Non-Markovian dynamics of a damped driven two-state system

Abstract

We study a driven two-state system interacting with a structured environment. We introduce the non-Markovian master equation ruling the system dynamics, and we derive its analytic solution for general reservoir spectra. We compare the non-Markovian dynamics of the Bloch vector for two classes of reservoir spectra: the Ohmic and the Lorentzian reservoir. Finally, we derive the analytic conditions for complete positivity with and without the secular approximation. Interestingly, the complete positivity conditions have a transparent physical interpretation in terms of the characteristic timescales of phase diffusion and relaxation processes. Comment: 12 pages, 5 figures, final version

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arXiv:1001.3564v2 [quant-ph] 14 Jun 2010
Non-Markovian dynamics of a damped driven two-state system
P. Haikka∗and S. Maniscalco†
Turku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, 20014 Turku, Finland
(Dated: June 15, 2010)
We study a driven two-state system interacting with a generic structured environment.
outline the derivation of the time-local microscopic non-Markovian master equation, in the limit of
weak coupling between the system and the reservoir, and we derive its analytic solution for general
reservoir spectra in the regime of validity of the secular approximation. We also consider the non-
Markovian master equation without the secular approximation and study the effect of nonsecular
terms on the system dynamics for two classes of reservoir spectra: the Ohmic and the Lorentzian
reservoir. Finally, we derive the analytic conditions for complete positivity of the dynamical map,
with and without the secular approximation. Interestingly, the complete positivity conditions have
a transparent physical interpretation in terms of the characteristic time-scales of phase diffusion and
relaxation processes.
We
PACS numbers: 03.65.Yz,03.65.Ta
I.INTRODUCTION
All quantum systems are open, i.e., they interact with
an environment. The interaction with the environment
leads to dissipation and decoherence due to a flow of en-
ergy and/or information from the system to the environ-
ment [1, 2]. The dynamics of dissipation and decoherence
in an open quantum system depends on the properties of
the environment, and therefore can be altered by modi-
fying characteristics of the environment such as its spec-
trum [3, 4].
A common example of environment is the quantized
electromagnetic field, typically modeled as an infinite
chain of non-interacting quantum harmonic oscillators.
The coupling of the quantum system to the environment
is described by the spectral density function. If the sys-
tem couples to all modes of the environment in an equal
way the spectrum of the reservoir is flat.
the spectral density function strongly varies with the fre-
quency of the environmental oscillators, the environment
is said to be structured. Structured environments arise
in many physical situations, e.g., in photonic band-gap
materials and lossy optical cavities [5, 6]. In these sys-
tems the reservoir memory effects induce a feedback of
information from the environment into the system. We
call these systems non-Markovian [7].
In this paper we study, to second order in perturba-
tion theory with respect to system-reservoir coupling, the
non-Markovian dynamics of a driven two-state system in
presence of a structured reservoir with a generic spec-
tral density. One of the first studies on the general dy-
namical properties of this model dates back to almost
twenty years ago, when Lewenstein and Mossberg stud-
ied a driven atom inside both an optical cavity with a
Lorentzian spectral density and a microwave cavity with
a step function spectral density [8]. Further studies on
If, instead,
∗pmehai@utu.fi; www.openq.fi
†sabrina.maniscalco@utu.fi; www.openq.fi
the laser-induced modification of the spontaneous decay
of an atom embedded in a structured reservoir are given
in Refs. [9–14].
We extend these results in several ways. First of all we
present the time-local non-Markovian master equation
for the dynamics and its solution, valid in the limit of
weak couling between the system and the environment.
The memory effects due to the reservoir structure are
contained in time-dependent decay rates. We show that,
for short initial times, the dependence of the decay rates
on the driving laser is more complicated than the one
presented in Ref. [8]. For these short initial times, in-
deed, the spontaneous decay of the atom can not only be
suppressed or enhanced, but also partly reversed, when
non-Markovian oscillations induced by reservoir memory
effects are present.
The importance of the driven two-state model is es-
pecially pronounced in quantum computation and quan-
tum technologies, where one or more driven qubits consti-
tute the basic building block of quantum logic gates [15].
Different implementations of qubits for quantum logic
gates are subjected to different types of environmental
noise, i.e., to different environmental spectra [3]. In this
study we focus on two examples of a structured reser-
voir, namely the Ohmic and the Lorentzian reservoirs.
Owing to the microscopic approach that we adopt in this
paper, we can make a comparison of the microscopic pro-
cesses underlying the system dynamics for the two differ-
ent reservoirs. This knowledge can aid the search for
physical set-ups that best retain quantum properties un-
der dissipative dynamics.
We study the system dynamics with and without the
widely used secular approximation, singling out its lim-
its of validity. The time-scale for nonsecular phenomena
ranges typically from small to intermediate in comparison
with the time-scale for relaxation processes and therefore
the secular approximation may not be consistent with
studies of non-Markovian dynamics. A recent article by
Cummings and Hu further elucidates the importance of
nonsecular studies on open quantum systems [16].

