Journal of Quantum Information Science, 2013, 3, 27-33
http://dx.doi.org/10.4236/jqis.2013.31007 Published Online March 2013 (http://www.scirp.org/journal/jqis)
Non-Markovian D ynamics of an Open Two-Level
System with Amplitude-Phase Damping
Ning Tang, Guoyou Wang, Zilong Fan, Haosheng Zeng*
Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of
Education, Department of Physics, Hunan Normal University, Changsha, China
Received February 6, 2013; revised March 10, 2013; accepted March 19, 2013
ABSTRACT
By use of the measure, the backflow of information presented recently, we study the non-Markovianity of the dynamics
for a two-level system interacting with a zero-temperature structured environment via amplitude-phase coupling. In the
limit of weak coupling between the system and its reservoir, the time-local non-Markovian master equation for the re-
duced state of the system is derived. Under the secular approximation, the exact analytic solution is obtained. Numerical
simulations show that the amplitude and phase dampings can produce destructive interference to the backflow of infor-
mation, leading to the weaker non-Markovianity of the compound dynamics compared with the dynamics of a single
amplitude or phase damping model. We also study the characteristics of the initial-state pairs that maximize the back-
flow of information.
Keywords: Non-Markovianity; Amplitude-Phase Dampings; Destructive Interference
1. Introduction
The evolution of open quantum systems can be divided
into two basic types, i.e., Markovian and non-Markovian
processes. For memoryless Markovian processes, the
environment acts as a sink and the information that the
system released into the environment during their inter-
action no longer reflows to the system. However, this is
not the case in the non-Markovian processes, where the
lost information will return to the system at a later time,
so that the later evolution of the system is affected by its
past history, i.e., which appears memory effect.
Although almost all early works are devoted to the
study of Markovian processes [1], people recently found
that Many relevant physical systems, such as the quan-
tum optical system, quantum dot [2], superconductor sys-
tem [3], quantum chemistry [4] and biological system [5]
etc. can not be described simply by Markovian dynamics.
Quantum non-Markovian processes can lead to distinctly
different effects on decoherence and disentanglement [6,
7] of open systems compared with Markovian processes,
which are important both for the enriching of the basic
theory of quantum mechanics and for some practical ap-
plications, such as the quantum metrology [8] and quan-
tum key distribution [9]. Because of these distinct prop-
erties and extensive applications, more and more atten-
tions and interest have been devoted to the study of
non-Markovian processes of open systems, including the
measures of non-Markovianity [10-17], the positivity [18,
19], and some other dynamical properties [20-26] of non-
Markovian processes. Experimentally, the simulation
[27-30] of non-Markovian environment has been real-
ized.
The study of non-Markovian dynamics of open quan-
tum systems is typically very involved and often requires
some assumptions or approximations. In the previous re-
searches of non-Markovian dynamics, only a single cou-
pling way (amplitude-damping or dephase interaction) is
assumed. Further, the rotating wave approximation, that
is, neglecting the counter-rotating terms in the system-re-
servoir interaction Hamiltonian, is employed. These as-
sumptions and approximations limit the serviceable
range of the model. In this paper, we consider a complex
model which simultaneously consists of amplitude and
phase dampings to the environment. We also reserve the
counterrotating terms in the system-reservoir interaction
Hamiltonian. Our motivation is to compare the non-
Markovian features of the dynamics for amplitude and
phase dampings, and to observe the interference be-
tween the two non-Markovian dynamics induced by the
two damping ways.
The article is organized as follows. In Section 2, we
introduce the microscopic Hamiltonian model and derive
the non-Markovian time-local master equation for a two-
level system weakly coupled to a vacuum reservoir. In
*Corresponding author.
C
opyright © 2013 SciRes. JQIS

