Non-Markovian finite-temperature two-time correlation functions of system operators of a pure-dephasing model

Non-Markovian finite-temperature two-time correlation functions of system operators of a pure-dephasing model

Abstract

We evaluate the non-Markovian finite-temperature two-time correlation functions (CF’s) of system operators of a pure-dephasing spin-boson model in two different ways, one by the direct exact operator technique and the other by the recently derived evolution equations, valid to second order in the system-environment interaction Hamiltonian. This pure-dephasing spin-boson model that is exactly solvable has been extensively studied as a simple decoherence model. However, its exact non-Markovian finite-temperature two-time system operator CF’s, to our knowledge, have not been presented in the literature. This may be mainly due to the fact, illustrated in this article, that in contrast to the Markovian case, the time evolution of the reduced density matrix of the system (or the reduced quantum master equation) alone is not sufficient to calculate the two-time system operator CF’s of non-Markovian open systems. The two-time CF’s obtained using the recently derived evolution equations in the weak system-environment coupling case for this non-Markovian pure-dephasing model happen to be the same as those obtained from the exact evaluation. However, these results significantly differ from the non-Markovian two-time CF’s obtained by wrongly directly applying the quantum regression theorem (QRT), a useful procedure to calculate the two-time CF’s for weak-coupling Markovian open systems. This demonstrates clearly that the recently derived evolution equations generalize correctly the QRT to non-Markovian finite-temperature cases. It is believed that these evolution equations will have applications in many different branches of physics.

goan@phys.ntu.edu.tw

1Introduction

A quantum system is inevitably subject to the influence of its surroundings or environments [?]. An environment usually consists of a practically infinite number of degrees of freedom and acts statistically as a whole identity referred as a reservoir or bath of the open quantum system. Most often, one is concerned with only the system dynamics and the key quantity is the reduced system density matrix defined as the partial trace of the total system-plus-reservoir density operator over the reservoir degrees of freedom; i.e., . If the time evolution of the reduced density matrix that can be Markovian or non-Markovian is known, one is able to calculate the (one-time) expectation values or quantum average of the physical quantities of the system operators. But knowing the time evolution of the reduced density matrix is not sufficient to calculate the two-time (multiple-time) correlation functions (CF’s) of the system operators in the non-Markovian case [?].

In the Markovian case, an extremely useful procedure to calculate the two-time (multiple-time) CF’s is the so-called quantum regression theorem (QRT) [?] that gives a direct relation between the time evolution equation of the single-time expectation values and that of their corresponding two-time (multiple-time) CF’s. So knowing the time evolution of the system reduced density matrix allows one to calculate all of the two-time (multiple-time) Markovian CF’s. For the non-Markovian case, it is known that the QRT is not valid in general [?]. Recently, using the stochastic Schrödinger equation approach and the Heisenberg equation of system operator method, an evolution equation, valid to second order in system-environment coupling strength, for the two-time (multiple-time) CF’s of the system operators has been derived for an environment at the zero temperature and for a system in an initial pure state [?]. This evolution equation has been applied to calculate the emission spectra of a two-level atom placed in a structured non-Markovian environment (electromagnetic fields in a photonic band-gap material) [?]. In Ref. [?], an evolution equation for the reduced propagator of the system state vector, conditioned on an initial state of the environment differing from the vacuum, was derived using the stochastic Schrodinger equation approach. It is thus possible to use the reduced propagator to evaluate the expectation values and CF’s of the system observables for general environmental initial conditions, not necessarily an initial vacuum state for the environment [?]. By using another commonly used open quantum system technique, the quantum master equation approach [?], we are able to extend the two-time CF evolution equation to a non-Markovian finite-temperature environment for any initial system-environment separable state. The detailed derivation will be presented elsewhere [?] but the essential results will be summarized in Section 2. The derived evolution equation that generalizes the QRT to the non-Markovian finite-temperature case is believed to have applications in many different branches of physics.

