Phenomenological memory-kernel master equations and time-dependent Markovian processes

Abstract

Do phenomenological master equations with memory kernel always describe a
non-Markovian quantum dynamics characterized by reverse flow of information? Is
the integration over the past states of the system an unmistakable signature of
non-Markovianity? We show by a counterexample that this is not always the case.
We consider two commonly used phenomenological integro-differential master
equations describing the dynamics of a spin 1/2 in a thermal bath. By using a
recently introduced measure to quantify non-Markovianity [H.-P. Breuer, E.-M.
Laine, and J. Piilo, Phys. Rev. Lett. 103, 210401 (2009)] we demonstrate that
as far as the equations retain their physical sense, the key feature of
non-Markovian behavior does not appear in the considered memory kernel master
equations. Namely, there is no reverse flow of information from the environment
to the open system. Therefore, the assumption that the integration over a
memory kernel always leads to a non-Markovian dynamics turns out to be
vulnerable to phenomenological approximations. Instead, the considered
phenomenological equations are able to describe time-dependent and
uni-directional information flow from the system to the reservoir associated to
time-dependent Markovian processes.

Full text

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arXiv:1003.3817v2 [quant-ph] 3 Mar 2011
Phenomenological memory-kernel master equations and time-dependent Markovian
processes
L. Mazzola,1, ∗E.-M. Laine,1, †H.-P. Breuer,2S. Maniscalco,1and J. Piilo1
1Turku Centre for Quantum Physics, Department of Physics and Astronomy,
University of Turku, FI-20014 Turun yliopisto, Finland
2Physikalisches Institut, Universit¨ at Freiburg, Hermann-Herder-Strasse 3, D-79104 Freiburg, Germany
Do phenomenological master equations with memory kernel always describe a non-Markovian
quantum dynamics characterized by reverse flow of information? Is the integration over the past
states of the system an unmistakable signature of non-Markovianity? We show by a counterexam-
ple that this is not always the case. We consider two commonly used phenomenological integro-
differential master equations describing the dynamics of a spin 1/2 in a thermal bath. By using a
recently introduced measure to quantify non-Markovianity [H.-P. Breuer, E.-M. Laine, and J. Piilo,
Phys. Rev. Lett. 103, 210401 (2009)] we demonstrate that as far as the equations retain their phys-
ical sense, the key feature of non-Markovian behavior does not appear in the considered memory
kernel master equations. Namely, there is no reverse flow of information from the environment to the
open system. Therefore, the assumption that the integration over a memory kernel always leads to a
non-Markovian dynamics turns out to be vulnerable to phenomenological approximations. Instead,
the considered phenomenological equations are able to describe time-dependent and uni-directional
information flow from the system to the reservoir associated to time-dependent Markovian processes.
PACS numbers: 03.65.Yz, 03.65.Ta, 42.50.Lc
I.INTRODUCTION
The study of non-Markovian open quantum systems
has attracted extraordinary attention and efforts in re-
cent years [1]. Many analytical methods and numerical
techniques have been developed to treat non-Markovian
processes [2–11]. In addition to their importance in ad-
dressing fundamental questions [12], this is mainly due
to the applications non-Markovian systems find in many
branches of physics. Non-Markovian processes appear in
quantum optics [1, 13, 14], solid state physics [15], quan-
tum chemistry [16], quantum information processing [17],
and even in the description of biological systems [18].
Recently, several more rigorous definitions and quan-
tifications of non-Markovian behavior in open quantum
systems have been proposed [19–23]. In fact, in the past
the concept of non-Markovian dynamics has been quite
loosely defined. The term non-Markovian process could,
e.g., stand for: Not describable by a master equation with
Lindblad structure, or leading to non-exponential decay,
or characterized by a time-dependent generator, or in-
volving an integral over the past states of the system.