N. TANG ET AL.
28
Section 3, we solve the master equation under the so-
called secular approximation and present the analytic
expression for the calculation of non-Markovianity based
on the measure proposed by Breuer, Laine and Piilo
(BLP) [10] recently. In Section 4, we choose the Lor-
entzian spectra reservoir as an exemplary example and
simulate numerically the non-Markovianity of the system
dynamics. The non-Markovian effects produced by am-
plitude noise, phase noise and their combination are in-
vestigated. And finally the conclusion is arranged in Sec-
tion 5.
2. Microscopic Model
Consider a two-level atom with Bohr frequency 0

in-
teracting with a zero-temperature bosonic reservoir mod-
eled by an infinite chain of quantum harmonic oscillators.
The total Hamiltonian for this system in the Schrodinger
picture is given by.



0
1
2
.
z
kk kkzkk
kk
kkk
k
H
bbgb b
gbb
 




 


 (1)
where
z

and

 are the Pauli and inversion opera-
tors of the atom, ,
kk
b

and k b are respectively the
frequency, annihilation and creation operators for the
k-th harmonic oscillator of the reservoir. The atom cou-
ples to its environment via both amplitude and phase
interactions, k

g
is the coupling strength which is as-
sumed to be real for simplicity. The parameters

and

describe the relative strengthes of the two couplings
which satisfy . Note that we include the
counter-rotating terms, and
22
1  



bk
k
b

, in the interac-
tion Hamiltonian.
The time-convolutionless (TCL) projection operator
technique [1] is most effective in dealing with the dy-
namics of open quantum systems. In the limit of weak
coupling between the system and its environment, by
expanding the TCL generator to the second order with
respect to coupling strength, the non-Markovian master
equation describing the evolution of the reduced system,
in the interaction picture, can be written as
  
d,.
dLS
tiHtt Dt Dt
t
 
  

(2)
where
 
2,
LS
Ht StSt  

 

(3)
is the Lamb shift Hamiltonian which describes a small
shift in the energy of the eigenstates of the two-level
atom. In many theoretical researches [20], this term was
neglected usually. But in this paper, we will take it into
the consideration. The parameters and
SS

which
corresponds to respectively the Lamb shifts of levels 0
and 1 may be written as
  
0
0dd sin,
t
St J


 



 

 

(4)
with


1, 1 

 and


2
kk
k
Jg

  
 the
spectral distribution of the environment.
The dissipator


Dt 



 that describes the secular
motion of the system has the form
  
  
  
2
2
2
0
1,
2
1,
2
,
zz
Dtt tt
tt t
tt t
 


 
 


 


 


 



 
(5)
where the first term describes the dissipation of the atom
to its environment with time-dependent decay rate


t

, and the second term denotes the heating with a
rate


t

. The last term describes the purely dephasing
with a rate


0t. These time-dependent rates can be
written as,
 

0
0
2d dcos,
t
tJ


 



 

 (6)
with


1,1, 0   .
The dissipator


Dt 




 represents the contribution
of the so-called nonsecular terms, that is, terms oscillat-
ing rapidly with Bohr frequency 0

,



 
 

 
 
 
2
..,
z
z
z
z
z
z
Dt
ut ivtt
tit t
tit t
ti tt
ti tt
ttitit t
ttitit th


 
 
 
 
 
 



 
 
 
 


















 

c

(7)
here h.c. denotes the Hermitian conjugation, and the
time-dependent coefficients are defined as
 
0
0
2d dcoscos,
t
utJ tt   

 

 
 (8)
 
0
0
2d dcossin,
t
vtJ tt   

 

   (9)
  
0
0dd cos,
t
tJt



 
 (10)

0
0dd cos,
t
tJ

  

 t
(11)

N. TANG ET AL. 29
 
0
0dd sin,
t
tJt



 

 
 
t
(12)

0
0dd sin
t
tJ

 .




