2 2

t3 vu, 3

(116)

and rewrite Eq. (101) as

ˆ 0 (t )P˜(t ) P˜˙ (t ) = (σˆ z∗ (t , t )P˜(t ) + P˜(t )z(t , t )σˆ † ) − W ˆ † (t ) −P˜(t )W 0

(117)

Putting Eq. (116) into Eq. (117), we get

P˜˙ (t ) = (σˆ z∗ (t , t )P˜(t ) + P˜(t )z(t , t )σˆ † )

+2(g∗ )4 |g|2 σˆ σˆ † σˆ

t2 2

t3 vu]P˜(t ) 3

−P˜(t ) [|g|2 σˆ σˆ † ]t + [|g|2 z(t , t )σˆ † σˆ σˆ † ] +2(g∗ )4 |g|2 σˆ σˆ † σˆ



t3 vu † (t ) 3

t2 2 (118)

Eq. (117) represents first order position non Markovian master equation. This example allow us to apply both the perturbation approaches to any open quantum system by putting specified values of Hamiltonian (Hˆ ), Lindblad operator(Lˆ ) and correlation functions α (t − s ) and γ (t + s ) to get the post Markovian equations in terms of P˜ˆ(t ) which then can be easily solved. 7. Conclusion

λ(t, s )λ(s, u )(t − s )2 duds = [(g∗ )2 σˆ + |g|2 σˆ † ]2

= [(g∗ )2 σˆ + |g|2 σˆ † ]2 , ×

λ(t, s )λ∗ (s, u )λ(u, v )(t − s )2 dvduds



2

u, 2 (107)

λ(t, s )λ (s, u )(t − s )duds = [(g ) σˆ + |g| σˆ ] ∗ 2

λ(t, s )(t − s )2 ds = [(g∗ )2 σˆ + |g|2 σˆ † ] ,

 t

(106)

λ(t, s )λ(s, u )(t − s )duds = [(g ) σˆ + |g| σˆ ] ∗ 2

∗

0

−[[|g|2 σˆ σˆ † ]t + [|g|2 z(t , t )σˆ † σˆ σˆ † ]

λ(t, s )(t − s )ds = [(g∗ )2 σˆ + |g|2 σˆ † ] ,

0

0

(105) 2

 t

0

= [(g∗ )2 σˆ + |g|2 σˆ † ]3

∂t Wˆ 0 (t ) = [(g∗ )2 σˆ + |g|2 σˆ † ]q0 (t , t )σˆ + [σˆ z∗ (t , t ) ˆ † (t )W ˆ 02 (t ) ˆ 0 (t ) − W ˆ 0 (t ) − W +z(t , t )σˆ † ]W 0

t

 v

= [(g∗ )2 σˆ + |g|2 σˆ † ]3

(102)



f8 (t ) =

(101)

where

ˆ 0 (t ) = W

0

u

= [(g∗ )2 σˆ + |g|2 σˆ † ]3

P˜˙ (t ) = (σˆ z∗ (t , t )P˜(t ) + P˜(t )z(t , t )σˆ † ) − W0 (t )σˆ P˜(t ) −P˜(t )σˆ †W0∗ (t ) t

 t

= [(g∗ )2 σˆ + |g|2 σˆ † ]3

The linear position non Markovian equation of motion for this model is,



f7 (t ) =

63

(111)

In this work, we derive equation of motion and master equation corresponding to position non Markovian SSE in linear and non linear cases. In this side, we derive for every case four formulas of the position non Markovian master equation by using Nivokov theorem. When the functional derivative operator independent with noise, we can be derived the convolutionless linear position non Markovian master equations by taking the combined functional derivative for the probability operator without using SSE. Also we can say that consequently the master equation in numerically point of view be solvable. If the functional derivative

64

A. Pourdarvish et al. / Chaos, Solitons and Fractals 95 (2017) 57–64

operator depend on the noise, we can be derived the perturbation and post Markovian perturbation to position non Markovian equation of motion without using SSE. It means that, we present independent perturbation approach to the non Markovian equation of motion and we can apply to non Markovian master equation. By finding the zero and first orders stochastic equations with respect to perturbation approach, we determined the functional operator. As for post Markovian perturbation, we calculated the functional operator to solve non Markovian equation of motion. Finally, we present simple example to explain our theory. Acknowledgments We thanks Iraqi Ministry of Higher Education and Scientific Research, specifically Iraqi Cultural Relations and Scholarship Department and Cultural Attach in Tehran. References [1] Davies EB. Quantum theory of open systems. London: Academic Press; 1976. [2] Breuer H-P, Petruccione F. The theory of open quantum systems. Oxford: Oxford University Press; 2002. [3] Carmichael HJ. An open systems approach to quantum optics. Lect. Notes Phys, m18. Berlin: Springer-Verlag; 1993. [4] Nakajima S. On quantum theory of transport phenomena steady diffusion. Prog Theor Phys 1958;20:948. [5] Zwanzig R. Ensemble method in the theory of irreversibility. J Chem Phys 1960;33:1338.

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