# Decoherence in open quantum systems: influence of the intrinsic bath
dynamics

V. V. Ignatyuka***Email<EMAIL_ADDRESS>V. G. Morozovb †††Prof. V.G.
Morozov passed away soon after this article had been submitted to the
Condensed Matter Physics journal. This is my last paper prepared in
collaboration with my colleague and a good friend.
aInstitute for Condensed Matter Physics, Lviv, Ukraine
bMIREA-Russian Technological University, Moscow, Russia

###### Abstract

The non-Markovian master equation for open quantum systems is obtained by
generalization of the standard Zwanzig-Nakajima (ZN) projection technique. To
this end, a coupled chain of equations for the reduced density matrices of the
bath $\varrho_{B}(t)$ and of the system $\varrho_{S}(t)$ are written. Formal
solution of the equation for $\varrho_{B}(t)$ in the 2-nd approximation in
interaction yields a specific extra term, related to the intrinsic bath
dynamics. This term is nonlinear in the reduced density matrix
$\varrho_{S}(t)$, and vanishes in the Markovian limit. To verify the
consistence and robustness of our approach, we apply the generalized ZN
projection scheme to a simple dephasing model. We study the obtained kinetic
equation both in the Markovian approximation and beyond it (for the term
related to the intrinsic bath dynamics) and compare the results with the exact
ones.

##  1 Introduction

When studying dynamical processes such as relaxation, decoherence, buildup of
correlations due to the interaction of the open quantum system with its
environment, at the certain stage of investigation one inevitably faces the
question: do these phenomena exhibit the Markovian behaviour or not? Many
physical systems are believed to be described within Markovian approximation,
since the coupling to the environment is weak (Born approximation), and/or the
correlations in the bath decay fast on the characteristic time scale. However,
there are situations, when the memory effects in the bath cannot be neglected,
and the Markov assumption is not applicable anymore. This can be due to strong
system-environment couplings [1], correlations and entanglement in the initial
state [2], at the heat transport in nanostructures [3], or due to the specific
character of the finite reservoirs [4]. The last cases are of a particular
interest, because the environment of an open quantum system due to its
compactness frequently cannot be regarded as a thermal bath [5]. In such a
case, the dynamics of the reservoir $B$ has to be treated (at least at the
initial stage of evolution) on equal footing with that of the $S$-subsystem.

A powerful tool for dealing with such systems is provided by the projection
operator techniques [6, 7], which was developed by Nakajima [8], Zwanzig [9],
and Mori [10]. This approach has manifested its efficiency in the derivation
of generalized master equations and investigation of the non-Markovian
dynamics in initially correlated open quantum systems [11], in the spin star
systems [12] and many others.

However, the ZN scheme has some disadvantages. Though the generic master
equation is usually written down up to the 2-nd order in the interaction, the
time convolution in the kinetic kernels is determined by the full evolution
operators (including the interaction part $V$ of the total Hamiltonian). Thus,
to provide a consistent analysis of the system dynamics, one has to take into
account as many terms in the coupling constant as possible when expanding the
master equation in the series in $V$. This is a cornerstone of the time-
convolutionless equation [13] and the related projection technique, when one
moves from the retarded dynamics to the equations local in time with the time-
dependent generators and the subsequent systematic perturbative expansion
scheme.

In the decades that passed since the pioneering works of Nakajima and Zwanzig,
the projection methods technique has been improved significantly [14]. The
main efforts have been usually directed toward construction of correlated
projection superoperators [15, 16], which consider the relevant part of the
dynamics as a correlated system-environment state, rather than a tensor
product state $\varrho_{S}(t)\otimes\varrho_{B}$, what allows one to
reproduces the exact solutions already in lowest order in the interaction
[17]. In the very recent paper [18] the correlated projector technique has
been generalized, yielding the so-called adapted projection operators (APO).
The APO master equation encloses the full dependence on the initial
correlations in a homogeneous term, which is essential to avoid the problems
[13] dealt with an inhomogeneous time contribution arising from the
unfactorisable initial states. An application of the APO methods manifested
its efficiency in several cases, e. g. in investigation of the polarization
degrees of freedom of a photon going through a quartz plate – the system,
which can be described by a similar dephasing model. Another kind of
projections, namely, to the environment subspaces corresponding to a given
energy, was proposed in Ref. [19], which allowed the authors to derive the
equation for the time-dependent reduced density matrix of the environment
$\varrho_{B}(t)$ and to approach closely the problem of taking into account
the intrinsic bath dynamics (BD).

However, none of the above methods considers the dynamics of the environment
$B$ from the very beginning, on equal footing with that of the open quantum
system $S$. Usually, the evolution equation for the reduced density matrix of
the environment $\varrho_{B}(t)$ is ignored either. Though for the infinite
thermal baths this simplification seems to be reasonable, its validity could
be questioned, when one deals with the finite size reservoirs [4] or tries to
investigate the impact of running correlations [20]. In other words, the open
system dynamics has definitely to be in accord with the concept [21, 22],
claiming that an evolution towards local equilibrium is always accompanied by
an increase of the system-bath correlations.

In this paper, we generalize the standard ZN projection technique on our way
to derive the non-Markovian master equation for open quantum systems. In
Section 2, we start from the coupled set of equations for the reduced density
matrices of the system $\varrho_{S}(t)$ and of the environment
$\varrho_{B}(t)$. Then in Section 3 we insert a formal solution of the dynamic
equation for $\varrho_{B}(t)$ into the equation for the reduced density matrix
of the $S$-subsystem and obtain the generalized master equation in the second
order in interaction. The above generalization gives rise to the extra term of
a very specific structure: i) it is nonlinear in $\varrho_{S}(t)$ and ii)
vanishes in the Markovian limit. These two points make the situation very
similar to what one faces when studying the inset of running correlations: the
generic kinetic equations are strongly nonlinear, and the correlational
contribution to the collision integral tends to zero in the Markovian limit
[21, 23]. We also briefly compare our results with those following from the
standard ZN scheme.

To verify the consistency of our approach, in Section 4 we apply the
generalized projection scheme to a simple dephasing model [24] and obtain the
kinetic equation for the system coherence. It is pointed out that the
application of the Markovian approximation (MA) allows us to reproduce the
exact results reported in Ref. [24] in the lowest orders in the interaction.
On the other hand, going beyond the MA in the term dealt with the BD leads to
the renormalization of the phase shift and the generalized decoherence. We
demonstrate that in the certain domain of the state parameters the above
renormalization allows to describe the correlational contribution to the
generalized decoherence more precisely than the ZN scheme.

In the last Section, we make conclusions and discuss the obtained results in
their relation to the concept of the running correlations build-up.

##  2 Basic equations

Suppose that the composed system ($S+B$) consists of the open quantum system
(subsystem $S$) and its surrounding $B$, which usually can be considered as a
thermal bath. The total Hamiltonian of this system

$H(t)=H_{S}(t)+H_{B}+V$ (2.1)

consists of the term $H_{S}(t)$ corresponding to the open quantum system $S$
(which in the general case is allowed to depend on time $t$ due to the action
of the external fields), the summand $H_{B}$ related to the bath, and the
interaction term $V$.