Quite often it has been argued that the treatment of
non-Markovian dynamics necessarily requires solving an
integro-differential equation for the reduced density ma-
trix of the system. However, it has been shown that
master equations which are local in time can also repre-
sent the memory effects of a non-Markovian process (see,
e.g., Ref. [1] and references therein), without the need to
take into account a time integration over the past his-
∗Electronic address: laumaz@utu.fi
†Electronic address: emelai@utu.fi
tory of the system.
only that memory kernel master equations are not the
unique tool to treat non-Markovian dynamics, but we
also demonstrate that the presence of a memory kernel
alone does not guarantee the non-Markovian character
of the process associated to the reverse flow of informa-
tion from the system to the environment. This surprising
result is obtained by applying a recently proposed mea-
sure of non-Markovianity [20, 21] to two quite commonly
used non-local master equations: The generalized mem-
ory kernel master equation discussed by one of us [7] and
the post-Markovian Shabani-Lidar master equation [10],
both used to study the time-evolution of a spin 1/2 in
a thermal bath. Our results are connected to the issue
of phenomenological vs. microscopically derived master
equations in quantum optics which do not always pro-
duce coinciding results in all of the relevant parameter
regimes [24]. Here we show that there can be also qual-
itative differences in addition to the quantitative ones
when the two approaches are used.
Before proceeding with our treatment we would like to
emphasize that we do not intend to discourage from the
use of memory kernel master equations, which indeed in
many cases constitute a fundamental tool to study non-
Markoviansystems (see for example Ref. [6]). Instead, we
rather want to point out that the mathematical features
of phenomenological kernel equations do not guarantee
that the feedback effects from the environment and the
physical characteristics of non-Markovian dynamics are
taken into account.
We begin by briefly describing the measure for non-
Markovianity introduced by three of us in Refs. [20, 21].
Then we present the master equations under investiga-
tion and their solutions for the density matrix of the re-
duced system. Finally, we use these solutions to evaluate
Here we go a step forward: Not

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the degree of non-Markovianity for the two quantum pro-
cesses, concluding with a discussion and some remarks.
II.MEASURE FOR NON-MARKOVIANITY
The construction of the measure for the degree of non-
Markovianity in open systems is based on the definition
of Markovian processes as those that continuously reduce
the distinguishability of quantum states [20]. One can
interpret this loss of distinguishability as a flow of in-
formation from the open system to its environment. By
contrast, in a non-Markovian process there exists a pair
of states the distinguishability of which grows for certain
times. This growth of the distinguishability of states can
be interpreted as a reverse flow of information from the
environment to the open system which is defined to be
the essential feature of non-Markovianity [20, 21].
An appropriate measure for the distinguishability be-
tween two quantum states given by density matrices ρ1
and ρ2is the trace distance [25]
D(ρ1,ρ2) =1
2Tr|ρ1− ρ2|, (1)
where |A| =
ric on the space of physical states. It has the important
property that all quantum operations, i.e., all completely
positive and trace preserving (CPT) maps are contrac-
tions for this metric. Given a pair of initial states ρ1,2(0)
the rate of change of the trace distance under the time
evolution is defined by
√A†A. The trace distance represents a met-
σ(t,ρ1,2(0)) =d
dtD(ρ1(t),ρ2(t)). (2)
A given process is said to be Markovian if for all pairs
of initial states the rate of change of the trace distance
is smaller than zero for all times, i.e., σ(t,ρ1,2(t)) ≤ 0.
Thus, a process is defined to be non-Markovian if there
exists a pair of initial states ρ1,2(0) and a certain time
t at which the trace distance increases, σ(t,ρ1,2(t)) >
0. As shown in Ref. [20, 21] one can construct on the
basis of this definition a measure for non-Markovianity
which represents a functional N(Φ) of the corresponding
quantum dynamical map Φ. This measure is defined as
the maximum over all pairs of initial states of the total
increase of the distinguishability during the whole time-
evolution:
N(Φ) = max
ρ1,2(0)
?
σ>0
σ(t,ρ1,2(0))dt. (3)
In the following we are going to prove that for the
dynamics of a simplified spin-boson model generated by
the generalized memory kernel master equation [7] and by
the Shabani-Lidar post-Markovian master equation [10]
the rate of change of the distinguishability of any pair
of states is always negative, implying that the measure
of non-Markovianity is equal to zero. Thus, despite the
presence of the time-integral over the past history, the
two master equations do not describe any feedback of
information from the environment to the system and are
thus memoryless in this sense. However, it is important
to note that the treated master equations can describe
time-dependent uni-directional flow of information from
the system to the environment.