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Our investigation of the effects of nonsecular terms on
the dynamics of the driven two-state system brings to
light the existence of nonsecular oscillations in the pop-
ulation of the two-state system. These oscillations have
similar nature to those observed in the entanglement dy-
namics in Ref. [17]. Contrarily to the oscillations due
to non-Markovianity, nonsecular oscillations persist for
times much longer than the reservoir correlation time.
Moreover, our analysis shows that the nonsecular terms
affect also the asymptotic long time values of the Bloch
vector components, i.e., their impact exceeds the time-
scale of nonsecular oscillations.
Finally, an important result we present in the paper
is the analysis of conditions for complete positivity (CP)
of the system. All phenomenological or approximated
non-Markovian master equations may lead to unphysical
results for certain values of the parameters. In order to
guarantee the physicality of the solution of the master
equation, one needs to study the complete positivity of
the dynamical map. This is by no means an easy task.
Explicit conditions for CP have been up to now obtained
only for very simple systems [18, 19]. Here we study for
the first time the CP conditions for the non-Markovian
driven two-state system and show that they have a clear
physical interpretation in terms of the decay rates.
The paper is organized as follows. In Sec. II we intro-
duce the microscopic Hamiltonian model and the non-
Markovian nonsecular time-local master equation de-
scribing the dynamics of the driven two-level atom in a
generic structured reservoir. In Sec. III we derive the an-
alytic expressions of the non-Markovian time-dependent
decay rates for the special cases of a Lorentzian and an
Ohmic reservoir and we discuss the physical processes
characterizing the system dynamics. These results are
used to study the solutions of the optical Bloch equations
in Sec. IV in the secular and the non-secular regime. In
Sec. V we derive the necessary and sufficient conditions
for completely positivity. Finally Sec. VI summarizes
the results and presents conclusions.
II.THE MICROSCOPIC MODEL
We consider a two-level atom with Bohr frequency ωA
interacting with a driving laser of frequency ωL almost
resonant with the atomic transition, i.e., |∆| = |ωA−
ωL| ≪ ωA. The two-level atom is embedded in a zero-T
thermal bosonic reservoir modeled by an infinite chain
of quantum harmonic oscillators. In a frame rotating
with the laser frequency ωL the total Hamiltonian for
this system, in units of ¯ h, is given by
H = HS+ HE+ HI,(1)
where
HS=1
2(∆σz+ Ωσx),
?
k
(2)
HE=ωka†
kak,(3)
HI=
?
k
gke−iωLta†
kσ−+ g∗
keiωLtakσ+,(4)
are the free Hamiltonians of the system and the envi-
ronment and the interaction Hamiltonian, respectively,
σx,y,z are the Pauli operators, σ± the atomic inversion
operators and akthe annihilation operator of quanta in
the reservoir k-th mode.
The Rabi frequency Ω describes the strength of the inter-
action between the atom and the laser and it is taken to
be small compared to the atomic and laser frequencies,
Ω ≪ ωA,ωL. The interaction strength between the two-
level atom and the k-th mode of the reservoir is given
by gk. In the limit of a continuum of reservoir modes
?
function, characterizing the reservoir spectrum. In this
paper we focus on structured reservoirs, i.e., reservoirs
with a spectrum that varies sensibly with the environ-
mental oscillators frequency.
The description of a quantum system in a structured
reservoir requires non-Markovian approaches since the
reservoir correlation time is typically longer than other
time-scales of the system dynamics. In the following sub-
section we present the microscopic non-Markovian mas-
ter equation for the system introduced above. We will
see how useful information on the system dynamics can
be inferred already by looking at the form of the master
equation and in particular by studying the behavior of
the time-dependent decay rates appearing in the equa-
tions.
k|gk|2→?dωJ(ω), where J(ω) is the spectral density
A.Time-local master equation
We use the time-convolutionless (TCL) projection op-
erator technique to obtain the master equation for the
driven two-level atom starting from the microscopic
model of Eqs.(1)-(4) [1].
pling between the system and the environment the TCL
generator is expanded to second order with respect to a
coupling constant quantifying the strength of the inter-
action between the system and the environment. In Ref.
[20] one of us has demonstrated that the time-local non-
Markovian master equation describing the system under
study can be written in the form
In the limit of weak cou-
d¯ ρ(t)
dt
= −i[¯HS+¯HLS, ¯ ρ(t)] + D[¯ ρ(t)] + D′[¯ ρ(t)],(5)
where bars indicate that the operators are given in the
dressed state basis |ψ±? = ±?C±|e? +?C∓|g?, where
|e? and |g? are the atomic excited and ground state, and
the coefficients C±are
C±=∆ ± ω
2ω
,C0=
Ω
2ω,
(6)
with
ω =
?
∆2+ Ω2
(7)