The purpose of this article is twofold: (a) We show that in general the time evolution of the reduced density matrix of the system (or the reduced quantum master equation) alone is not sufficient to calculate the two-time CF’s of the system operators of non-Markovian open systems, even in the weak system-environment coupling case. We present an evaluation of an exactly solvable non-Markovian model, i.e., a pure-dephasing spin-boson model [?], to justify the statement. The exact non-Markovian finite-temperature two-time CF’s of the system operators of this model, to our knowledge, have not been presented in the literature. (b) This exactly solvable model allows us to test the validity of the derived non-Markovian finite-temperature evolution equation of two-time CF’s presented in Section 2. It will be shown that the two-time CF’s obtained using the evolution equation in the weak system-environment coupling limit [?] in Section 2 for the exactly solvable non-Markovian model happen to be the same as those obtained from the exact evaluation. However, these results significantly differ from the non-Markovian CF’s obtained by wrongly applying directly the QRT. This demonstrates clearly that the derived evolution equations generalize correctly the QRT to non-Markovian finite-temperature cases.

The article is organized as follows. We first summarize the important results of the newly obtained evolution equations [?] that generalizes the QRT to the non-Markovian finite-temperature case in Section 2. After brief description of the pure-dephasing spin-boson model in the beginning of Section 3, we calculate the exact time evolution of the reduced density matrix of the system and one-time expectation values in Section 3.1. The exact two-time CF’s are evaluated in subSection 3.2. In Section 4, we use the derived evolution equations in Ref. [?] to calculate the one-time and two-time CF’s. It is shown that the results obtained in Section 4 are the same as those by the exact evaluation in Section 3. This demonstrates the validity and practical usage of the derived evolution equations in Ref. [?]. Numerical results and discussions are presented in Section 5. A short conclusion is given in Section 6.

2Evolution equation of non-Markovian finite-temperature two-time CF’s

A class of systems considered in [?] is modeled by the Hamiltonian

where and are system and environment Hamiltonians, respectively, and stands for the Hamiltonian that describes the interaction between the system and the environment. So acts on the Hilbert space of the system, and are creation and annihilation operators on the environment Hilbert space, and and are the coupling strength and the frequency of the th environment oscillator, respectively. The derivations of the non-Markovian evolution equations of the two-time (multitime) CF’s for the general Hamiltonian model (Equation 1) in Refs. [?] (Eq. (6) in Ref. [?], Eq. (31) in Ref. [?] and Eq. (60) in Ref. [?]) are presented for an environment at the zero temperature and for a system state in an initial pure state. It was mentioned in Ref. [?] that it is possible to use the reduced stochastic system propagator that corresponds to an initial state of the environment different from the vacuum to evaluate the single-time expectation values and multitime CF’s with more general initial conditions. But only a master equation that is conditioned on initial bath states and is capable of evaluating the single-time expectation values of system observables for general initial conditions, both for an initial pure state and mixed state, was derived [?]. In Refs. [?], calculations of the two-time CF’s of system observables for dissipative spin-boson models in thermal baths are, however, presented even though in their derivations of the two-time (multitime) evolution equations, the bath CF’s are given in its zero-temperature form. This is possible due to the reason that for a system-environment model with a Hermitian system operator coupled to the environment, the linear finite-temperature stochastic Schrödinger equation could be written in a simple form of the zero-temperature equation [?] if the zero-temperature bath CF is replaced with its corresponding effective finite-temperature bath CF. As a result, the evolution equation of thermal two-time (multitime) CF’s for a Hermitian coupling operator also becomes equal to its zero-temperature counterpart with the replacement of the zero-temperature bath CF with its effective finite-temperature bath correlation kernel. It is for this reason that the dissipative spin-boson model with a thermal environment can be studied with the two-time (multitime) evolution equations derived in Refs. [?], since in that model . But this reduction of the finite-temperature evolution equation to its zero-temperature form [?] is not valid for more general non-Markovian finite-temperature cases where the system coupling operators are not Hermitian, i.e., . In other words, if the system operator coupled to the environment is not Hermitian , the two-time (multitime) differential evolution equations presented in Refs. [?] are valid for a zero-temperature environment only.

By using another commonly used open quantum system technique, the quantum master equation approach [?], it is possible to obtain in the weak system-environment coupling limit a two-time evolution equation for non-Markovian finite-temperature environments with both Hermitian and non-Hermitian system coupling operators and for any initial system-environment separable states. The detailed derivation will be presented elsewhere [?] but the important results are summarized here. The second-order evolution equations of the single-time expectation values for the class of systems modeled by the Hamiltonian (Equation 1) is

and that of the two-time CF’s can be obtained as

Here is the system operator in the interaction picture with respect to , and

are known as the environment CF’s: and , where and are the reservoir operators in the interaction picture.