The density matrix $\varrho(t)$ of the composed system obeys the quantum
Liouville equation (hereafter we put $\hbar=1$).

$\frac{\partial\varrho(t)}{\partial t}=-i\left[H(t),\varrho(t)\right].$ (2.2)

For further convenience, let us pass to the interaction picture
$\widetilde{\varrho}(t)$ for the total density matrix, which is defined as

$\widetilde{\varrho}(t)=U^{\dagger}(t)\varrho(t)U(t),$ (2.3)

where we have introduced the unitary evolution operators

$\begin{array}[]{c}U(t)=U_{S}(t)U_{B}(t),\\\\[10.0pt] \displaystyle
U_{S}(t)=\exp_{+}\left\\{-i\int^{t}_{0}dt^{\prime}\,H_{S}(t^{\prime})\right\\},\qquad
U_{B}(t)=\exp\left(-itH_{B}\right).\end{array}$ (2.4)

In Eq. (2.4), the expression $\exp_{+}\left\\{\ldots\right\\}$ denotes the
time ordered exponent, which takes a much simpler form
$U_{S}(t)=\exp\left(-itH_{S}\right)$ if the open system Hamiltonian $H_{S}$
does not depend on time. The unitary operator $U(t)$ and its Hermitian
conjugate counterpart $U^{\dagger}(t)$ obey the evolution equations

$\frac{dU(t)}{dt}=-i\left\\{H_{S}(t)+H_{B}\right\\}U(t),\qquad\frac{dU^{\dagger}(t)}{dt}=i\,U^{\dagger}(t)\left\\{H_{S}(t)+H_{B}\right\\},$
(2.5)

wherefrom it is easy to present the quantum Liouville equation (2.2) in the
interaction picture,

$\frac{\partial\widetilde{\varrho}(t)}{\partial
t}=-i\left[\widetilde{V}(t),\widetilde{\varrho}(t)\right].$ (2.6)

Hereafter, operators in the interaction picture are defined as

$\widetilde{A}(t)=U^{\dagger}(t)AU(t).$ (2.7)

If $A_{S}$ ($A_{B}$) is an operator acting in the Hilbert space of the
subsystem $S$ (bath $B$), then the corresponding interaction representation
can be written down as follows:

$\widetilde{A}_{S}(t)=U^{\dagger}_{S}(t)A_{S}U_{S}(t),\qquad\widetilde{A}_{B}(t)=U^{\dagger}_{B}(t)A_{B}U_{B}(t).$
(2.8)

The reduced density matrices for the system $S$ and the environment are
introduced in the usual way,

$\varrho_{S}(t)=\text{Tr}_{B}\varrho(t),\quad\varrho_{B}(t)=\text{Tr}_{S}\varrho(t),$
(2.9)

by taking a trace over the environment (the system) variables. The
corresponding time averages for operators $A_{S}$ ($A_{B}$) can be introduced
as follows:

$\begin{array}[]{l}\langle
A_{S}\rangle^{t}\equiv\text{Tr}_{SB}\left\\{A_{S}\varrho(t)\right\\}=\text{Tr}_{S}\left\\{A_{S}\varrho_{S}(t)\right\\},\qquad\langle
A_{B}\rangle^{t}\equiv\text{Tr}_{SB}\left\\{A_{B}\varrho(t)\right\\}=\text{Tr}_{B}\left\\{A_{B}\varrho_{B}(t)\right\\}.\end{array}$
(2.10)

The reduced density matrix $\widetilde{\varrho}_{S}(t)$ of the open quantum
system is defined as

$\widetilde{\varrho}_{S}(t)=U^{\dagger}_{S}(t)\varrho_{S}(t)U_{S}(t).$ (2.11)

Analogously, in the interaction picture the reduced density matrix
$\widetilde{\varrho}_{B}(t)$ of the environment reads

$\widetilde{\varrho}_{B}(t)=U^{\dagger}_{B}(t)\varrho_{B}(t)U_{B}(t).$ (2.12)

It is obvious that
$\widetilde{\varrho}_{S}(t)=\text{Tr}_{B}\widetilde{\varrho}(t)$ and
$\widetilde{\varrho}_{B}(t)=\text{Tr}_{S}\widetilde{\varrho}(t)$.

To derive the evolution equations for the operators
$\widetilde{\varrho}_{S}(t)$ and $\widetilde{\varrho}_{B}(t)$, we apply the
following decomposition for the total density matrix:

$\varrho(t)=\varrho_{S}(t)\varrho_{B}(t)+\Delta\varrho(t),$ (2.13)

where the correlation term satisfies the relations

$\text{Tr}_{S}\,\Delta\varrho(t)=0,\quad\text{Tr}_{B}\,\Delta\varrho(t)=0.$
(2.14)

In the interaction picture, Eqs. (2.13)-(2.14) are converted into the similar
relations,

$\widetilde{\varrho}(t)=\widetilde{\varrho}_{S}(t)\widetilde{\varrho}_{B}(t)+\Delta\widetilde{\varrho}(t)$
(2.15)

and

$\text{Tr}_{B}\,\Delta\widetilde{\varrho}(t)=0,\qquad\text{Tr}_{S}\,\Delta\widetilde{\varrho}(t)=0.$
(2.16)

Taking the trace $\text{Tr}_{B}$ of both sides of Eq. (2.6), we get

$\frac{\partial\widetilde{\varrho}_{S}(t)}{\partial
t}=-i\text{Tr}_{B}\left[\widetilde{V}(t),\widetilde{\varrho}(t)\right].$
(2.17)

Using the decomposition (2.15), one can rewrite Eq. (2.17) as

$\frac{\partial\widetilde{\varrho}_{S}(t)}{\partial
t}=-i\left[\widetilde{V}_{S}(t),\widetilde{\varrho}_{S}(t)\right]-i\text{Tr}_{B}\left[\widetilde{V}(t),\Delta\widetilde{\varrho}(t)\right],$
(2.18)

where

$\widetilde{V}_{S}(t)=\text{Tr}_{B}\left\\{\widetilde{V}(t)\widetilde{\varrho}_{B}(t)\right\\}.$
(2.19)

In a similar way, taking the trace $\text{Tr}_{S}$ of Eq. (2.6), one obtains

$\frac{\partial\widetilde{\varrho}_{B}(t)}{\partial
t}=-i\left[\widetilde{V}_{B}(t),\widetilde{\varrho}_{B}(t)\right]-i\text{Tr}_{S}\left[\widetilde{V}(t),\Delta\widetilde{\varrho}(t)\right],$
(2.20)

where

$\widetilde{V}_{B}(t)=\text{Tr}_{S}\left\\{\widetilde{V}(t)\widetilde{\varrho}_{S}(t)\right\\}.$
(2.21)

To derive the equation of motion for the correlational part
$\Delta\widetilde{\varrho}(t)$ of the total density matrix, let us rewrite
Eqs. (2.6), (2.18), and (2.20) in the form