III. MEMORY KERNEL AND
POST-MARKOVIAN MASTER EQUATIONS
We first present a paradigmatic example of a phe-
nomenological memory kernel master equation, describ-
ing the dynamics of a spin 1/2 interacting with a bosonic
reservoir at temperature T under the rotating-wave ap-
proximation,
dρ(t)
dt
=
?t
0
k(t′)Lρ(t − t′)dt′. (4)
Here, ρ(t) represents the reduced density matrix of the
spin, k(t) is a memory kernel function containing infor-
mation about the properties of the reservoir, and L is
a Markovian superoperator. This superoperator is given
by
Lρ =γ0(N + 1)
+γ0N
2
(2σ−ρσ+− σ+σ−ρ − ρσ+σ−)
2
(2σ+ρσ−− σ−σ+ρ − ρσ−σ+),
(5)
with γ0being the phenomenological dissipation constant,
N the mean number of excitations of the reservoir, and
σ±the usual raising and lowering operators of the spin.
We consider a widely used form for the memory kernel
function, namely an exponential function
k(t) = γe−γt. (6)
The memory kernel master equation (4) can be solved
using the method of the damping basis [26]. In Ref. [8]
the solution for the components of the Bloch vectors was
derived from which one easily obtains the solution for the
spin density matrix ρ(t) corresponding to a generic initial
state. Using the basis {|0?,|1?} of the eigenstates of σz
the elements of ρ(t) can be written as
ρ11(t) = u(t)ρ11(0) + v(t)ρ00(0),
ρ00(t) = (1 − u(t))ρ11(0) + (1 − v(t))ρ00(0),
ρ10(t) = z(t)ρ10(0),
(7)
where u(t), v(t), and z(t) depend on the damping matrix
Λ = diag{λ1,λ2,λ3} and the translation vector− →
(T1,T2,T3) as u(t) = (1+T3+λ3)/2, v(t) = (1+T3−λ3)/2
and z(t) = λ1. The damping matrix elements and the
translation vector components can in turn be expressed
T =

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as [7]
λ1= λ2= ξM(R/2,t),
λ3= ξM(R,t),
T1= T2= 0,
T3=ξM(R,t) − 1
2N + 1
,
(8)
where the function ξM(R,t) is given by
ξM(R,t) =e−γt/2
?
1
√1 − 4Rsinh
?γt
?γt
2
√1 − 4R
?
+ cosh
2
√1 − 4R
??
,
(9)
with R = γ0(2N + 1)/γ.
An interesting master equation which interpolates be-
tween the generalized measurement interpretation of the
Kraus operators and the continuous measurement inter-
pretation of the Markovian dynamics is the Shabani-
Lidar post-Markovianmaster equation. The general form
of this master equation is [10]
dρ
dt= L
?t
0
k(t′)exp(Lt′)ρ(t − t′)dt′, (10)
where once more ρ(t) is the density matrix of the reduced
system, k(t) the Shabani-Lidar memory kernel, and L is
the Markovian superoperator. In the following L is taken
to be of the form of Eq. (5), and the kernel function
is again an exponential function given by Eq. (6). The
solution for the density matrix elements of the spin can
be written again in the general form of Eq. (7), where
u(t), v(t), and z(t) depend in the same way as before on
the damping matrix elements and the translation vector
components [7]. They in turn have the same analytic
expressions of Eqs. (8) with the exception that ξM(R,t)
has to be replaced by the quantity ξP(R,t) which is given
by
ξP(R,t) =exp
?
−R + 1
1
?1 − r(R)sinh
+ cosh
2
γt
?
×
?
??
1 − r(R)(R + 1)γt
2
?
??
1 − r(R)(R + 1)γt
2
??
,
(11)
with r(R) = 4R/(R + 1)2and R = γ0(2N + 1)/γ.
