Lindblad rate equations

Lindblad rate equations

Adrián A. BudiniInstituto de Biocomputación y Física de Sistemas Complejos, Universidad de Zaragoza, Corona de Aragón 42, (50009) Zaragoza, Spain

and Consejo Nacional de Investigaciones Científicas y Técnicas, Centro Atómico Bariloche, Avenida E. Bustillo Km 9.5 (8400)Bariloche, Argentina*

�Received 16 July 2006; published 20 November 2006�

In this paper we derive an extra class of non-Markovian master equations where the system state is writtenas a sum of auxiliary matrixes whose evolution involve Lindblad contributions with local coupling between allof them, resembling the structure of a classical rate equation. The system dynamics may develop strongnonlocal effects such as the dependence of the stationary properties with the system initialization. Theseequations are derived from alternative microscopic interactions, such as complex environments described in ageneralized Born-Markov approximation and tripartite system-environment interactions, where extra unob-served degrees of freedom mediates the entanglement between the system and a Markovian reservoir. Condi-tions that guarantee the completely positive condition of the solution map are found. Quantum stochasticprocesses that recover the system dynamics in average are formulated. We exemplify our results by analyzingthe dynamical action of nontrivial structured dephasing and depolarizing reservoirs over a single qubit.

DOI: 10.1103/PhysRevA.74.053815 PACS number�s�: 42.50.Lc, 03.65.Ta, 03.65.Yz, 05.30.Ch

I. INTRODUCTION

The description of open quantum systems in terms oflocal-in-time evolutions is based on weak coupling and Mar-kovian approximations �1,2�. When these approximations arevalid, the dynamics can be written as a Lindblad equation�1–4�. The evolution of the density matrix �S�t� of the systemof interest then reads

d�S�t�dt

=− i

��Heff,�S�t�� − �D,�S�t��+ + F��S�t�� , �1�

where Heff is an effective Hamiltonian, �¯�+ denotes an an-ticonmutation operation, and

D =1

2��,�

a��V�†V�, F�·� = �

�,�a��V� · V�

† . �2�

Here, the sum indexes run from 1 to �dim HS�2, wheredim HS is the system Hilbert space dimension. The set �V��corresponds to a system operator base and a�� denotes asemipositive Hermitian matrix that characterize the dissipa-tive time scales of the system.

Outside the weak coupling and Markovian approxima-tions, it is not possible to establish a general formalism fordealing with non-Markovian system-environment interac-tions �5–12�. Nevertheless, an increasing interest in describ-ing open quantum system dynamics in terms of non-Markovian Lindblad equations exists �13–23�. Here, thedensity matrix �S�t� of the system evolves as

d�S�t�dt

=− i

��Heff,�S�t�� + �

0

t

d�K�t − ��L��S�� , �3�

where L�·�=−�D , · �++F�·� is a standard Lindblad superop-erator. The memory kernel K�t� is a function that may intro-

duces strong non-Markovian effects in the system decay dy-namics.

The study and characterization of this kind of dynamics istwofold: on the one hand, there is a general fundamentalinterest in the theory of open quantum systems to extend thewell-developed methods and concepts for Markovian dy-namics to the non-Markov case. On the other hand, there aremany new physical situations in which the Markov assump-tion usually used is not fulfilled and then non-Markoviandynamics has to be introduced. Remarkable examples aresingle fluorescent systems hosted in complex environments�24–28�, superconducting qubits �29,30�, and band gap ma-terials �31,32�.

Most of the recent analysis on non-Markovian Lindbladevolutions �13–19� focused on the possibility of obtaining anonphysical solution for �S�t� from Eq. �3�. This problemwas clarified in Refs. �14,15�, where mathematical con-straints on the kernel K�t� that guarantees the completelypositive condition �2–4� of the solution map �S�0�→�S�t�were found. Furthermore, in Ref. �15� the completely posi-tive condition was associated with the possibility of finding astochastic representation of the system dynamics.

There also exist different analysis that associate evolu-tions such as Eq. �3� with microscopic system environmentinteractions �19–22�. In Ref. �21� the microscopic Hamil-tonian involves extra stationary unobserved degrees of free-dom that modulate the dissipative coupling between the sys-tem of interest and a Markovian environment. This kind ofinteraction leads to a Lindblad equation characterized by a“random rate.” A similar situation was found in Ref. �22� byconsidering a complex environment whose action can be de-scribed in a “generalized Born-Markov approximation”�GBMA�. This approach relies on the possibility of splittingthe environment into a “direct sum” of subreservoirs, eachone being able to induce a Markovian system evolution byitself. When the system-environment interaction does notcouple the different subspaces associated to each subreser-voir, the system dynamics can also be written as a Lindbladequation with a random dissipative rate. After performing the*Present address.

PHYSICAL REVIEW A 74, 053815 �2006�

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http://dx.doi.org/10.1103/PhysRevA.74.053815

average over the random rate, the system dynamics can bewritten as a nonlocal evolution with a structure similar to Eq.�3�. In addition to the theoretical interest, the GBMA wasfound to be an useful tool for describing and modeling spe-cific physical situations, such as the fluorescence signal scat-tered by individual nanoscopic systems host in condensedphase environments �28�.