$\frac{\partial\widetilde{\varrho}(t)}{\partial
t}=-i{\mathcal{L}}(t)\widetilde{\varrho}(t),$ (2.22)
$\frac{\partial\widetilde{\varrho}_{S}(t)}{\partial
t}=-i{\mathcal{L}}_{S}(t)\widetilde{\varrho}_{S}(t)-\text{Tr}_{B}\left\\{i{\mathcal{L}}(t)\,\Delta\widetilde{\varrho}(t)\right\\},$
(2.23) $\frac{\partial\widetilde{\varrho}_{B}(t)}{\partial
t}=-i{\mathcal{L}}_{B}(t)\widetilde{\varrho}_{B}(t)-\text{Tr}_{S}\left\\{i{\mathcal{L}}(t)\,\Delta\widetilde{\varrho}(t)\right\\}.$
(2.24)

Here we have introduced the Liouville operators ${\mathcal{L}}(t)$,
${\mathcal{L}}_{S}(t)$, and ${\mathcal{L}}_{B}(t)$ via the corresponding
commutators:

${\mathcal{L}}(t)A=[\widetilde{V}(t),A],\qquad{\mathcal{L}}_{S}(t)A=[\widetilde{V}_{S}(t),A],\qquad{\mathcal{L}}_{B}(t)A=[\widetilde{V}_{B}(t),A].$
(2.25)

Using the decomposition (2.15) and taking into account Eqs. (2.22)-(2.24), one
can obtain the equation of motion for the correlation contribution to the
total density matrix,

$\displaystyle\frac{\partial\,\Delta\widetilde{\varrho}}{\partial
t}=-i{\mathcal{L}}\widetilde{\varrho}-\widetilde{\varrho}_{S}\left\\{-i{\mathcal{L}}_{B}\widetilde{\varrho}_{B}-\text{Tr}_{S}\left\\{i{\mathcal{L}}\,\Delta\widetilde{\varrho}\right\\}\right\\}-\widetilde{\varrho}_{B}\left\\{-i{\mathcal{L}}_{S}\widetilde{\varrho}_{S}-\text{Tr}_{B}\left\\{i{\mathcal{L}}\,\Delta\widetilde{\varrho}\right\\}\right\\}.$
(2.26)

The above equation can be rewritten in a more compact form,

$\left(\frac{\partial}{\partial
t}+{\mathcal{Q}}(t)i{\mathcal{L}}(t){\mathcal{Q}}(t)\right)\Delta\widetilde{\varrho}(t)=-{\mathcal{Q}}(t)i{\mathcal{L}}(t)\widetilde{\varrho}_{S}(t)\widetilde{\varrho}_{B}(t),$
(2.27)

using the superoperators

${\mathcal{Q}}(t)=1-{\mathcal{P}}(t),\qquad{\mathcal{P}}(t)A=\widetilde{\varrho}_{S}(t)\,\text{Tr}_{S}A+\widetilde{\varrho}_{B}(t)\,\text{Tr}_{B}A.$
(2.28)

It can be shown that, when acting on operators with zero trace,
$\text{Tr}A=0$, the superoperator ${\mathcal{P}}(t)$ satisfies the relation
${\mathcal{P}}^{2}(t)={\mathcal{P}}(t)$, i.e., it is a projector. A formal
solution of Eq. (2.27) is

$\Delta\widetilde{\varrho}(t)=-i\int^{t}_{0}dt^{\prime}\,{\mathcal{U}}(t,t^{\prime}){\mathcal{Q}}(t^{\prime}){\mathcal{L}}(t^{\prime})\widetilde{\varrho}_{S}(t^{\prime})\widetilde{\varrho}_{B}(t^{\prime}),$
(2.29)

where the superoperator ${\mathcal{U}}(t,t^{\prime})$ satisfies the equation
of motion

$\frac{\partial{\mathcal{U}}(t,t^{\prime})}{\partial
t}=-i{\mathcal{Q}}(t){\mathcal{L}}(t){\mathcal{Q}}(t){\mathcal{U}}(t,t^{\prime}),\qquad{\mathcal{U}}(t^{\prime},t^{\prime})=1,$
(2.30)

and we have introduced the evolution operator

${\mathcal{U}}(t,t^{\prime})=\exp_{+}\left\\{-i\int^{t}_{t^{\prime}}d\tau\,{\mathcal{Q}}(\tau){\mathcal{L}}(\tau){\mathcal{Q}}(\tau)\right\\}.$
(2.31)

Substituting expression (2.29) into Eqs. (2.23) and (2.24), we arrive at the
closed system of equations for the reduced density matrices,

$\frac{\partial\widetilde{\varrho}_{S}(t)}{\partial
t}=-i{\mathcal{L}}_{S}(t)\widetilde{\varrho}_{S}(t)-\int^{t}_{0}dt^{\prime}\,\text{Tr}_{B}\left\\{{\mathcal{L}}(t){\mathcal{U}}(t,t^{\prime}){\mathcal{Q}}(t^{\prime}){\mathcal{L}}(t^{\prime})\widetilde{\varrho}_{S}(t^{\prime})\widetilde{\varrho}_{B}(t^{\prime})\right\\},$
(2.32) $\frac{\partial\widetilde{\varrho}_{B}(t)}{\partial
t}=-i{\mathcal{L}}_{B}(t)\widetilde{\varrho}_{B}(t)-\int^{t}_{0}dt^{\prime}\,\text{Tr}_{S}\left\\{{\mathcal{L}}(t){\mathcal{U}}(t,t^{\prime}){\mathcal{Q}}(t^{\prime}){\mathcal{L}}(t^{\prime})\widetilde{\varrho}_{S}(t^{\prime})\widetilde{\varrho}_{B}(t^{\prime})\right\\}.$
(2.33)

However, Eq. (2.32) cannot be considered yet as a master equation for the open
quantum system since it depends on the reduced density matrix of the
environment. Solving Eq. (2.33) with respect to the density matrix of the
environment $\widetilde{\varrho}_{B}(t)$ seems unrealistic. Thus the only way
is to use some approximations for $\widetilde{\varrho}_{B}(t)$ obtainable from
Eq. (2.33). This is a subject of the next Section.

##  3 Master equation for $\widetilde{\varrho}_{S}(t)$: weak coupling
approximation

Let us start this Section with two assumptions:

  * •

If the subsystem $S$ is small as compared to the environment $B$, it is
reasonable to suppose that, for sufficiently small time $t$, the state of the
environment is close to $\varrho_{B}(0)$. Based on the above assumption, we
write the solution of Eq. (2.33) as

$\widetilde{\varrho}_{B}(t)\approx\varrho_{B}(0)-\int^{t}_{0}dt^{\prime}\,i{\mathcal{L}}_{B}(t^{\prime})\varrho_{B}(0).$
(3.1)

The above assumption means that we restrict the equation for
$\widetilde{\varrho}_{B}(t)$ by the first order in the interaction.

  * •

On the right-hand side of Eq. (2.32), we set ${\mathcal{U}}(t,t^{\prime})=1$
and $\widetilde{\varrho}_{B}(t^{\prime})=\varrho_{B}(0)$, ensuring the
integral terms to be precisely of the 2-nd order in interaction.