We note here that for a Hermitian coupling operator the finite-temperature evolution equations (Equation 2) and (Equation 3) reduce, respectively, to their zero-temperature counterparts but with the effective bath CF given by [?]. This was pointed out to occur in general for -time CF’s in Refs. [?].

3Exact evaluations of pure dephasing spin-boson model

Here we consider an exactly solvable pure dephasing model of

to test the evolution equations (Equation 2) and (Equation 3). This pure dephasing spin-boson model in which has been extensively studied as a simple decoherence model in the literature [?]. But most of the studies focus on the discussion of the time evolution of the reduced density matrix of the spin, or other one-time expectation values of the spin system operators. Recently, the two-time CF’s of the system operators at the zero temperature for this model was reported in Ref. [?]. Nevertheless, to demonstrate the validity and practical usage of the finite-temperature non-Markovian evolution equation of the two-time CF’s (Equation 3), we present a detailed evaluation of the exact finite-temperature two-time CF’s for this simple model. These exact non-Markovian finite-temperature two-time CF’s of the system operators, to our knowledge, have not been presented in the literature.

3.1Reduced density matrix and one-time expectation values

Before we derive the two-time CF’s, we evaluate the exact time evolution of the reduced density matrix and one-time expectation values for the non-Markovian spin-boson model. In the interaction picture, the total density matrix of the combined (spin plus bath) system at time is given by

where the time evolution operator is

Here , and is the time-ordering operator which arranges the operators with the earliest times to the right. >From Eqs. (Equation 1) and (Equation 5), a simple calculation gives

This result allows us to calculate the time evolution operator to be (see Appendix A for details)

The time integrations in the exponents in Eq. (Equation 9) can be easily and analytically carried out. But we keep them in those forms in Eq. (Equation 9) so it will be easier to identify them with the results in Ref. [?]. If the time-ordering operation in Eq. (Equation 7) for were not performed, one could have just obtained the first term (line) of Eq. (Equation 9) for . Thus the second and third terms (lines) of Eq. (Equation 9) can be considered as the correction terms due to the time-ordering operation.

The reduced density matrix can be obtained by tracing over the reservoir’s degrees of freedom: . Suppose initially the state is factorized, where and are initial system and thermal reservoir(environment) density operators, respectively, and . Then the reduced density matrix elements in the interaction picture can be written as

where , , and the states of the two-level system are defined as , . To evaluate Eq. (Equation 10), the well known formula of

valid for operators and both commuting with the commutator , can be used to combine the evolution operators together. One then obtains

Then a useful identity [?] for the average over the thermal reservoir (environment) density operator, , can be employed:

where , are complex numbers, and stands for the thermal mean occupation number of the environment oscillators. As a result, we obtain

where

and and are defined in Eqs. ( ?) and ( ?), respectively. It is easy to show that and . Thus, using these results for the reduced density matrix elements Eq. (Equation 10) in the interaction picture and then transforming them back to the Schrödinger picture , we obtain the exact reduced density operator in the matrix form of

with . The same result was obtained in Ref. [?] using the stochastic Schrödinger equation approach.

With the exact time evolution of the reduced density matrix, the one-time expectation value of the system operators

can be calculated exactly, where represents a general system Heisenberg operator(s) and is the reduced Schrödinger density matrix operator at time . We may also write in the interaction picture,

where is defined in Eq. (Equation 6), and , and . For a general system operator , we obtain exactly from either Eq. (Equation 18) or Eq. (Equation 19)

3.2Two-time correlation functions

In contrast to the Markovian case in which the QRT is valid, the time evolution of the reduced density matrix of a non-Markovian open system alone is not sufficient to obtain the two-time system operator CF’s. This can be understood as follows. The two-time CF’s of system operators for can be written as

where the Heisenberg evolution operators and . If the environment is Markovian so the environment operator CF at two different times is correlated in time, then we may regard that the environment operator in is not correlated with that in . So the trace over the environment degrees of freedom for operator and operator can be performed independently or separately. Thus one may first trace over the environment degrees of freedom to obtain the reduced density matrix . Equation (Equation 21) in this case can be written as

where is the effective reduced density matrix at time with the initial condition . Thus knowing the time evolution of the reduced density matrix in the Markovian case, one is able to calculate the two-time CF’s of the system operators. This is also the reason why the QRT works in the Markovian case. But the situation differs for a non-Markovian environment as the environment operator in may, in general, be correlated with that in .