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the energy separation between the eigenstates of the
driven atom. The unitary part of Eq. (5) is governed
by the Hamiltonians
¯HS=ω
2¯ σz,
¯HLS= λ−(t)C2
+ λ0(t)C2
(8)
−¯ σ−¯ σ++ λ+(t)C2
0¯ σ2
+¯ σ+¯ σ−
z,(9)
namely the free Hamiltonian and the Lamb shift Hamilto-
nian, respectively. The latter one describes a small shift
in the energy of the eigenstates of the two-level atom.
This term has no qualitative effect on the dynamics of
the system and therefore will be neglected in the follow-
ing.
The dissipator in Eq. (5) has been written as the sum of
two terms, D and D′. The first term is
?
?
?
D[¯ ρ(t)] = C2
+γ+(t)¯ σ−¯ ρ(t)¯ σ+−1
¯ σ+¯ ρ(t)¯ σ−−1
¯ σz¯ ρ(t)¯ σz−1
2{¯ σ+¯ σ−, ¯ ρ(t)}
?
?
+ C2
−γ−(t)
2{¯ σ−¯ σ+, ¯ ρ(t)}
+ C2
0γ0(t)
2{¯ σz¯ σz, ¯ ρ(t)}
?
. (10)
The second term has a more complicated form and con-
tains the contribution of the so-called nonsecular terms,
i.e., terms oscillating rapidly with respect to the dressed
atom characteristic time τS= ω−1,
D′[¯ ρ(t)]=
?γ−(t)
2
− iλ−(t)
?¯ σ+¯ ρ(t)¯ σ+− ¯ σ+¯ σ+¯ ρ(t)??
− iλ+(t)
?¯ σ−¯ ρ(t)¯ σ−− ¯ σ−¯ σ−¯ ρ(t)??
− iλ0(t)
?¯ σz¯ ρ(t)¯ σ+− ¯ σ+¯ σz¯ ρ(t)??
??
C−C0
?¯ σ+¯ ρ(t)¯ σz− ¯ σz¯ σ+¯ ρ(t)?
+ C+C−
+
?γ+(t)
2
??
C+C0
?¯ σ−¯ ρ(t)¯ σz− ¯ σz¯ σ−¯ ρ(t)?
+ C+C−
+
?γ0(t)
2
??
C−C0
?¯ σz¯ ρ(t)¯ σ−− ¯ σ−¯ σz¯ ρ(t)?
+ h.c.+ C+C0
(11)
where h.c. denotes Hermitian conjugation.
As for all time-local master equations,
Markovian effects are contained in the coefficients γξ(t)
and λξ(t), with ξ ∈ {−,0,+}, which arise from the real
and imaginary part of the reservoir correlation function,
respectively [20]. These coefficients read
the non-
γξ(t) = 2
?t
0
dτ
?
d˜ ωJ(˜ ω)cos[(ωL+ ξω − ˜ ω)τ], (12)
λξ(t) =
?t
0
dτ
?
d˜ ωJ(˜ ω)sin[(ωL+ ξω − ˜ ω)τ].(13)
For times longer than the reservoir correlation time τC
the decay rates attain their stationary Markovian values
γM
ξ
≡ limt→∞γξ(t) and λM
the first non-Markovian corrections on the dynamics
ξ
≡ limt→∞λξ(t). Therefore
of the driven two-level atom are visible only for small
initial times t ≃ O(τC).
We observe that the dynamics of the driven two-level
atom comprises of three different dynamical effects
occurring at three different respective time-scales. In-
deed, the dynamics of both dissipation and decoherence
occur at a time-scale of the order of the relaxation
time-scale τR, which is defined by the properties of
the reservoir.Nonsecular terms cause oscillations,
occurring over the typical time-scale of the system,
τS = ω−1= (∆2+ Ω2)−1/2, for the driven two-level
atom. Finally, as mentioned above, the non-Markovian
memory effects happen for times shorter or of the order
of the reservoir correlation time-scale τC.
B. The secular approximation
Conventionally the nonsecular terms, contained in the
dissipator D′, are neglected in the secular approxima-
tion when τS ≪ τR [1]. However, as one might expect,
a non-Markovian description of the short-time dynam-
ics is often incompatible with the secular approximation.
In order to investigate the effects of the nonsecular terms
on the non-Markovian dynamics, we focus instead on two
regimes identified by the mutual relationship between the
characteristic time-scale τSand the reservoir correlation
time-scale τC.
The first regime we call the secular regime, characterized
by the condition τS≪ τC. In this regime the nonsecular
terms are negligible even at short non-Markovian time
scales and we can make the secular approximation. The
second regime is the nonsecular regime, characterized by
the opposite condition, i.e., τC ≪ τS. In this case we
must retain the nonsecular terms to correctly describe
the non-Markovian dynamics.
When the secular approximation is valid, and the dissipa-
tor D′can be neglected, the coefficients λξ(t) appear only
in the Lamb-shift Hamiltonian of Eq. (9), and therefore
they describe a time-dependent renormalization of the
dressed atomic energy. Moreover, the master equation
is in time-dependent Lindblad form and the coefficients
γξ(t) are proportional to the decay rates associated to
transitions between the atomic dressed states described
by the operators ¯ σ+, ¯ σ− and ¯ σz. In this case, the dy-
namics of the system can be inferred from the time evo-
lution of the decay rates for different types of reservoirs
in terms of direct and reversed quantum jumps between
dressed states, as suggested by the Non-Markovian quan-
tum jump (NMQJ) method of Ref. [21, 22]. We will dis-
cuss this point further in Sec. III.
When the secular approximation is not valid, the mas-
ter equation is not in Lindblad-type form. Both γξ(t)
and λξ(t) appear now in the dissipator D′. In this case
it is not possible to extract from the master equation
the jump operators, describing transitions between the
dressed states, and the associated time-dependent decay

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4
rates. However, as we will see in Sec. IV, the nonsecular
terms give rise to interesting effects not only at interme-
diate times but also in the asymptotic long time regime.
III.TIME-DEPENDENT COEFFICIENTS AND
NMQJ INTERPRETATION
In the following we specify our study to two exemplary
reservoir spectra widely used in the literature, namely
the Lorentzian and the Ohmic spectra. Our aim is to
investigate how both the system dynamics and the va-
lidity of the secular approximation depend on the prop-
erties of the reservoir spectrum. We give analytic ex-
pression for the time-dependent coefficients and use the
NMQJ method to compare the microscopic dynamics of
the driven two-state system in the two exemplary reser-
voirs.
A.Lorentzian reservoir
As a first example we consider a Lorentzian spectral
density characterizing, e.g., one quantized mode of the
electromagnetic field inside a cavity,
JLor(ω) =α2
2π
λ2
(ω − ω0)2+ λ2, (14)
where ω0is the frequency of the mode supported by the
cavity and λ is the width of the distribution quantify-
ing leakage of photons through the cavity mirrors. The
reservoir correlation time is given by τC= λ−1. The cou-
pling constant α2has frequency dimensions; the limit of
weak coupling between the system and the environment,
assumed in this paper, is valid when α2is smaller than
the smallest relevant frequency in the system.
1.Time-dependent coefficients
The time-dependent coefficients for the two-level atom
in a Lorentzian reservoir can be calculated explicitly us-
ing Eqs. (12)-(13), and are given by
γξ(T) =
α2
2(1 + q2
ξ)
?1 − e−TcosqξT + e−TqξsinqξT?, (15)
λξ(T) =
α2
1 + q2
ξ
?−qξ+ e−TqξcosqξT + e−TsinqξT?,
(16)
where T = λt and qξ = s − ξp with ξ = {−,0,+}. We
introduce two important parameters
p =τC
τS
=ω
λ,
(17)
s =ω0− ωL
λ
.(18)
The former of these two parameters identifies the region
of validity of the secular approximation, more precisely
p ≫ 1 corresponds to the secular regime and p ≪ 1 cor-
responds to the nonsecular regime. The latter parameter
s = (ω0−ωL)/λ ≈ (ω0−ωA)/λ indicates how far detuned
is the peak of the Lorentzian spectrum from the atomic
and/or laser frequency in units of λ.
In Fig. 1 we plot, as an example, the dynamics of γξ(T)
for different values of the parameters p and s. We note
in passing that the coefficient γ0(t) does not depend on
p but only on s.A first look at the plots of Fig.
shows that for small values of p (upper row), i.e., in the
nonsecular regime, the time-dependent coefficients γ+(t),
γ−(t) and γ0(t) coincide. Indeed, a power series expan-
sion with respect to p shows that γ±(t) ≃ γ0(t) when
p ≪ 1. In this case, however, the master equation con-
tains the nonsecular dissipator of Eq. (11) and hence it
is not in time-dependent Lindblad form so we cannot de-
scribe the dynamics in terms of the NMQJ approach. We
will numerically investigate the dynamics of this regime
of parameters in Sec. IV. For intermediate and large val-
ues of p the three coefficients are clearly distinct, as can
be seen in the second and third row of Fig. 1. We now
focus on the secular regime p ≫ 1 described by the last
row in Fig. 1.
1
2.Non-Markovian Quantum Jumps
In the secular regime the nonunitary dynamics is de-
scribed only by the dissipator of Eq. (10). The decay
rates associated to transitions between the dressed states
are given by C2
±γ±(t) while the decay rate associated to
phase flips in the dressed state basis is given by C2
When the laser is resonant with the atomic transition,
i.e., ∆ = 0, then C2
Fig. 1 shows that, for all values of s, γ0(t) is the domi-
nant decay rate so the main contribution to the system
dynamics is given by phase flips in the dressed states.
A similar conclusion holds for ∆/Ω ≪ 1, since in this
case C+/C0 ≃ C−/C0 ≃ 1.
∆/Ω ≫ 1, one gets C+/C0 ≃ 2∆/Ω and C−/C0 ≃ 0.
In this case the dynamics is dominated by transitions be-
tween dressed states, in particular those described by the
jump operator ¯ σ+, occurring at a rate C2
We note also that for increasing values of s the station-
ary Markovian value of the time dependent coefficients
decreases, due to a smaller effective coupling with the
reservoir. Moreover, high values of s are characterized
by oscillatory behavior of all the three coefficients, inde-
pendently from the value of p. For large enough values of
s the decay rates take temporarily negative values. This
feature is typical of time-local non-Markovian open quan-
tum systems and generally occurs when the characteristic
frequency of the open quantum system, here ωA≃ ωL, is
detuned from the peak of the reservoir spectrum [23, 24].
Negative values of the decay rates are interpreted, in the
NMQJ formalism, as reversed jumps canceling the jumps
0γ0(t).
+= C2
−= C2
0= 1/4. In this case
On the other hand, for
+γ+(t) ≃ γ+(t).