Based on these approximations, we get

$\frac{\partial\widetilde{\varrho}_{S}(t)}{\partial
t}=-i{\mathcal{L}}_{S}(t)\widetilde{\varrho}_{S}(t)-\int^{t}_{0}dt^{\prime}\,\text{Tr}_{B}\left\\{{\mathcal{L}}(t){\mathcal{Q}}^{(0)}(t^{\prime}){\mathcal{L}}(t^{\prime})\widetilde{\varrho}_{S}(t^{\prime})\varrho_{B}(0)\right\\},$
(3.2)

where

${\mathcal{Q}}^{(0)}(t)A=A-\widetilde{\varrho}_{S}(t)\,\text{Tr}_{S}A-\varrho_{B}(0)\,\text{Tr}_{B}A.$
(3.3)

After some manipulations with the r.h.s. of Eq. (3.2), which are described in
detail in the Appendix A of our recent paper [25], we obtain the final master
equation in the interaction picture:

$\displaystyle\frac{\partial\widetilde{\varrho}_{S}(t)}{\partial
t}=-i{\mathcal{L}}^{(0)}_{S}(t)\widetilde{\varrho}_{S}(t)-\int^{t}_{0}dt^{\prime}\,\left[R_{S}(t,t^{\prime}),\widetilde{\varrho}_{S}(t)-\widetilde{\varrho}_{S}(t^{\prime})\right]$
$\displaystyle{}-\int^{t}_{0}dt^{\prime}\,\left\\{\text{Tr}_{B}\left\\{{\mathcal{L}}(t){\mathcal{L}}(t^{\prime})\widetilde{\varrho}_{S}(t^{\prime})\varrho_{B}(0)\right\\}-{\mathcal{L}}^{(0)}_{S}(t){\mathcal{L}}^{(0)}_{S}(t^{\prime})\widetilde{\varrho}_{S}(t^{\prime})\right\\},$
(3.4)

where we have used the denotation

$R_{S}(t,t^{\prime})=\text{Tr}_{B}\left\\{\widetilde{V}(t){\mathcal{L}}_{B}(t^{\prime})\varrho_{B}(0)\right\\}=\text{Tr}_{B}\left\\{\left[\widetilde{V}(t),\widetilde{V}_{B}(t^{\prime})\right]\varrho_{B}(0)\right\\}$
(3.5)

for the kernel of the first integral in Eq. (3), while the operator
${\mathcal{L}}^{(0)}_{S}(t)$ is defined as

${\mathcal{L}}^{(0)}_{S}(t)A=\left[\widetilde{V}^{(0)}_{S}(t),A\right],\qquad\widetilde{V}^{(0)}_{S}=\text{Tr}_{B}\left\\{\widetilde{V}(t)\varrho_{B}(0)\right\\}.$
(3.6)

Some comments on the structure of master equation (3) are quite pertinent at
this stage:

  * •

The first term on the r.h.s. vanishes if $\varrho_{B}(0)$ is the equilibrium
density matrix of the bath and

$\text{Tr}_{B}\left\\{V\varrho_{B}(0)\right\\}=0.$

  * •

In the above case the term
${\mathcal{L}}^{(0)}_{S}(t){\mathcal{L}}^{(0)}_{S}(t^{\prime})\widetilde{\varrho}_{S}(t^{\prime})$
also vanishes. It is shown in Section 4 that the terms with
${\mathcal{L}}^{(0)}_{S}$ describe an influence of the initial correlations in
the open quantum systems. In particular, the non-integral term contributes to
the non-dissipative properties, whereas the integral term represents a new
channel of dissipation related the initial preparation of the system.

  * •

The second term in the r.h.s. has some interesting properties:

1) It vanishes in the Markovian limit 333One should distinguish two kinds of
approximation for the integrand. The first one looks as
$\int_{0}^{t}\mathcal{K}(t-\tau)f(\tau)d\tau\approx\int_{0}^{\mathbf{t}}\mathcal{K}(t-\tau)d\tau
f(t).$ This approximation is often called the MA of the 1-st kind [14] or the
Redfield approximation (dealt with the corresponding Redfield equation [13]
for the reduced density matrix of the subsystem $S$). The other approximation,
$\int_{0}^{t}\mathcal{K}(t-\tau)f(\tau)d\tau\approx\int_{0}^{\mathbf{\infty}}d\tau\mathcal{K}(t-\tau)d\tau
f(t)$ is known as the MA of the 2-nd kind [14], very often [21] referred to
just as the Markovian approximation. In this paper, we use the MA of the 1-st
kind only. , when dynamics of the kinetic kernel $R_{S}(t,t^{\prime})$ is
considered fast in comparison with the time evolution of the density matrix;
then we set
$\widetilde{\varrho}_{S}(t^{\prime})\approx\widetilde{\varrho}_{S}(t)$.

2) This term is nonlinear in $\widetilde{\varrho}_{S}$ since the operator
$\widetilde{V}_{B}(t^{\prime})$ appearing in Eq. (3.5) depends on
$\widetilde{\varrho}_{S}(t^{\prime})$ (see also Eq. (2.21)). A nonlinearity
appears due to consideration of the intrinsic dynamics of the environment by
means of the time evolution of the reduced density matrix
$\widetilde{\varrho}_{B}(t)$. There is some resemblance with the basic points
following from the quantum kinetic theory [21, 23], where the running
correlations yield nonlinear terms in the kinetic equations, vanishing in the
Markovian limit.

It has been mentioned at the end of previous Section that the system of
equations (2.32)-(2.33) for the reduced density matrices is unmanageable, and
obtaining of the master equation for $\widetilde{\varrho}_{S}(t)$ in higher
orders in interaction is possible only via some iterative scheme. The eventual
master equation would be too cumbersome to be manipulated with. Nevertheless,
some possible conclusions can be anticipated by remembering the results, which
follow from the quantum kinetic theory. For instance, a contribution of the
dynamical correlations is known to remain in the Markovian limit [21] if high
order approximation in the coupling strength is performed. The origin of the
above dynamical correlations [26] can be mapped to our concept of the
intrinsic bath dynamics, so we could expect the presence of the BD term even
in the Markovian limit, as opposed to the above mentioned zeroth contribution
following in MA from Eq. (3).

On the other hand, as it will be shown in Section 4, the obtained non-
Markovian master equation (3), though being formally of the second order in
${\cal L}(t)$, contains explicitly all the orders in the interaction, if one
is going to pass to its convolutionless counterpart. Thus the latter approach
seems to be much easier but not less effective, and generally there is no
particular need to construct the non-Markovian master equations of higher
order in the coupling strength.

Now it would be instructive to compare the obtained results with those, which
follow from the traditional ZN projection scheme. In this scheme, one usually
uses a different decomposition for the total density matrix instead of (2.15),

$\widetilde{\varrho}(t)=\widetilde{\varrho}_{S}(t)\varrho_{B}(0)+\delta\widetilde{\varrho}(t).$
(3.7)

It is important to note that (c.f. Eqs. (2.16))

$\text{Tr}_{B}\,\delta\widetilde{\varrho}(t)=0,\qquad\text{Tr}_{S}\,\delta\widetilde{\varrho}(t)\not=0.$
(3.8)

In fact, decomposition (3.7) means that one neglects the intrinsic dynamics of
the environment by setting the reduced density matrix of the bath to be equal
to its initial value at any time, $\varrho_{B}(t)\equiv\varrho_{B}(0)$. 444It
should be noted that in the original ZN scheme [13] it is supposed that
$[H_{B},\varrho_{B}(0)]=0$, i.e., the bath is initially in thermal
equilibrium. If this initial state differs from thermal equilibrium, it is
natural to write the decomposition of $\varrho(t)$ in the form
$\varrho(t)=\varrho_{S}(t)\varrho^{(0)}_{B}(t)+\delta\varrho(t)$, where
$\varrho^{(0)}_{B}(t)=U_{B}(t)\varrho_{B}(0)U^{\dagger}_{B}(t)\equiv{\rm
e}^{-iH_{B}t}\varrho_{B}(0){\rm e}^{iH_{B}t}.$ Note that the initial state
$\varrho_{B}(0)$ of the environment does not necessarily have to be
equilibrium: it may contain information about the initial system–bath
correlations arising due to the selective [24] quantum measurements, as it
occurs in the case under study (see Section 4) or non-selective ones [13, 30,
31].