The two-time CF’s of the system operators for the pure-dephasing spin-boson model can also be evaluated exactly. To evaluate the two-time CF of system operators for , we express it in terms of the interaction picture operators as

where again an operator with a tilde on the top indicates that it is an operator in the interaction picture with respect to the free Hamiltonian . Compared with Eq. (Equation 19), Eq. (Equation 23) for general non-Markovian open systems can not be expressed as a product of the reduced density matrix and system operators. So again, the reduced density matrix alone is not sufficient to obtain the non-Markovian two-time system operator CF’s.

As we want to compare the results by the direct evaluation with those by the evolution equation (Equation 3), we calculate, in the following, the two-time CF’s for different cases of system operators and . The structure of the evolution equations in Ref. [?] or Eqs. (Equation 2) and (Equation 3) in this article depends on the commutation relations of operator and operator (or ), and on the commutation relations of operator and operator (or ), where is the system operator in the interaction picture with respect to . For the pure-dephasing spin-boson model, , , and then . So we will discuss the two-time CF’s in the following three cases and the trivial case of is obvious due to .

Case 1. and . In this case, let us set , and . Then and . It is easy to see from Eq.(Equation 9) that commutes with but anticommutes with , i.e., and . Using these results and the fact that , we obtain from Eq. (Equation 23)

Substituting the result of Eq. (Equation 14) into Eq. (Equation 24), we arrive at the exact two-time CF’s

Case 2. and . In this case, let , and . Similar to the calculations in Case 1, we obtain

The exact two-time CF’s of Eqs. (Equation 25) and (Equation 26) depend on only one time variable, or , respectively, since one of the system operator is time-independent.

Case 3. and . Suppose , and . In this case, both and anticommute with both and . Furthermore, . Thus we can obtain from Eq. (Equation 23)

It is obvious from Eq. (Equation 27) that to evaluate the general two-time CF, we need to take into account the correlations of the reservoir operators of the evolution operators between different time periods of and before the trace over the environment is performed. Using Eqs. (Equation 11), (Equation 12), and (Equation 13), we get

where

The term in Eq. (Equation 28) describes the cross-time contribution of the environment CF’s of the reservoir operators in the evolution operators and [or and ] of the two different time periods and . We can see this from of Eq. (Equation 29) and in Eq. (Equation 28) that the environment CF , defined in Eq. (Equation 16), has the time variable in and the time variable in . On the other hand, the time evolution of the reduced density matrix (Equation 10) is involved with the reservoir operator CF’s in the evolution operators of only the same time interval. As a result, it, alone, cannot provide us with the full information to evaluate the non-Markovian two-time CF, even in the weak system-environment coupling case. Similarly, we find that has the same result as Eq. (Equation 28). Substituting these results into Eq. (Equation 27), finally we arrive at the two-time CF

This non-Markovian finite-temperature two-time CF, to our knowledge, has not been presented in the literature.

4Evaluation by derived non-Markovian finite-temperature evolution equations

In this section, we will use the derived evolution equations in Ref. [?] to compute the one-time expectation values and two-time CF’s to compare with the exact expressions evaluated in Section 3. Despite the fact that the evolution equations in Ref. [?] derived perturbatively, the results obtained this way for the pure-dephasing spin-boson model happen to be the same as the exact expressions by the direct evaluation.

4.1Quantum master equation and one-time expectation values

Before going to calculate the CF’s, it is instructive to derive the master equation of the reduced system density matrix for the model. After some calculations, we obtain for the Hamiltonian in the form of Eq. (Equation 1) a time-covolutionless non-Markovian master equation [?] valid to second order in the system-environment interaction strength

where and are defined in Eqs. ( ?) and ( ?) respectively, H.c. indicates the Hermitian conjugate of previous terms, and an operator with a tilde on the top indicates that it is an operator in the interaction picture. For the pure-dephasing spin-boson model, Eq. (Equation 31) gives the master equation of the reduced system density matrix

where is defined in Eq. (Equation 15). It is not difficult to show that the exact expression of the density matrix (Equation 17) is the solution of the master equation (Equation 32) although the master equation is derived perturbatively. Non-Markovian dynamics usually means that the current time evolution of the system state depends on its history, and the memory effects typically enters through integrals over the past state history. However, the non-Markovian system dynamics of some class of open quantum system models may be summed up and expressed as a time-local, convolutionless form [?] where the dynamics is determined by the system state at the current time only. This time-local, convolutionless class of open quantum systems may be treated exactly without any approximation. The quantum Brownian motion model or the damped harmonic oscillator bilinearly coupled to a bosonic bath of harmonic oscillators [?] is a famous example of this class. The pure-dephasing spin-boson model considered here also belongs to this class, and the non-Markovian effect in the master equation (Equation 32) is taken into account by the time-dependent coefficient instead of memory integral. This time-local, convolutionless property and the fact of allow the exact system density matrix Eq. (Equation 17) to be obtained from Eq. (Equation 32).