In Ref. [7] the conditions for the positivity and com-
plete positivity of the dynamical maps associated to the
master equations (4) and (10) were studied. There it
was found that 4R ≤ 1 is a necessary and sufficient con-
dition for the positivity of the dynamical map associated
to the memory kernel master equation, while complete
positivity is satisfied only for moderate and high temper-
atures of the reservoir. On the other hand, the dynamical
map corresponding to the post-Markovian master equa-
tion (10) is always completely positive.
are in agreement with Ref. [8] where it was noticed that
the memory kernel master equation can be derived from
the post-Markovian one in the limit in which the phe-
nomenological dissipation constant γ0 is much smaller
than the reservoir correlation decay rate γ, suggesting
that the post-Markovian master equation is somehow
more fundamental than the former one. Therefore, while
for the post-Markovian Shabani-Lidar master equation
we can freely investigate the non-Markovianity for the
whole range of parameters, in the case of the memory
kernel master equation we are restricted to the condi-
tions 4R ≤ 1 and N ≫ 1.
Having constructed the solution of the two master
equations we can now determine the rate of change of
the distinguishability given by Eq. (2), which leads to
These results
σ(t,ρ1,2(0)) =a(t)d
dta(t) + |b(t)|d
?a2(t) + |b(t)|2
11(t) and b(t) = ρ1
dt|b(t)|
, (12)
where a(t) = ρ1
represent the differences of the populations and the co-
herences of the density matrices ρ1(t) and ρ2(t). It is easy
to see that these function are equal to a(t) = λ3(ρ1
ρ2
with λ3 = ξM(P)(R,t) and λ1 = ξM(P)(R/2,t) for the
memory kernel and the post-Markovian master equation,
respectively. The derivative of the trace distance can thus
be written as
11(t) − ρ2
10(t) − ρ2
10(t)
11(0)−
11(0)) = λ3a0 and b(t) = λ1(ρ1
10(0) − ρ2
10(0)) = λ1b0,
σ(t) =a2
0ξ(R,t)d
dtξ(R,t) + |b0|2ξ(R/2,t)d
?a2
dtξ(R/2,t)
0ξ(R,t)2+ |b0|2ξ(R/2,t)2
,
(13)
with ξ(R,t) = ξM(P)(R,t) in the two cases. At this point
we need to study the properties of these two functions.
Let us consider first the memory kernel master equa-
tion: Under the condition 4R ≤ 1, ξM(R,t) is a posi-
tive, monotonically decreasing function which means that
d
dtξM(R,t) < 0 and
are obviously positive for any pairs of states we thus
have σ(t) ≤ 0.
of the distinguishability of any pair of initial states al-
ways decreases and, hence, the flow of information from
the system to the environment is never inverted during
the dynamics and non-Markovian effects do not appear.
Analogously, also in the case of the post-Markovian mas-
ter equation ξP(R,t) is a positive, monotonically decreas-
ing function such that σ(t) ≤ 0. This implies that also
the post-Markovian master equation does not describe
any feedback of information from the environment to the
open system. In contrast to purely Markovian evolution
fulfilling the semigroup property, the dynamics generated
by the memory-kernel and the post-Markovian master
equations can be classified as time-dependent Markovian
processes since they do not fulfill the semigroup property
while the information flow is nevertheless uni-directional
from the system to the reservoir. This is demonstrated
d
dtξM(R/2,t) < 0. Since a2
0and |b0|2
We conclude that the rate of change

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by the time-dependence of the decay rates in the corre-
sponding time-local description given below.
Indeed, both master equations studied here can be
written in the time-convolutionless form
dρ(t)
dt
=
γ1(t)
2
+γ2(t)
(2σ−ρσ+− σ+σ−ρ − ρσ+σ−)
2
(2σ+ρσ−− σ−σ+ρ − ρσ−σ+)
+γ3(t)
2
(2σzρσz− σzσzρ − ρσzσz), (14)
where
γ1(t) = −N + 1
2N + 1
d
dtξ(R,t)
ξ(R,t)
, (15)
γ2(t) = −
N
2N + 1
?
d
dtξ(R,t)
ξ(R,t)
,
γ3(t) =
1
2
d
dtξ(R,t)
2ξ(R,t)−
d
dtξ(R/2,t)
ξ(R/2,t)
?
,
and ξ(R,t) = ξM(P)(R,t) in the two cases. Thus we
see that the integro-differential master equations given
by Eqs. (4)-(6) and by Eq. (10) can be transformed into
a form which is local in time and does not involve any
time-integration over a memory kernel.
The decay rate γ3(t) in Eq. (15) is always negative.