It has been pointed out in Ref. [27] that the ZN projection technique can be
viewed as a particular version of the non-equilibrium statistical operator
method (NSO), when the interaction term is not taken into account in the
relevant distribution $\widetilde{\varrho}_{\rm{rel}}(t)$. The relevant
statistical operator is usually considered as the distribution, relating the
initial state of the system to the NSO $\widetilde{\varrho}(t)$ through the
decomposition

$\displaystyle\widetilde{\varrho}(t)=\widetilde{\varrho}_{\rm{rel}}(t)+\Delta\widetilde{\varrho}(t)$
(3.9)

(c.f. Eqs.(2.15) and (3.7)). Since in the limit $t\to\infty$ the system
attains the equilibrium state [21], the correlational part
$\Delta\widetilde{\varrho}(t)$ of the total density matrix approaches zero
only if the interaction $V$ is included into the relevant distribution.
Otherwise, like in the case (3.7) of the ZN projection method and its
generalization (2.15),
$\lim\limits_{t\to\infty}\delta\widetilde{\varrho}(t)\not=0$ and
$\lim\limits_{t\to\infty}\Delta\widetilde{\varrho}(t)\not=0$, since the
equilibrium distribution of the composite system is not factorisable in the
operators, belonging to the different subsystems $S$ and $B$. Here we leave
aside all peculiarities dealt with tending the system to equilibrium,
referring readers to the papers [27, 26].

Secondly, the decomposition (3.8) renders the main disadvantage of the ZN
projection method quite obvious: the state of the environment is postulated to
be described by an equilibrium distribution plus some correction
$\mbox{Tr}_{S}\delta\widetilde{\varrho}(t)$, which seems reasonable only in
the case of small interactions. On the contrary, the generalization of the ZN
method implies that the reduced density matrix of the environment has its own
dynamics, completely incorporated in the distribution
$\widetilde{\varrho}_{B}(t)$, see Eqs. (2.13) and (2.17) in Ref. [28], whereas
approximation (3.1) can be treated as its linearized version.

In the weak coupling limit (see Appendix B in Ref. [25] for details) the ZN
master equation looks as follows:

$\displaystyle\frac{\partial\widetilde{\varrho}_{S}(t)}{\partial
t}=-i{\mathcal{L}}^{(0)}_{S}(t)\widetilde{\varrho}_{S}(t){}-\int^{t}_{0}dt^{\prime}\,\left\\{\text{Tr}_{B}\left\\{{\mathcal{L}}(t){\mathcal{L}}(t^{\prime})\widetilde{\varrho}_{S}(t^{\prime})\varrho_{B}(0)\right\\}-{\mathcal{L}}^{(0)}_{S}(t){\mathcal{L}}^{(0)}_{S}(t^{\prime})\widetilde{\varrho}_{S}(t^{\prime})\right\\}.$
(3.10)

Note that the above reasoning should not be misleading about a formal
resemblance of the equation for the density matrix of the environment
$\widetilde{\varrho}_{B}(t)=\varrho_{B}(0)+\mbox{Tr}_{S}\delta\widetilde{\varrho}(t)$,
which follows from the relations (3.7)-(3.8) of the ZN method, and the
approximation (3.1) in our generalized projecting scheme. The final results
for the master equations (3) and (3.10) are completely different, since they
are derived starting from two different basic points: there is only one
equation for the reduced density matrix of the open quantum system in the
framework of the ZN projection technique, while taking the BD into
consideration one deals with the coupled chain of equations for
$\widetilde{\varrho}_{S}(t)$ and $\widetilde{\varrho}_{B}(t)$, see Eqs.
(2.32)-(2.33). As a consequence, in contrast to Eq. (3), the master equation
(3.10) does not contain the nonlinear term with $R_{S}(t,t^{\prime})$ due to
exclusion of the intrinsic dynamics of the environment from consideration.
Nevertheless, in the Markovian limit, both master equations are identical.
This is quite obvious, since the influence of BD (considered in terms of
$\widetilde{\varrho}_{B}(t)$) is expected to manifest itself only in the
initial stage of the system evolution.

To check out the consistency and robustness of the generalized projection
scheme, one has to use the master equation (3) to derive the quantum kinetic
equations for observables in some simple models (preferably, exactly solvable
ones). It is done in the next Section, where we obtain the dynamic equation
for coherence, considering the purely dephasing model [24].

##  4 Non-Markovian quantum kinetic equation for the dephasing model

We consider a simple version of the spin-boson model describing the two-state
system ($S$) coupled to the bath ($B$) of harmonic oscillators [24, 29]. In
the spin representation for a qubit, the total Hamiltonian of the model is
written as

$\displaystyle
H=H_{S}+H_{B}+V=\frac{\omega_{0}}{2}\sigma_{3}+\sum\limits_{k}\omega_{k}b^{\dagger}_{k}b_{k}+\sigma_{3}\sum\limits_{k}(g_{k}b^{\dagger}_{k}+g^{*}_{k}b_{k}),$
(4.1)

where $\omega_{0}$ is the energy difference between the excited state
$|1\rangle$ and the ground state $|0\rangle$ of the qubit, and $\sigma_{3}$ is
the 3-rd Pauli matrix $\sigma_{i}$, $i=\\{1,2,3\\}$. Bosonic creation and
annihilation operators $b^{\dagger}_{k}$ and $b_{k}$ correspond to the $k$-th
bath mode with the frequency $\omega_{k}$.

Since $\sigma_{3}$ commutes with Hamiltonian (4.1), it does not evolve in
time, $\sigma_{3}(t)=\sigma_{3}$. Hence, the interaction operator can be
easily expressed as

$\displaystyle\tilde{V}(t)\equiv\sigma_{3}\sum\limits_{k}{\cal
F}_{k}(t)=\sigma_{3}\sum\limits_{k}\left\\{g_{k}(t)b^{\dagger}_{k}+g^{*}_{k}(t)b_{k}\right\\},\qquad
g_{k}(t)=g_{k}\exp(i\omega_{k}t).$ (4.2)

Let us introduce in a usual fashion the spin inversion operators
$\sigma_{\pm}=(\sigma_{1}\pm i\sigma_{2})/2$, which obey the permutation
relations $\left[\sigma_{3},\sigma_{\pm}\right]=\pm 2\sigma_{3}$. Our task is
to obtain the quantum kinetic equations for the mean values
$\langle\widetilde{\sigma}_{\pm}\rangle_{S}^{t}=\mbox{Tr}_{S}\\{\tilde{\rho}_{S}(t)\sigma_{\pm}\\}$
dealt with the system coherence. For simplicity, we consider the case with
$\langle\widetilde{\sigma}_{+}\rangle_{S}^{t}$, since the equation for
$\langle\widetilde{\sigma}_{-}\rangle_{S}^{t}$ can be easily obtained in a
similar manner.