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5
0246810
0.0
0.5
1.0
1.5
2.0
2.5
3.0
T
0246810
0.0
0.5
1.0
1.5
T
0246810
-0.1
0.0
0.1
0.2
0.3
T
p = 0.01
s = 0.1
p = 0.01
s = 1
p = 0.01
s = 10
02468 10
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Lorentzian time-dependent coefficients
T
0246810
0.0
0.5
1.0
1.5
2.0
2.5
3.0
T
0246810
-0.1
0.0
0.1
0.2
0.3
T
p = 1
s = 0.1
p = 1
s = 1
p = 1
s = 10
02468 10
0.0
0.5
1.0
1.5
2.0
2.5
3.0
T
0246810
0.0
0.5
1.0
1.5
T
02468 10
-0.1
0.0
0.1
0.2
0.3
T
0.000.020.040.060.080.10
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
p = 100
s = 0.1
0.000.02 0.040.060.080.10
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
0.000.020.040.060.080.10
-0.03
-0.02
-0.01
0.00
0.01
0.02
0.03
p = 100
s = 1
p = 100
s = 10
FIG. 1. (Colors online) Lorentzian time-dependent coefficients γ+(T)/α2(dot-dashed blue line), γ−(T)/α2(dashed red line)
and γ0(T)/α2(solid black line) for p = 0.01,1,100 and s = 0.1,1,10. The insets show very short time scale dynamics.
previously occurred in the same channel when the decay
rate was positive. So the presence of oscillations around
zero in the decay rates indicates non-Markovian dynam-
ics induced by the reservoir memory and describing the
feedback of information and/or energy from the reser-
voir into the system [7]. It is worth noticing that non-
Markovian oscillations occur in γ±(t) for all values of s,
in the secular regime p ≫ 1.
In the next subsection we present the analytic expressions
of the same coefficients for a different reservoir spectral
density, namely the Ohmic one, and study their time evo-
lution. Comparing the behavior of γξ(t) for the two types
of spectra we will see which features are common and
which ones vary significantly when changing the spec-
trum.
B.Ohmic reservoir
We now focus on the Ohmic spectral density with ex-
ponential cut-off function
JOhm(ω) = α2ωexp
?
−ω
ωC
?
,(19)
where ωCis the cut-off frequency and α is a dimensionless
coupling constant in the limit of weak coupling between
the system and the environment, i.e., α ≪ 1. The inverse
of the cut-off frequency is the Ohmic reservoir correlation
time τC= ω−1
C.
1.Time-dependent coefficients
The non-Markoviantime-dependent coefficients for the
two-level atom in an Ohmic reservoir take the form
2
1 + T2
+ qoe−qo(π − iCi¯ z + iCiz + Si¯ z + Siz)
λξ(T) =α2ωC
21 + T2
+ qoe−qo?
+ iSi¯ z + iSiz
??
where T = ωCt, z = qo(T +i), qo= sO+ξpO, and the pa-
rameters sOand pOcorrespond to the s and p parameters
introduced in the previous subsection, but now adapted
to the Ohmic reservoir spectral density, i.e., sO= ωL/ωC
and pO = ω/ωC. Moreover, in Eqs. (20)-(21), the bar
is used to denote complex conjugation, Ci and Si are the
cosine and the sine integrals and Chi and Shi are the
hyperbolic cosine and hyperbolic sine integrals, respec-
tively.
Similarly to the Lorentzian case of the previous subsec-
tion, pO ≫ 1 corresponds to the secular regime and
pO≪ 1 corresponds to the nonsecular regime. We note,
however, that, in the Ohmic case, differently from the
Lorentzian case, the parameters pOand sOare no longer
γξ(T) = α2ωC
?
?
T cos(qoT) + sin(qoT)
?
(20)
?
,
1
??
cos(qoT) + T sin(q0T) − 1 − T2?
2Chi(qo) + 2Shi(qo) − Ci¯ z − Ciz
,(21)