 (13)
3. Solution and Non-Markovian Measure
We now consider the case where the nonsecular term

Dt 

 
can be neglected. Just as pointed out by
Maniscalco [31], this kind of secular approximation that
used after tracing over the bath degrees of freedom is
different from the rotating wave approximation before
the tracing. It is a more precise approximation that con-
sists in an average over rapidly oscillating terms, but
does not wash out the effect of the counter-rotating terms
present in the coupling Hamiltonian. Under the secular
approximation, the master Equation (2) has the Lindblad-
like form with time-dependant decay rates. It is straight-
forward and easy to show that the corresponding Bloch
equation may be written as


22 22
0
14
2,
x
x
bb  
 
 y SSb(14)


22 22
0
14,
2
y
yx
bbS  
 
 

Sb(15)

22
,
zz
bb 
 
  (16)
where the three components of the Bloch vector are
defined as
 
=
j
j
bt trt



 
with and
,,jxyz
j

the Pauli operators. For compactness we omit the
argument of all the time-dependent coefficients.
Employing the method proposed by Hall [32], the
corresponding Bloch Equations (14)-(16) can be solved
exactly which gives


 
e0cos 0sin
t
xx y
btbt bt 
 

,



, 


, 

(17)


 
e0sin 0cos
t
yx y
btbt bt 
 

 (18)



 
 
2
0
e0ede
t
tts
zz
btbss s 
 

  


(19)
with
  
22 2
0
0
1d4,
2
t
tss ss  


 






(20)
 
2
0d,
t
ts ss 

 

 (21)
and
 
2
0d.
t
tsSsSs 



 (22)
In this paper, we employ the (BLP) measure [10] to
describe the non-Markovianity of the considered system.
Note that Markovian processes always tend to continu-
ously reduce the trace distance between any two states of
a quantum system, thus an increase of the trace distance
during any time interval implies the emergence of non-
Markovianity. BLP further linked the change of the trace
distance to the flow of information between the system
and its environment, and concluded that the backflow of
information from environment to the system is the key
feature of a non-Markovian dynamics. In quantum in-
formation science, the trace distance for quantum states
1

and 2

is defined as [33]

121 2
1
,
2
Dtr,
 
 (23)
with trace norm defined as
A
AA

. For a given pair
of initial states


1,2 0

of the system, the change of the
dynamical trace-distance can be described by its time
derivative
    
1, 212
d
,0 ,
d
tDt
t
  

,t (24)
where


1,2 t
 are the dynamical states of the system
with the initial states


1,2 0
 . For Markovian processes,
the monotonically reduction of the trace distance implies




1,2
,0t  0

for any initial states


1,2 0
 and at
any time . If there exists a pair of initial states of the
system such that for some evolutional timet,
t




1,2
,0t  0

, then the information takes backflow
from environment to the system, and the process is non-
Markovian. In order to describe the degrees of non-Mark-
ovianity of the whole dynamical process, the quantity,



1,2
1,2
0
0
maxd,0,tt




 (25)
is introduced. Where the time integration is extended
over all intervals in which

is positive, and the ma-
ximum is taken over all initial-state pairs of the system.
For any Markovian process, . The larger the
quantity is, the higher the non-Markovianity of the
process is.
0
For our considered open system and by use of the so-
lution of Equations (17)-(22), Equations (24) becomes









222 2
0
2
22
22
1e4
4
2e ,
t
t
xy z
Gt
bb b






 

 


(26)
where


 




12
2
22
22
ee
tt
xy z
Gtbbb
 



 and




12
0
jj j
bb b 0 is the difference between the two
Bloch components that correspond to the initial states


1,2 0


N. TANG ET AL.
30
4. Numerical Simulation
In order to demonstrate quantitatively the non-Markovian
characteristics of the system dynamics, we specify our
study to a particular reservoir spectra, Lorentzian spectra,


2
0
22
0
,
2π
J

 


(27)
which describes the interaction of an atom with an im-
perfect cavity and is widely used in literatures. Where
0

denotes the transition frequency of the atom,
0

  is the frequency detuning between the atom
and the cavity mode.