Using the basic equation (3) for the reduced density matrix
$\tilde{\rho}_{S}(t)$ and taking a trace over the system variables, one can
derive the following kinetic equation:

$\displaystyle\frac{\partial\langle\widetilde{\sigma}_{+}\rangle_{S}^{t}}{\partial
t}=i\left\langle\left[\tilde{V}_{S}^{(0)}(t),\sigma_{+}\right]\right\rangle^{t}_{S}+\int\limits_{0}^{t}dt^{\prime}\biggl{\\{}\left\langle\left[R_{S}(t,t^{\prime}),\sigma_{+}\right]\right\rangle^{t}_{S}-\left\langle\left[R_{S}(t,t^{\prime}),\sigma_{+}\right]\right\rangle^{t^{\prime}}_{S}\biggr{\\}}$
$\displaystyle-\int\limits_{0}^{t}dt^{\prime}\biggl{\\{}\left\langle\left\langle\left[\tilde{V}(t),\left[\tilde{V}(t^{\prime}),\sigma_{+}\right]\right]\right\rangle_{B}\right\rangle^{t^{\prime}}_{S}+\left\langle\left[\tilde{V}_{S}^{(0)}(t),\left[\tilde{V}_{S}^{(0)}(t^{\prime}),\sigma_{+}\right]\right]\right\rangle^{t^{\prime}}_{S}\biggr{\\}},$
(4.3)

where we denote the bath averaging as
$\langle\ldots\rangle_{B}\equiv\mbox{Tr}_{B}\left\\{\varrho_{B}(0)\ldots\right\\}$.

Having calculated all commutators in Eq. (4) and performed thermal averaging
(see Appendixes C and D in Ref. [25] for details), we can write down the final
kinetic equation for the generalized coherence:

$\displaystyle\frac{d\left\langle\widetilde{\sigma}_{+}\right\rangle^{t}_{S}}{dt}=iA_{\rm{init}}\int\limits_{0}^{\infty}\frac{J(\omega)}{\omega}\cos\omega
t\,d\omega\left\langle\widetilde{\sigma}_{+}\right\rangle^{t}_{S}$
$\displaystyle-i\langle\sigma_{3}\rangle\int\limits_{0}^{t}\Biggl{\\{}\int\limits_{0}^{\infty}J(\omega)\sin[\omega(t-t^{\prime})]d\omega\Biggr{\\}}\left(\langle\widetilde{\sigma}_{+}\rangle^{t}_{\widetilde{S}}-\langle\widetilde{\sigma}_{+}\rangle^{t^{\prime}}_{S}\right)dt^{\prime}$
$\displaystyle-\int\limits_{0}^{t}\Biggl{\\{}\frac{1}{2}\int\limits_{0}^{\infty}J(\omega)\coth\left(\frac{\omega}{2k_{B}T}\right)\cos[\omega(t-t^{\prime})]d\omega$
$\displaystyle-2(A_{\rm{init}}^{2}-1)\int\limits_{0}^{\infty}\frac{J(\omega)}{\omega}\cos\omega
t\,d\omega\int\limits_{0}^{\infty}\frac{J(\omega^{\prime})}{\omega^{\prime}}\cos\omega^{\prime}t^{\prime}\,d\omega^{\prime}\Biggr{\\}}\left\langle\widetilde{\sigma}_{+}\right\rangle^{t^{\prime}}_{S}dt^{\prime},$
(4.4)

where the quantity

$\displaystyle
A_{\rm{init}}=\frac{\exp(\beta\omega_{0}/2)(1-\langle\sigma_{3}\rangle)-\exp(-\beta\omega_{0}/2)(1+\langle\sigma_{3}\rangle)}{\exp(\beta\omega_{0}/2)(1-\langle\sigma_{3}\rangle)+\exp(-\beta\omega_{0}/2)(1+\langle\sigma_{3}\rangle)}$
(4.5)

is related to the initial correlations (in other words, it is dealt with the
initial preparation of the system due to some kind of the quantum measurement
[24, 30]).

In the kinetic equation (4) the qubit-bath coupling is modelled by the
spectral weight function $J(\omega)$ taken in its standard form [24]

$\displaystyle J(\omega)=\lambda\Omega^{1-s}\omega^{s}\exp(-\omega/\Omega).$
(4.6)

The parameter $\lambda\sim|g_{k}|^{2}$ in Eq. (4.6) denotes a dimensionless
coupling constant, while $\Omega$ is the cut-off frequency. Depending on the
exponent $s$, we can distinguish three coupling regimes: the sub-Ohmic at
$0<s<1$, the Ohmic at $s=1$, and the super-Ohmic at $s>1$.

Let us discuss some peculiarities of the quantum kinetic equation (4). First
of all, we remind that the second term in the r.h.s of (4) vanishes in the
Markovian limit. On the other hand, we can always pass from the mean values in
the interaction picture to the averages taken with the statistical operator
$\rho_{S}(t)$ according to the rule
$\left\langle\sigma^{+}\right\rangle^{t}_{\widetilde{S}}\rightarrow\exp(-i\omega_{0}t)\left\langle\sigma^{+}\right\rangle^{t}_{S}$.
Then the second term in the r.h.s of (4), being an imaginary one, along with
the first (non-dissipative or ‘‘quasi-free’’) term should add up to
$\omega_{0}t$, yielding the corresponding phase shift [24].

Second, the initial state of the qubit contributes to the quasi-free term as
well as to the last summand in the r.h.s of Eq. (4) via $A_{\rm{init}}$.
Besides, while the kinetic kernels in the 2-nd and the 3-rd terms have the
usual form ${\cal K}(t-t^{\prime})$, this is not the case for the last term
dealt with the initial correlations in the ‘‘qubit+bath’’ system. It is not
surprising: the dynamics of initial correlations is not a stationary process
with a typical convolution dependence; the initial correlations decay in time
(aging effect [32]). Third, the contribution of the initial correlations is of
the 4-th order in interaction.

In our recent paper [25] we have solved the generic non-Markovian equation (4)
numerically. We have considered the Ohmic and super-Ohmic (with $s=3/2$)
qubit-bath coupling and studied the qubit dynamics at both low and high
temperatures. The results were not quite satisfactory from the physical point
of view: taking the BD into account led to an uncontrollable increase of
$\mbox{Im}\langle\widetilde{\sigma}_{+}(t)\rangle$ at low temperatures. The
picture was not improved much even at high temperatures, since the imaginary
part of the system coherence tended to saturation. This contradicts the
conclusions made in Ref. [24] that the partial decoherence is admitted only at
the strong super-Ohmic regime with $s>2$, while at smaller values of the
ohmicity index the system coherence always tends to zero at large times. Since
no systematic analysis can be proposed in the non-Markovian regime but to
solve the kinetic equations numerically, and the reasons leading to the above
mentioned unphysical behaviour of the coherence were formulated only
qualitatively, we perform the MA 555Formally, it corresponds to the ZN
approach in a time convolutionless picture. of Eq. (4) to obtain exact
analytical expressions and to study the system dynamics more carefully.