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6
02468
0.0
0.2
0.4
0.6
0.8
1.0
1.2
02468
10
5
0
5
10
02468
-10
-5
0
5
10
Ohmic time-dependent coefficients
T
T
T
pO = 0.01
sO = 1
pO = 1
sO = 100
pO = 100
sO = 10 000
x10-2
x10-4
0.000.100.200.30
-10
0
10x10-2
0.00 0.100.200.30
-10
0
10
x10-4
FIG. 2. (Colors online) Ohmic time-dependent coefficients
γ+(T)/α2(dot-dashed blue line), γ−(T)/α2(dashed red line)
and γ0(T)/α2(solid black line) for p = 0.01,1,5. The insets
show very short time scale dynamics.
independent. Our model of a driven two-level atom is
valid when the Rabi frequency Ω and the detuning be-
tween the atom and the laser |∆| = |ωA− ωL| are small
compared to the atomic frequency ωA. This imposes a
restriction on the relative values of pOand sO, in partic-
ular we must have
pO≪ sO.(22)
The Ohmic time-dependent coefficients for the secular,
nonsecular and intermediate regimes are shown in Fig.
2. As in the Lorentzian case, in the nonsecular regime,
the three decay rates coincide. In this case, therefore,
similar considerations as those done in Sec. IIIA apply.
Again, in the nonsecular and intermediate regimes, the
master equation is not in the Lindblad form, hence little
can be said about the dynamics from the behavior of the
decay rates only.
2.Non-Markovian Quantum Jumps
In the secular regime, p ≫ 1, the Ohmic coefficients
display oscillatory behavior, similarly to the Lorentzian
case of Fig. 1 (last row). For the Ohmic reservoir, how-
ever, all the three time-dependent coefficients are of sim-
ilar order of magnitude. For short times they coincide, as
one can see from the inset in Fig. 2, but as time passes
they start oscillating out of phase. As a consequence,
when the laser is resonant with the atomic transition,
i.e., ∆ = 0, both quantum jumps between dressed states
and phase flip jumps contribute to the dynamics, contrar-
ily to the Lorentzian case where the phase flips between
dressed states were dominant.
Summarizing, in this section we have explored the dif-
ferences in the dynamics due to different reservoir spec-
tra. We have seen that for both the Lorentzian and the
Ohmic spectral density, in the nonsecular regime, the
three coefficients γξ(t) appearing in the Lindblad-type
master equation have the same time dependency.
Nonetheless, different spectral distributions correspond-
ing to different physical environments do give rise to no-
ticeable differences. As an example, we have seen that,
in the secular regime and in the case of resonance ∆ = 0,
the type of quantum jumps occurring in the systems do
depend on the spectral properties: in the Lorentzian case
phase flips between different eigenstates dominate the dy-
namics while in the Ohmic case quantum jumps between
different dressed states also occur. Moreover, not all the
spectra are similarly compatible with the assumptions on
which our model rely. The Ohmic reservoir, for example,
imposes some limitations on the value of the physical pa-
rameters characterizing the dynamics.
IV. BLOCH VECTOR DYNAMICS
An alternative way to describe the dynamics of a
driven two-state system is by means of the Bloch vec-
tor R(t) whose components are defined as
Ri(t) = Tr[ρ(t)σi],(23)
with i = x,y,z. The equation describing the dynamics of
the Bloch vector, known as optical Bloch equation, can
be obtained straightforwardly from Eq. (5)
dR(t)
dt
= [D(t) + D′(t)]R(t) + d(t) + d′(t),(24)
with D(t)+D′(t) the damping matrix and d(t)+d′(t) the
drift vector, whose explicit forms are given in Appendix
A. Note that, also in this case, we separate the contri-
bution of the nonsecular terms, contained in the primed
quantities, from the contribution of the secular terms. As
we will see in Sec. V, the optical Bloch equations prove to
be particularly useful for studying the conditions under
which the dynamical map is completely positive. More-
over, they provide us with a clear physical picture of the