It follows that the dynamical map Φ corresponding to
the master equation is nondivisible [21]. On the other
hand, we have found above that the process is Markovian.
Thus we have an explicit example of a nondivisible quan-
tum process with zero measure for non-Markovianity,
N(Φ) = 0. The existence of such processes was already
conjectured in Ref. [21]. Physically this means that the
influence of the decay channel with a negative rate is over-
compensated by the effect of the other channels with pos-
itive rates, such that the distinguishability of quantum
states is still monotonically decreasing. We include these
non-divisible processes which have uni-directional infor-
mation flow into the class of time-dependent Markovian
processes. This class also includes processes whose de-
cay rates are time-dependent and positive quantities [21].
We emphasize that the time-dependent uni-directional
(time-dependent Markovian) processes and the reversed
information flow (non-Markovian) processes have impor-
tant fundamental differences, as described recently, e.g.,
in Refs. [3, 20, 21].
Going back to the memory kernel master equation (4),
it is also interesting to notice that the non-appearance of
memory effects depends on the restrictions of the range
of parameters imposed by the requirement of positiv-
ity. In fact, when positivity breaks down for 4R > 1,
the hyperbolic sine and cosine of Eq. (9) are replaced
by trigonometric sine and cosine. The function ξM(R,t)
then shows damped oscillations, its derivative has no def-
inite sign, and, consequently, there can be intervals of
time in which the rate of change of the trace distance
σ(t) becomes positive implying that non-Markovian ef-
fects appear. We mention that a violation of the posi-
tivity of the dynamical map in phenomenological master
equations was previously studied by Barnett and Sten-
holm in Ref. [27]. There it was shown that the intro-
duction of an exponential memory kernel function in the
dynamics of a damped harmonic oscillator can lead to
blatantly non-physical behavior.
IV.DISCUSSION AND CONCLUSIONS
We have applied a recently developed measure for
the degree of non-Markovianity of quantum processes to
the dynamical solutions of a simplified spin-boson model
given by two widely used integro-differential master equa-
tions. It has been demonstrated that, as long as the re-
quirement of the positivity of the associated dynamical
maps is fulfilled, no non-Markovian behavior occurs, i.e.,
the measure of non-Markovianity is equal to zero. This
means that the phenomenological memory kernel mas-
ter equations considered here are not able to describe a
genuine non-Markovian behavior involving a backflow of
information from the environment to the open system.
Recently, the exact memory kernel master equation for
a two-state system coupled to a zero temperature reser-
voir has been constructed [28], showing that in this case
the structure of the master equation given by Eqs. (4) and
(5) is incompatible with a non-perturbative treatment of
the underlying microscopic system-reservoir model. The
perturbation expansion of the exact memory kernel re-
veals that in higher orders a new decay channel appears
in the superoperator (5) which is not present in the stan-
dard Born approximation. Thus, while the exact mem-
ory kernel master equation describes correctly all non-
Markovian features of the model, approximation schemes
and phenomenological models can lead to strong restric-
tions in the treatment of non-Markovianity.
Generally, one might be tempted to think that the
introduction of a memory kernel necessarily leads to a
dynamics with non-Markovianity and memory effects.
However, our results demonstrate that one needs to be
cautious when characterizing the physical properties of
open systems only through the mathematical structure
of their equations of motion. The presence of an integral
over the past history in a phenomenological or approx-
imate master equation does not necessarily guarantee a
proper description of memory effects, namely, the feed-
back of information from the environment to the open
system. Even though memory kernel master equations
certainly provide a very useful tool for the description
of non-Markovian quantum processes, our results lead to
the following questions: Which of the commonly used
phenomenological or approximate memory kernel master
equations are able to reproduce the key features of non-
Markovianity? Can one formulate general conditions for
the structure of the memory kernels which guarantee the
presence of these features in the dynamics?

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Acknowledgments
We thank the Magnus Ehrnrooth Foundation, the
Turku University Foundation, the Emil Aaltonen Foun-
dation, the Finnish Cultural Foundation, the Turku Col-
legium of Science and Medicine, the National Gradu-
ate School of Modern Optics and Photonics, and the
Academy of Finland (project 133682) for financial sup-
port.
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