The first term in the the r.h.s of Eq. (4) is proportional to the time
derivative of the phase shift,

$\displaystyle\chi(t)=\frac{\sinh(\beta\omega_{0}/2)-\langle\sigma_{3}\rangle\cosh(\beta\omega_{0}/2)}{\cosh(\beta\omega_{0}/2)-\langle\sigma_{3}\rangle\sinh(\beta\omega_{0}/2)}\Phi(t),\qquad\qquad\Phi(t)=\int\limits_{0}^{\infty}d\omega
J(\omega)\frac{\sin\omega t}{\omega^{2}},$ (4.7)

which is nothing but the linearized over $\Phi(t)$ version of the expression
(32) from Ref. [24]. Furthermore, taking the generalized coherence out of the
integral in the third term and integrating the kinetic kernel over
$t^{\prime}$, we come to the conclusion that it is dealt with the time
derivative of

$\displaystyle\gamma(t)=\gamma_{\rm{vac}}(t)+\gamma_{\rm{th}}(t),$ (4.8)
$\displaystyle\gamma_{\rm{vac}}(t)=\int\limits_{0}^{\infty}d\omega
J(\omega)\frac{1-\cos\omega
t}{\omega^{2}},\qquad\gamma_{\rm{th}}(t)=\int\limits_{0}^{\infty}d\omega
J(\omega)\left[\coth(\beta\omega/2)-1\right]\frac{1-\cos\omega
t}{\omega^{2}},\qquad$

where $\gamma_{\rm{vac}}(t)$ ($\gamma_{\rm{th}}(t)$) denotes the vacuum
(thermal) contribution to the generalized decoherence.

In a similar way, performing the MA, we verify that the last term of Eq. (4)
is related to the time derivative of the correlational contribution to the
generalized decoherence,

$\displaystyle\gamma_{\rm{cor}}(t)=\frac{1-\langle\sigma_{3}\rangle^{2}}{2[\cosh(\beta\omega_{0}/2)-\langle\sigma_{3}\rangle\sinh(\beta\omega_{0}/2)]^{2}}\,\Phi(t)^{2}.$
(4.9)

Again, the expression (4.9) is nothing but the series expansion of Eq. (31)
from Ref. [24] up to the fourth order in interaction.

Now let us examine more thoroughly the role of the intrinsic BD, which is
described by the second term in the r.h.s. of the kinetic equation (4). This
term has a very specific structure: it does not depend on the temperature in
spite of having been generated by the qubit environment. In some sense, it is
similar to the vacuum contribution $\gamma_{\rm{vac}}(t)$ to the decoherence
function. It is obvious that the above similarity is rather coincidental,
being a cumulative effect of the applied approximation (3.1) for
$\tilde{\rho}_{B}(t)$ and a specific feature (non-ergodicity) of the dephasing
model. To get some more information about the role of BD, let us go beyond the
MA and expand the difference of the generalized coherences up to the first
order in $\tau=t-t^{\prime}$:

$\displaystyle\langle\widetilde{\sigma}_{+}\rangle^{t}_{\widetilde{S}}-\langle\widetilde{\sigma}_{+}\rangle^{t^{\prime}}_{S}=\frac{\langle\widetilde{\sigma}_{+}\rangle^{t}_{\widetilde{S}}}{dt}(t-t^{\prime})+o(t-t^{\prime})^{2}.$
(4.10)

Since the time derivatives of both the phase shift and the generalized
coherence are of the 2-nd order in the coupling constant, the second term in
the r.h.s. of (4) is proportional to $|g_{k}|^{4}$. Thus taking (4.7)-(4.10)
into account, it is straightforward to solve Eq. (4) and to obtain the
generalized coherence in its usual form

$\displaystyle\langle\tilde{\sigma}_{+}\rangle^{t}_{S}=\langle\tilde{\sigma}_{+}\rangle^{t=0}_{S}\exp[i\bar{\chi}(t)-\bar{\gamma}(t)].$
(4.11)

Here the renormalized values of the phase shift $\bar{\chi}(t)$ and the
generalized decoherence $\bar{\gamma}(t)$ up to the 4-th order in interaction
read as follows:

$\displaystyle\bar{\chi}(t)=\chi(t)+\int\limits_{0}^{t}\,dt^{\prime}F(t^{\prime})\frac{d[\gamma_{\rm{vac}}(t^{\prime})+\gamma_{\rm{th}}(t^{\prime})]}{dt^{\prime}},$
(4.12)
$\displaystyle\bar{\gamma}(t)=\gamma_{\rm{vac}}(t)+\gamma_{\rm{th}}(t)+\gamma_{\rm{cor}}(t)-\int\limits_{0}^{t}\,dt^{\prime}F(t^{\prime})\frac{d\chi(t^{\prime})}{dt^{\prime}},$
(4.13)

where the function

$\displaystyle
F(t)=\langle\sigma_{3}\rangle\int\limits_{0}^{t}d\tau\,\tau\left\\{\int\limits_{0}^{\infty}J(\omega)\sin\omega\tau\,d\omega\right\\}$
(4.14)

can be calculated explicitly, but is too cumbersome to be presented here. In
the integrand of (4.12) we have omitted $\gamma_{\rm{cor}}(t)$, since it is
already of the 4-th order in the coupling constant. At the chosen order in the
coupling constant, the BD contribution to the decoherence function vanishes
for the initially uncorrelated subsystems (see the Eq. (4.13)), since the
phase shift is zero in such a case [24]. It can be said that the initial
correlations and the intrinsic BD are working together in the composite
quantum system to enhance or decrease the coherence, depending on the sign of
$\langle\sigma_{3}\rangle$. Since in the dephasing model the exact value of
the correlational contribution to the decoherence function is known [24],

$\displaystyle\gamma_{\rm{cor}}^{\rm{exact}}=-\frac{1}{2}\ln\left\\{1-\frac{(1-\langle\sigma_{3}\rangle^{2})\sin^{2}[\Phi(t)]}{[\cosh(\beta\omega_{0}/2)-\langle\sigma_{3}\rangle\sinh(\beta\omega_{0}/2)]^{2}}\right\\},$
(4.15)

it is reasonable to treat the last term in (4.13) as a renormalization of
(4.9) and to compare

$\displaystyle\bar{\gamma}_{\rm{cor}}(t)=\gamma_{\rm{cor}}(t)-\int\limits_{0}^{t}\,dt^{\prime}F(t^{\prime})\frac{d\chi(t^{\prime})}{dt^{\prime}}\vspace{-7mm}$
(4.16)

with the above exact result (4.15).

As it is seen in Figs. 1 and 2, there is a domain of the state parameters (the
temperature $1/\beta$ and the mean levels’ inversion
$\langle\sigma_{3}\rangle$), where the renormalized values (4.16), obtained
within our generalized projection technique, reproduce the exact result (4.15)
better than those derived using the ZN approach (4.9). This region is found to
be characterized by the values $-0.2<A_{\rm{init}}<0$, meaning a slight qubit
frequency downshift.