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7
012345
0.99998
0.99999
1.00000
The z-component of the Bloch vector
T
FIG. 3. Dynamics of the z-component of the Bloch vector as
a function of T = λt for p = 100 and s = 0.1. We have set
α2/ωA = 0.01 and Ω/ωA = 0.01.
dynamics in terms of dephasing and dissipation phenom-
ena, as described later in this section.
A. Secular regime
When p ≫ 1 the secular approximation is valid and
the dynamics of the z-component of the Bloch vector
Rz, in the dressed state basis, decouples from the x- and
y-components. In this case the non-Markovian optical
Bloch equations have a simple solution for any initial
state R(0) = (x0,y0,z0):
Rx(t) = exp[−Γ(t)](x0cosωt − y0sinωt),
Ry(t) = exp[−Γ(t)](y0cosωt + x0sinωt),
Rz(t) = e−Λ(t)
z0+
0
where
?t
0
?t
0
and the time dependent coefficients γξ(t), with ξ =
{+,0,−}, are given by Eq. (12).
It should be stressed that the solution of the non-
Markovian Bloch equation, given by Eqs. (25)-(27), is
valid for any form of the spectral density function J(ω)
and therefore it can be used to describe the dynamics of
a driven two-level system in any structured reservoir in
the secular regime and in weak coupling.
As an example of the dynamics, we plot in Fig. 3 the
time evolution of the z-component of the Bloch vector for
the initial state R(0) = (0,0,1) in the Lorentzian case.
For times of the order of the reservoir correlation time
the Markovian exponential decay of Rz is replaced by
rapid non-Markovian oscillations, occurring at the same
frequency of the oscillations of γ+(t) and γ−(t). Physi-
cally these oscillations correspond to a rapid exchange of
energy and information between the two-level atom and
the environment due to the reservoir memory.
(25)
(26)
?
?t
dseΛ(s)?C2
−γ−(s) − C2
+γ+(s)??
(27)
Γ(t) =1
2
ds?C2
ds?C2
+γ+(s) + C2
−γ−(s) + 4C2
0γ0(s)?,(28)
Λ(t) =
+γ+(s) + C2
−γ−(s)?, (29)
1.Markovian limit
For times longer than the reservoir correlation time
τC the time-dependent decay rates γξ(t) approach their
Markovian stationary values γM
Bloch equations reduces to the well known Markovian
one [25]
ξ
and the solution of the
Rx(t) = e−t/τD(x0cosωt − y0sinωt),
Ry(t) = e−t/τD(y0cosωt + x0sinωt),
Rz(t) = e−t/τR(z0− z∞) + z∞,
(30)
(31)
(32)
where
z∞=C2
−γM
C2
−− C2
−γM
+γM
+γM
+
−+ C2
+
(33)
is the z-component of the stationary Bloch vector R∞≡
(0,0,z∞) and γM
+are the Markovian stationary val-
ues of γ−(t) and γ+(t), respectively. In Eqs. (30)-(32),
the Markovian relaxation time τRand decoherence time
τDare
=1
2
τ−1
−, γM
τ−1
R
?C2
+γM
+γM
++ C2
−γM
−+ 4C2
0γM
0
?
(34)
D= C2
++ C2
−γM
−.(35)
When the driven two-state system interacts with a struc-
tured environment, the relaxation and decoherence rates
become time-dependent, the time dependency being de-
termined by the form of the reservoir spectrum.
The equations above show that, in the Markovian limit,
the well known relationship 2τR ≥ τD is satisfied. A
close inspection to Eqs. (28)-(29) shows that, even in
the non-Markovian case, the time dependent decay rates
always satisfy the relation 2Γ(t) ≥ Λ(t), if the time de-
pendent coefficients γξ(t) are positive at all times. We
have seen however, that the time dependent coefficients
may oscillate taking temporarily negative values (See
Figs. 1 and 2). In this case it is not a priori guaran-
teed that the inequality 2Γ(t) ≥ Λ(t) holds. Stated an-
other way, at certain time instants, the generalized time-
dependent relaxation and decoherence times may violate
the 2τR(t) ≥ τD(t) inequality. We will explore in detail
this issue in Sec. V.
It should be noted that, in the secular regime and in
the limit of weak coupling between the system and the
environment, the non-Markovian effects are small and
occur at a time-scale that is many orders of magnitudes
smaller than the relaxation time of the two-level system.
However, as we will see in the following subsection, in
the nonsecular regime, strong oscillations may charac-
terise the dynamics of the Bloch vector also at longer
time scales.
B.Nonsecular regime
When p ≪ 1 we cannot neglect the nonsecular terms in
the dynamics of the driven two-level system. Due to the

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8
050100150200 250
0.0
0.2
0.4
0.6
0.8
1.0
012345
0.88
0.92
0.96
1.00
T
0
246810 1214
0.5
0.6
0.7
0.8
0.9
1.0
T
012345
0.998
0.999
1.000
The z-component of the Bloch vector
(a)
(b)
x103
FIG. 4. The z-component of the Bloch vector in the Lorentzian case and in the nonsecular regime as a function of T = λt for
p = 0.01 and (a) s = 0.1, (b) s = 10. In each plot we have set α = 0.01ωA and Ω = 0.01ωA. The insets show the dynamics in
the short, non-Markovian time-scale.
0 50100150200250
0.0
0.2
0.4
0.6
0.8
02468 1012 14
- 0.2
0.0
0.2
0.4
T
T
x- and y-components of the Bloch vector
x
y
x
y
x103
FIG. 5. The x and y-components of the Bloch vector in the Lorentzian case and in the nonsecular regime as a function of
T = λt for p = 0.01 and (a) s = 0.1, (b) s = 10. In each plot we have set α = 0.01ωA and Ω = 0.01ωA.
presence of these terms the equation of motion for the
z-component of the Bloch vector does not decouple any-
more from the equations for the x- and y- components.
Therefore, we can no longer obtain an analytical solution
for the optical Bloch equations. Furthermore, the master
equation of the driven two-state system is no longer in
the time-dependent Lindblad form and the microscopic
description of the dynamics of the system in terms of the
NMQJ method is not straightforward.
A numerical study of the solution of the full non-
Markovian optical Bloch equations shows, however, sub-
stantial differences in the dynamics compared to the secu-
lar regime. In Fig. 4 we plot the time evolution of Rz(T)
for p = 0.01 when the reservoir spectrum is Lorentzian,
for two exemplary values of s. Interestingly, while for
small s Rz(T) decays monotonically, for large s strong
oscillations are present and last for long times. These
oscillations are due to the nonsecular terms and have to
be distinguished from the short-time non-Markovian os-
cillations. The latter ones, occurring at the correlation
time τC, are shown in the inset of Fig. 4 (b).
The behavior of Rz(T) can be traced back to the dynam-
ics of the time-dependent coefficients γξ(T). We recall
that, in the nonsecular regime, these three coefficients
coincide, i.e., γ±(T) = γ0(T) ≡ γ(T). As shown in Fig.
1 (first row), for p = 0.01 and s = 0.1, all three decay
rates are positive hence we do not expect short time non-
Markovian oscillations in the dynamics of Rz(T). The
initial quadratic decay of Fig. 4(a) is due to the fact that
γ(T), for T ≤ 1, is smaller then its Markovian stationary
value and consequently the decay of Rz(T) is slower than
the one predicted by the Markovian theory.
For p = 0.01 and s = 10, on the contrary, Fig. 1 shows
that γ(T) oscillates taking negative values. These oscil-
lations are responsible for the non monotonic behavior
of Rz(T) at short times, see inset of Fig. 4 (b). The
nonsecular oscillations occur on a time-scale comparable
to the relaxation time. Similar oscillations can be seen
in the dynamics of the x- and y-components of the Bloch
vector, see Fig. 5(b).
A second difference with the secular dynamics, well visi-
ble from Fig. 5, is that the stationary states of Rx(t) and
Ry(t) are now no longer zero, at contrast with the predic-
tion of the secular equations (25)-(26). In the bare state
basis this corresponds to a non-zero steady state value of
all three components of the Bloch vector, as one can see
from Eq. (A5). The nonzero stationary value of RB
effect known as vacuum-field dressed state pumping [8],
z, an