Since the exact solution of the system density matrix (Equation 17) can be calculated from the perturbatively derived master equation (Equation 32), one may expect that the exact non-Markovian finite-temperature one-time expectation values and two-time CF’s of the pure-dephasing model can be obtained from the evolution equation (Equation 3) We show below that this is indeed the case, and at the same time the agreement of the results demonstrates the validity and practical usage of the evolution equation (Equation 3).

For the pure-dephasing spin-boson model, , , and we have . Taking , , in Eq. (Equation 2), we obtain straightforwardly the evolution equations of the single-time expectation values as

with defined in Eq. (Equation 15). With proper chosen values for , , , and of a general operator for , one can verify that the exact expression of the expectation value of in Eq. (Equation 20) satisfies Eqs. (Equation 33)–( ?).

4.2Two-time correlation functions

Before using Eq. (Equation 3) to calculate the non-Markovian finite-temperature two-time CF’s, we discuss briefly below the relation between the QRT and the evolution equation (Equation 3). If the last two terms of Eq. (Equation 3) vanish, then the single-time and two-time evolution equations (Equation 2) and (Equation 3) will have the same form with the same evolution coefficients and thus the QRT will be applicable. The last two terms of Eq. (Equation 3) or more generally the last term of Eq. (17) in Ref.[?] involve(s) the propagation from to , and these terms would vanish for the CF’s as in this case. So the QRT is valid to calculate the CF’s of both Markovian and non-Markovian open systems, where the system-environment density matrix is separable at . The QRT is also valid and is often applied to calculate, in the Markovian weak system-environment coupling case, more general CF’s or equivalently with . For example, the QRT is used to calculate the Markovian steady-state CF’s, and in this case is set to any of the large times when the steady state is reached. This is because in the Markovian case, the last two terms of Eq. (Equation 3) vanish since the time integration of the corresponding -correlated reservoir CF’s, and , over the variable in the domain from to is zero as . On the other hand, the QRT cannot be blindly applied to calculate with in a general non-Markovian open system due to the non-vanishing contributions of the cross correlation of the reservoir operators at two different times: a later time and an earlier time in the period between and (see the last two terms of Eq. (Equation 3) and also Figure 1). In other words, in contrast to the Markovian case, not only the initial condition for the two-time evolution equation (Equation 3) but also the equation (Equation 3) itself may depend on the choice of the starting time of the non-Markovian finite-temperature two-time CF’s. In the steady state, the situation may change when is in any of the large times where the state and system expectation values do not change with time any more. In this case, the contributions from the last two terms of Eq. (Equation 3) saturate and do not depend on where time is set in the steady state, and thus both the Markovian and non-Markovian CF’s may depend only on the time difference (see also Figure 2). But the nonvanishing contributions from the last two terms of Eq. (Equation 3) would still make the non-Markovian CF’s deviate from that obtained wrongly using the QRT in the non-Markovian case or obtained using the QRT in the Markovian case (see also Figure 3).

For the time evolutions of system two-time CF’s of the pure-dephasing spin-boson model, we also consider the following three cases as in Section 3. Note that .

Case 1. and . In this case, let and . By using Eq. (Equation 3), it is easy to obtain

In this case, one can see that the evolution equations of single-time expectation values , Eqs. (Equation 33) and ( ?), have the same forms as the evolution equations of two-time CF’s , Eqs. (Equation 34) and ( ?), respectively. Hence the QRT is valid in this case. It is easy to check that taking the derivative of Eq. (Equation 25) with respect to with ( i.e. A= ) and ( i.e. A= ), one can obtain the evolution equations for and , exactly the same as Eqs. (Equation 34) and ( ?), respectively.

Case 2. and . In this case, let and . By using Eq. (Equation 3), we then easily obtain

Indeed, Eq. (Equation 26) satisfies Eqs. (Equation 35) and ( ?), and , independent of .

Case 3. and . In this case, let and . Eq. (Equation 3) straightforwardly yields