Figure 1: Time dependence of the correlational contribution to the generalized
decoherence, Ohmic coupling. Model parameters: $\omega_{0}/\Omega=1$,
$\beta\omega_{0}=0.1$, $\langle\sigma_{3}\rangle=0.2$, $\lambda=1/3$. The
solid, dashed and dotted curves correspond to the exact (4.15), renormalized
(4.16) and the ZN (4.9) expressions.

Figure 2: The same as in Fig. 1 at $\beta\omega_{0}=5$ and
$\langle\sigma_{3}\rangle=0.99$.

It should be noted that such a behaviour of $\gamma_{\rm{cor}}(t)$ is typical
both in the high (Fig. 1) and low (Fig. 2) temperature limits. It could be
also shown that in the moderate super-Ohmic regime, the obtained dynamics has
tendencies similar to those presented at $s=1$. However, at the sub-Ohmic
coupling, the behaviour of $\gamma_{\rm{cor}}^{\rm{exact}}(t)$ is known to be
non-monotonic in time [24], and none of the approximations (4.9) or (4.16) can
reproduce the true dynamics of the correlational contribution to the
decoherence function at large times due to specific kind of coupling [14].

On the other hand, the lowest series expansion in (4.10) does not allow us to
estimate the correction to the phase shift reliably: according to Eq. (4.12),
it arises even at $\chi(t)=0$, contradicting the well-known statement [24],
which attributes the qubit frequency renormalization solely to the appearance
of the initial correlations in the composite $S+B$ system. Going beyond the
4-th order in the interaction in calculations of the renormalized values of
$\chi(t)$ and $\gamma_{\rm{cor}}(t)$ could eventually improve the situation.
But it also implies taking into account the next terms in the series expansion
(4.10) that automatically would deviate the expected result from the exact
exponential form (4.11).

Nevertheless, the results presented in Figs. 1 and 2 show that the proposed
generalization of the projection technique can be considered as a good
alternative to other perturbative methods of solving the quantum master
equations [13], which perceptibly improves the ZN results already in the
lowest order in the coupling strength.

To conclude this Section, we mention that all the results are obtained for the
system that has been initially prepared by the selective quantum measurement.
The quantity (4.5), entering all the expressions dealt with initial
correlations, is uniquely defined by the above preparation measurement.
However, there is a much more interesting situation, when the system is
prepared due to non-selective measurement scheme [13]. In such a chase, not
only the decoherence induced by the initial correlations is observed, but also
the inverse process (recoherence, or the coherence enhancement) is possible.
In Ref. [30], we proposed some kinds of measurements leading to the permanent
coherence growth at small times $t\sim 1/\Omega$. Obviously, at times $t\gg
1/\Omega$, the other dissipation channels (thermal and vacuum ones) prevail,
yielding a decrease of the system coherence. There is also a possibility to
prepare the system in such an initial stage, when the periods of recoherence
and decoherence are alternating. In the very recent paper [33], the evolution
of quantum coherence for the same dephasing model has been studied in detail
over short and long times.

Thus we are in a position to perform a similar analysis of the BD contribution
to the system, which is initially prepared after the non-selective quantum
measurements, since the general ideas leading to the ‘‘renormalized’’
functions (4.12)-(4.13) are valid in this case as well, and the general
expression (4.11) for the system coherence remains unaltered. However, the
expression for the ‘‘renormalized’’ correlational contribution
$\bar{\gamma}_{{\rm cor}}(t)$ to the decoherence/recoherence rate would
consist of too many parameters to be analyzed reliably. As far as our case of
selective preparation measurements is concerned, the short or long time
analysis of the renormalized function (4.16) can hardly be informative or
useful, because its contribution to the (permanent!) system decoherence is
small. Therefore, we prefer to focus our efforts on investigation of a measure
of its deviation from the exact result (4.15) to verify the validity and
robustness of our generalized projection scheme beyond the Zwanzig-Nakajima
framework.

##  5 Conclusions and outlook

In this paper, we generalize the ordinary ZN projection scheme for derivation
of the master equation for the open quantum system weakly coupled with its
surrounding. We start from the chain of equations for the density matrices of
the $S$\- and $B$-subsystems and obtain the non-Markovian master equation for
$\widetilde{\varrho}_{S}(t)$. We simplify this equation, restricting ourselves
by the lowest approximation in the coupling constant. The consideration of the
intrinsic BD yields an extra term in the master equation, which is found to be
nonlinear in $\widetilde{\varrho}_{S}(t)$ and vanishing in the Markovian
limit. When the dynamic equation for $\widetilde{\varrho}_{B}(t)$ is
neglected, we come to the standard ZN projecting scheme [13].

In order to test the developed method we have applied it to the open quantum
system, described by a very simple dephasing model. The non-Markovian quantum
kinetic equation for the generalized coherence has been derived. In the MA,
which means nothing but the time convolutionless ZN result, its solution up to
the 2-nd order in interaction coincides with the exact one obtained in Ref.
[24]. We have also analyzed the kinetic kernel related to the BD, which is
found to be temperature independent. When going beyond the MA in the above
mentioned term, we obtain the renormalized values of the phase shift and the
generalized decoherence. We have compared the renormalized correlational
contribution to the decoherence function with the exact result [24] and with
that of the ZN approach. It is found that consideration of the intrinsic BD
brings the corresponding curves closer to the exact data as compared to the
results obtained within the ordinary ZN projection scheme.

However, some model peculiarities, e.g. its non-ergodicity do not allow us to
take the full advantages of our method: in the lowest order in interaction the
kernel, dealt with the BD, turns out to be temperature independent, which
seems a bit strange if one speaks about the influence of the qubit
surrounding. Other difficulties arise from the necessity of taking the higher
order terms in consideration that inevitably causes significant computational
complications666In this context it should be noted that a formal application
of the perturbation theory in the interaction can sometimes be misleading: in
the case of the dephasing model under study, any eventual higher-order
corrections to the vacuum $\gamma_{\rm{vac}}(t)$ and thermal
$\gamma_{\rm{th}}(t)$ terms have to be rejected as contradicting the exact
solution. and, from the mathematical point of view, can even deviate the
solution for the generalized coherence from its exact exponential form [24].

There is also a different approach to treat the influence of BD, which is
based upon consideration of the dynamical correlations in the composite system
[20]. These long-lived dynamical correlations, which are associated with the
total energy conservation, play an important role in the transition to the
Markovian regime and subsequent equilibration of the system. In our recent
paper [26], we have applied this approach to the dephasing model and obtained
nonlinear kinetic equations, involving the so-called correlational
temperature. Though the above method does not admit consideration of the
intrinsic BD explicitly, it allows to study the build-up of the dynamic
correlations between the qubit and the bath, as well as its time evolution
from the initial stage up to the system equilibration. It is interesting that
the dynamic correlations act in a similar way, renormalizing the qubit
frequency. Thus we dare to say that the studies of the BD influence on the
state of open quantum systems, the build-up of the system–bath correlations,
as well as a choice of the most suitable models for verification of these
approaches, are still far from completeness and should be continued in the
future.

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