Page 9

9
has been observed experimentally in Ref. [26]. On the
other hand, the nonzero stationary value of RB
indicates a stationary value of the atomic dipole moment
different from zero, leading to substantial changes in the
resonance fluorescence spectrum [8].
xand RB
y
V.COMPLETE POSITIVITY
Theoretical descriptions of non-Markovian open quan-
tum systems are often based on a series of assump-
tions and approximations without which it would not
be possible to tackle the problem of the description of
the dynamics in simple analytic terms. In the case de-
scribed in the paper, e.g., the main assumptions and ap-
proximations are the factorized initial condition, i. e.,
ρ(t = 0) = ρS(t = 0) ⊗ ρR(t = 0), with ρS and ρR the
system and reservoir density matrices, respectively, the
weak coupling approximation and, in some cases, the sec-
ular approximation.
The non-Markovian master equation we have used in the
paper is not in Lindblad form, since even in the secu-
lar regime, the time-dependent coefficients γξ may tem-
porarily take negative values. Therefore, both positivity
and complete positivity (CP) of the dynamical map that,
for Markovian systems in Lindblad form, are automat-
ically guaranteed by the Lindblad-Gorini-Kossakowski-
Sudarshan theorem [27, 28], can here be violated indi-
cating that our solution no longer describe a physical
state of the system.
In this section we will present the first study of complete
positivity for the non-Markovian driven two-state model.
We will derive explicit conditions for CP, and therefore
positivity, of the dynamical map and we will see how
these conditions have a clear and important physical in-
terpretation. Once again, in the following subsections
we will distinguish between the secular and nonsecular
regimes.
A. Secular regime
Let us begin with the case in which the secular ap-
proximation is valid and we can neglect the nonsecular
damping matrix D′(t) and the nonsecular drift vector
d′(t) in Eq. (24). In this case the damping matrix is in
block diagonal form, see Eq. (A1). In the secular regime
we can directly use the CP conditions presented in Ref.
[29]. The details of the calculation are presented in Ap-
pendix B.
We find that the necessary and sufficient condition for CP
for the driven two-state system, in the secular regime and
in weak coupling, is given by
2Γ(t) ≥ Λ(t) ≥ 0. (36)
Note that the physical meaning of Eq. (36) is straight-
forward since the inverse of Γ(t) and Λ(t) are the non-
Markovian deoherence and relaxation times respectively.
Therefore the necessary and sufficient condition for CP,
in the secular approximation, is that the decoherence rate
is at each time instant twice as big as the relaxation rate.
In the Markovian limit the conditions of Eq. (36) reduces
to the well known condition
2τR≥ τD≥ 0. (37)
Recall from Section IV A that the Markovian condition
is always satisfied by the driven two-state system for any
spectral density. Interestingly, inserting Eqs. (28)-(29)
into Eq. (36) one sees that the necessary and sufficient
condition for CP is equivalent to
?t
0
dsγ0(s) ≥ 0. (38)
It is worth noting that the condition above does not de-
pend on ω =√∆2+ Ω2, as one can see from Eq. (12).
B. Nonsecular regime
The method used to study the CP condition in the
secular regime is no longer applicable in the nonsecular
regime, when the damping matrix is not in a block di-
agonal form anymore. We use a more general method,
based on the positivity of the Choi matrix, and make use
of the weak coupling limit [30]. Again the details of the
calculation are given in Appendix B.
We find that the necessary and sufficient condition for
CP for the driven two-state system, in the nonsecular
regime and in weak coupling, is given by
Λ(t) + 2Γ(t) ≥ 0. (39)
We note in passing that in the nonsecular regime Λ(t)
and Γ(t) cannot be interpreted anymore as decoher-
ence and relaxation non-Markovian rates, since the form
of the master equation is now much more complicated
and no simple analytical solution for the optical Bloch
equation can be found. However, recall that in the
nonsecular regime the three decay rates coincide, i.e.,
γ±(t) = γ0(t) ≡ γ(t). Therefore, we find that the CP
condition takes a form very similar to the one valid for
the secular regime, given by Eq. (38),
?t
0
dsγ(s) ≥ 0.(40)
VI.CONCLUSIONS
In this paper we have studied the non-Markovian
dynamics of a driven two-state system immersed in
a structured environment.
Markovian master equation and the optical Bloch
equations both with and without the secular approxima-
tion, and we have presented the solution in terms of the
We have derived the non-

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10
Bloch vector dynamics for general reservoir spectra.
We have compared the dissipative dynamics of the
driven two-state system for two different reservoirs,
namely the Lorentzian and the Ohmic reservoir, and
we have discovered that it is strongly influenced by the
spectral properties. For example, in the secular regime
and on resonance, in the Lorentzian case the dynamics
is dominated by phase jumps in the eigenstate basis,
while in the Ohmic case the dominant quantum jumps
describe transitions between the dressed states.
We have discovered the existence of strong and long
living nonsecular oscillations in all components of the
Bloch vector in some regions in the parameter space.
The nonsecular terms were also discovered to have a
significant effect on the stationary quantum state of our
system. An interesting open question we will consider
next is whether the nonsecular oscillations describe a
feedback of information/energy from the system into the
environment as measured, e.g., by the non-Markovianity
measure proposed in Ref. [7].
We have also studied the validity of the secular approxi-
mation and how it depends on the spectral properties. In
particular, our results show that in the Ohmic reservoir
the use of the secular approximation is more subtle than
in the Lorentzian case.That is, in the Ohmic case,
one cannot always perform the secular approximation
whenever pO ≪ 1, since the model is not valid for
pO>
∼sO, but instead both conditions have to be met
before the secular approximation can be applied.
Finally, we have investigated in detail the complete pos-
itivity condition in both the secular and the nonsecular
regime.We have discovered that this condition can
be traced back to the behavior of the time-dependent
coefficients appearing in the master equation and pro-
portional, in the secular regime, to the decay rates of the
system. Moreover, we have discovered that whenever the
system is in time-dependent Lindblad form, i.e., when
the secular approximation is valid, the CP condition
consists of an inequality linking the non-Markovian
decoherence and relaxation rates.
the non-Markovian generalization of the well known
condition 2τR≥ τD.
The dissipative driven two-state system is one of the
most fundamental models of the theory of open quantum
systems. Since most of our results are independent from
the specific form of the spectral distribution, they can
be straightforwardly applied to many different physical
contexts where non-Markovian approaches are necessary,
e.g., for implementations of quantum computing and
other quantum technologies.
Such inequality is
ACKNOWLEDGMENTS
This work was supported by the Emil Aaltonen
foundation, the Finnish Cultural foundation, and by the
Turku Collegium of Science and Medicine (S.M.). We
acknowledge stimulating discussions with J. Piilo.
Appendix A: Non-Markovian optical Bloch
equations
In section IV we introduced the optical Bloch equations
describing the dynamics of the driven two-state system.
More explicitly the damping matrix and the drift vector
in the optical Bloch equations of Eq. (24), expressed in
terms of the time-dependent decay rates, are as follows:
D(t) =