is the width of Lorentzian dis-
tribution, which is connected to the reservoir correla-
tion time 1
R



. The parameter 0

can be re-
garded as the decay rate for the excited atom in the
Markovian limit of flat spectrum which is related to
the relaxation time 1
0 S



. For the Lorentzian spec-
tra, all the time-dependent coefficients can be calcu-
lated analyticcally,


 


0
22
esin cos
2
t
Stt t


,













(28)
  

0
22
esin cos,
t
tt




t






 

(29)
with ,
 
0
1,1,1,1, 0,  


2
00

  
00c
 , and



.
In Figure 1, we plot the non-Markovianity as a func-
tion of the weight factor

and the dimensionless de-
Figure 1. (Color online) Non-Markovianity as a
function of the weight factor

and the dimensionless
detuning

0
 for 


0
50 and .


0
02
0
. Except for
the orange domain in which , all the other pa-
rameter domains have nonzero non-Markovianity.
tuning 0


, where the parameters are taken as
00
5



and 0 0.2



. It is shown that except for the
orange domain in which , all the other parameter
domains (i.e., the red and blue domains) have nonzero
non-Markovianity.
0
1


denotes the amplitude dam-
ping noise in which 0

for a small section of the
beginning part of 0


, and then rises with the
increase of the detuning, but the change is not monoto-
nous. The changing regulation is very similar to that of
the damped Jaynes-Cummings model case [10]. But here
the effect of the counter-rotating terms is considered
which induces some tiny oscillations of the non-Mark-
ovianity (see the dashed line in Figure 2) compared with
the damped Janes-Cummings model case. 0


de-
notes the purely dephasing noise in which increases
firstly from nonzero and then decreases when the detun-
ing increases. Interestingly, for the detunings up to
00.79


 which indeed covers all the reasonable de-
tunings of practical experiments, the non-Markovianity
induced by the purely dephasing noise is apparently
lager than the one induced by purely amplitude-damping
noise (see Figure 2), that is, in the same coupling strength,
dephasing is more preferable to induce the backflow of
information than dissipation. This is in accordance with
the prediction [34] that the main contribution to the
non-Markovianity stems from the evolution of the system
coherence, because dephase interaction may be more
preferable to induce the time evolution of the off-diago-
nal elements of the system density matrix than dissipa-
tion. Further we find that the superimposed interaction
of phase and amplitude damplings

can sup-
press the backflow of information, leading to smaller
non-Markovianity for measure . It means that in

10  
0 1 2 3 4
5
0
0.05
0.1
0.15
Δ/γ0
N
α=0
α=1
Figure 2. Non-Markovianity as a function of the di-
mensionless detuning

0

for amplitude and phasing
dampings for 


0
50 and .


0
02 . For the detunings
up to .5.


 
00
397 0 079

, the non-Markovianity
induced by the phase damping is lager than that in-
duced by amplitude damping.

N. TANG ET AL. 31
practice we may, not only by the engineering of envi-
ronmental structure but also by the adjusting of the cou-
pling ways between the system and its environment, ma-
nipulate the non-Markovian dynamics of open quantum
systems.
In Figure 3, we plot the measure as a function of
the weight factor

and the dimensionless decay rate
0

 , where the parameters is chosen as 025

  and
5

 . We find that for a given weight factor

, the
measure increases monotonously with decay rate
0

 . This may be understood easily: The parameter

is the inverse of the reservoir correlation time. The de-
crease of

indicates an increase of the reservoir cor-
relation time. The parameter 0

is the inverse of the
system relaxation time. Decreasing the system relaxation
time is equivalent to increasing the reservoir correlation
time. Both the two changes can enhance the non-Mark-
ovianity of the system dynamics [35]. Note that for
1

, the result is similar to the one of damped
Jaynes-cummings model with resonant interaction [36].
Further we also find that the dephase noise

0 


can
give rise to much more non-Markovianity than ampli-
tude-damping noise

, and their superposition
can reduce the value of the measure .
These results once again demonstrate the dominating
rules of the quantum coherence to the non-Marko-
vianity of open quantum system and the effect of de-
structive interference between amplitude and phase
dampings.