−1
2
?C2
+γ+(t) + C2
−γ−(t) + 4C2
ω
0
0γ0(t)?
−ω
−γ(t) + 4C2
0
0
0−1
2
?C2
+γ+(t) + C2
0γ0(t)?
−C2
−γ−(t) − C2
+γ+(t)

,
(A1)
D′(t) =


1
2C+C−[γ+(t) + γ−(t)]
C+C−[λ+(t) − λ−(t)] −1
C0[C++ C−]γ0(t)
C+C−[λ+(t) − λ−(t)]
2C+C−[γ+(t) + γ−(t)] 2C0[C+λ+(t) − C−λ−(t)]
2C0(C−− C+)λ0
C0[C−γ−(t) + C+γ+(t)]
0

,(A2)
d(t) =?0,0,C2
−γ−(t) − C2
+γ+(t)?,(A3)
d′(t) =
?
C0{C+[γ0(t) + γ+(t)] − C−[γ0(t) + γ−(t)]},2C0{C−[λ−(t) − λ0(t)] + C+[λ+(t) − λ0(t)]},0
?
. (A4)
Note that the optical Bloch equations are given in the
dressed state basis of the driven two-level atom. The
transformation between the Bloch vector in the dressed
state basis R(t) and the Bloch vector in the bare state

Page 11

11
basis RB(t) is given by
RB(t) =


cosθ 0 −sinθ
01
sinθ 0
0
cosθ

R(t), (A5)
where θ = arctan(Ω/∆), i.e., the change of basis amounts
to a rotation of the Bloch vector.
Appendix B: Complete positivity
In Section V we give the necessary and sufficient con-
ditions for the complete positivity of the dynamics of the
driven two-state system. Here we describe in more detail
the derivation of the CP conditions.
1. Secular regime
The necessary condition for CP, for the driven two-
state system, is given by the following two inequalities
[29]
Λ(t) ≥ 0,
2Γ(t) ≥ Λ(t).
(B1)
(B2)
The sufficient condition for CP is also given by two in-
equalities. The first one coincides with Eq. (B2) and the
second one can be expressed in the form
1 + ϕ(t)2− χ(t) − 2|ϕ(t) − χ(t)| − ψ(t)2≥ 0,
where we have introduced the following auxiliary func-
tions
(B3)
ϕ(t) = e−Λ(t)
χ(t) = e−2Γ(t)
ψ(t) = Rz(t) − e−Γ(t)z0,
(B4)
(B5)
(B6)
When the condition (B2) holds then the second sufficient
condition simplifies to
[1 − ϕ(t)]2+ χ(t) − ψ(t)2≥ 0.
In the Markovian limit, having in mind Eq. (33), one sees
that Eq. (B7) is equivalent to requiring that the station-
ary Bloch vector is contained inside the Bloch sphere.
In the non-Markovian time-scale we make use of the fact
that the master equation (5), and the corresponding op-
tical Bloch equation, are valid to second order in the
coupling constant α. Expanding Eqs. (B4)-(B6) with
respect to α Eq. (B3) becomes
(B7)
1 − 2Γ(t) + O(α4) ≥ 0. (B8)
For large values of t, Γ(t), given by Eq.
without bound. In the short non-Markovian time-scales
(28), grows
we are interested in, however, Eq. (B8) is always valid.
2.Nonsecular regime
In the nonsecular regime we use a method based on the
positivity of the Choi matrix [30]. For a two-level system
whose dynamics is given by a master equations with dis-
sipator D, the Choi matrix is computed as follows:
(1) Compute the auxiliary matrix L defined by
Lij=1
2Tr[σiD(σj)],(B9)
where i,l ∈ {0,1,2,3}, σ0is the identity matrix and we
number the Pauli matrices as σ1,2,3= σx,y,z, respectively.
(2) Define a second auxiliary matrix F as
F(t) = T exp
??t
0
dsL(s)
?
,(B10)
where T is the time-ordering operator.
(3) The Choi matrix is now defined as
Sab=1
4
3
?
i,j=0
FijTr[σjσaσiσb],(B11)
with Fij matrix elements of F(t). The dynamics of the
two-level system is completely positive if and only if the
Choi matrix S is positive semi-definite, i.e., its eigenval-
ues are positive.
Using the fact that our master equation is valid to second
order in perturbation theory with respect to α, we write
the matrix˜L(t) ≡?t
˜L =˜L0+ α2˜L2,
0dsL(s) as
(B12)
where˜L0is a matrix containing all elements of˜L inde-
pendent of α and α2˜L2 contains all elements of˜L pro-
portional to α2. Then
F(t) = exp
?˜L(t)
?
= exp
?˜L0(t)
??
I + α2˜L2(t)
?
+ O(α3).
(B13)
The eigenvalues of the Choi matrix, calculated neglecting
all the terms of order greater than the second in α, are
ǫ1,2= 0
ǫ3,4= 1 ±
?
1 − [Λ(t) + 2Γ(t)]. (B14)
The eigenvalues ǫ3,4 are real whenever Λ(t) + 2Γ(t) ≤
1. This condition is always satisfied in the short non-
Markovian time-scale when α ≪ 1. This ensures that
ǫ3≥ 0. The condition of non-negativity of the last eigen-
value, i.e., ǫ4≥ 0, is satisfied whenever Λ(t)+2Γ(t) ≥ 0.

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