1  

0 

1
We also investigate the problem of what initial state
pairs can maximize the measure . Interestingly, in
our numerical simulations, we find that there exist only
two types of such initial state pairs. One pair is 0 and
1 which in Figures 1 and 3 applies to the red parame-
Figure 3. (Color online) Non-Markovianity as a func-
tion of the weight factor and the dimensionless decay
rate


0 for 


025 and 

5

.
ter domain, the other pair is

101
2
 

and

101
2


which applies to the blue parameter
domain. Both the initial state pairs have the largest trace
distance. For the orange parameter domain in Figure 1,
the measure 0

. Note that the initial state pair


0,1 agree with the result [10] for the damped
Jaynes-Cummings model, though here we have con-
sidered the effect of counter-rotating terms in the interac-
tion Hamiltonian. However, the initial state pair


,

 does not coincide with the result


,0
of reference [36]. The former is composed by two or-
thogonal states with maximal coherence. While the later
contains the ground state and a maximal coherence state
which have no the largest trace distance.
5. Conclusions
In conclusion, we have studied the non-Markovianity of
the dynamics for a two-level system interacting with a
zero-temperature structured environment via amplitude-
phase damping. In the limit of weak coupling between
the system and the reservoir, we have derived the time-
local non-Markovian master equation for the reduced
state of the system. The exact analytic solution under the
secular approximation has been gained and the non-
Markovian properties for the system dynamics based on
the BLP measure have been studied. We found, in the
same coupling strength between the system and its envi-
ronment, that the dephasing interaction is more beneficial
to induce the backflow of information from the environ-
ment to the system compared with amplitude-damping mo-
del, leading to more stronger non-Markovianity for the
system dynamics. This result further proves the previous
viewpoint that the main contribution to the non-Marko-
vianity stems from the evolution of the system coherence.
We also found that the information backflows induced by
amplitude and phase damplings can take destructive in-
terference so as to suppress the non-Markovianity of the
system dynamics.
For a given system-reservoir coupling model (a fixed

), the non-Markovian measure increases monotonously
with 0

 , but it is not monotonous with dimensionless
detuning 0


. The physical explanation for the former
case is clear, but for the later case the reason is not clear
yet and requires further studying.
In the definition of BLP non-Markovianity measure, a
maximization over all possible initial state pairs is in-
volved. This maximization process usually requires to
spend much time or effort on the practical calculations.
In our numerical simulations, we found that there are
only two pairs of such initial states. One pair was found
previously but another pair is new. For various different

N. TANG ET AL.
32
kinds of dynamical models, looking for the characteris-
tics of such initial state pairs is perhaps a problem worthy
of consideration [37,38].
The measure of non-Markovianity is a fundamental
problem in the study of open quantum system dynamics.
In the numerical simulations, we have only considered
the Lorentzian environment. Actually, our analytic re-
sults also adapt to other structured environments, such as
the Ohmic reservoir, the photonic band-gap material [39],
etc. By properly engineering the structure of the envi-
ronment, one can selectively alter the non-Markovian dy-
namical property of the open quantum system, so as to
effectively control the evolution of some interesting phy-
sical quantities, such as the quantum coherence, quan
tum entanglement and discord. Therefore, our work will
also be helpful for the researches of the related problems.
6. Acknowledgements
This work is supported by the National Natural Science
Foundation of China (Grant Nos. 11275064, 11075050),
Specialized Research Fund for the Doctoral Program of
Higher Education (Grant No. 20124306110003) the Cons-
truct Program of the National Key Discipline, the Pro-
gram for Changjiang Scholars and Innovative Research
Team in University under Grant No. IRT0964, and Hu-
nan Provincial Natural Science Foundation under Grant
No. 11JJ7001.
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