The aim of the present work is to go beyond previousresults �13–23�, and present an alternative kind of evolutionthat induces strong nonlocal effects, providing in this way anextra framework for studying and characterizing non-Markovian open quantum system dynamics. In the presentapproach, the system density matrix can be written as

�S�t� = �R

�R�t� , �4�

where the unnormalized states �R�t� have associated an ef-fective Hamiltonian HR

eff, and their full evolution is definedby

d

dt�R�t� =

− i

��HR

eff, �R�t�� − �DR, �R�t��+ + FR��R�t��

− �R�

R��R

�DR�R, �R�t��+ + �R�

R��R

FRR��R��t�� , �5�

subject to the initial conditions

�R�0� = PR�S�0� . �6�

The positive weights PR satisfy �RPR=1. On the other hand,the diagonal superoperator contributions are defined by

DR =1

2��,�

aR��V�

†V�, FR�·� = ��,�

aR��V� · V�

† , �7�

while the nondiagonal contributions reads

DR�R =1

2��,�

aR�R�� V�

†V�, FRR��·� = ��,�

aRR�� V� · V�

† . �8�

For convenience, we have introduced different notations forthe diagonal and nondiagonal terms. As in standard Lindbladequations �Eq. �1��, the matrixes aR

�� and aR�R�� characterize

the dissipative rate constants. The structure of the nondiago-nal terms in Eq. �5� resemble a classical rate equation �33�.Therefore, we call this kind of evolution a “Lindblad rateequation.”

Our main objective is to characterize this kind of equa-tions by finding different microscopic interactions that leadsto this structure. Furthermore, we find the conditions thatguarantees that the solution map �S�0�→�S�t� is a com-pletely positive one.

While the evolution of �S�t� can be written as a nonlocalevolution �see Eq. �61��, the structure Eq. �5� leads to a kindof non-Markovian effects where the stationary propertiesmay depend on the system initialization. In order to under-stand this unusual characteristic, as in Refs. �15,22�, we alsoexplore the possibility of finding a stochastic representationof the system dynamics.

We remark that specific evolutions such as Eq. �5� werederived previously in the literature in the context of differentapproaches �10,12,22�. The relation between those results isalso clarified in the present contribution.

The paper is organized as follows. Is Sec. II we derive theLindblad rate equations from a GBMA by considering inter-actions Hamiltonians that has contribution terms between thesubspaces associated to each subreservoir. An alternativederivation in terms of tripartite interactions allows one tofind the conditions under which the dynamic is completelypositive. A third derivation is given in terms of quantumstochastic processes. In Sec. III we characterize the resultingnon-Markovian master equation. By analyzing some simplenontrivial examples that admit stochastic reformulation, weexplain some nonstandard general properties of the non-Markovian dynamics. In Sec. IV we give the conclusions.

II. MICROSCOPIC DERIVATION

In this section we present three alternative situationswhere the system dynamics is described by a Lindblad rateequation.

A. Generalized Born-Markov approximation

The GBMA applies to complex environments whose ac-tion can be well described in terms of a direct sum of Mar-kovian subreservoirs �22�. This hypothesis implies that thetotal system-environment density matrix, in contrast with thestandard separable form �1,2�, assumes a classical correlatedstructure �4� �see Eq. �6� in Ref. �22��. In our previous analy-sis, we have assumed a system-environment interactionHamiltonian that does not have matrix elements between thesubspaces associated to each subreservoir. Therefore it as-sumes a direct sum structure �see Eq. �5� in Ref. �22��. Byraising up this condition, i.e., by taking in account arbitraryinteraction Hamiltonians without a direct sum structure, it ispossible to demonstrate that the GBMA leads to a Lindbladrate equation �5�.

As in the standard Born-Markov approximation, the deri-vation of the system evolution can be formalized in terms ofprojector techniques �11�. In fact, in Ref. �12� Breuer andcollaborators introduced a “correlated projector technique”intended to describe situations where the total system-environment density matrix does not assume an uncorrelatedstructure. Therefore, the system dynamics can be alterna-tively derived in the context of this equivalent approach. Themain advantage of this technique is that it provides a rigor-ous procedure for obtaining the dynamics to any desired or-der in the system-environment interaction strength �11,12�.Here, we assume that the system is weakly coupled to theenvironment. Therefore, we work out the system evolutionup to second order in the interaction strength.

We start by considering a full microscopic Hamiltoniandescription of the interaction of a system S with its environ-ment B

HT = HS + HB + HI. �9�

The contributions HS and HB correspond to the system andbath Hamiltonians, respectively. The term HI describes theirmutual interaction.

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The system density matrix follows after tracing out theenvironment degrees of freedom �S�t�=TrB��T�t��, where thetotal density matrix �T�t� evolves as

d�T�t�dt

=− i

��HT,�T�t�� LT��T�t�� . �10�

Now, we introduce the projector P defined by

P�T�t� = �R

�R�t� ��R

TrB��R�, �11�

where �R is given by

�R �R�B�R, �12�

with �B being the stationary state of the bath, while the sys-tem states �R�t� are defined by

�R�t� TrB��R�T�t��R� . �13�

We have introduced a set of projectors �R=��R��R��R,which provides an orthogonal decomposition of the unit op-erator �IB� in the Hilbert space of the bath, �R�R= IB, with�R�R�=�RR,R�. The full set of states �R� corresponds tothe base where �B is diagonal, which implies �R�R=�B.

It is easy to realize that P2=P. In physical terms, thisprojector takes in account that each bath-subspace associatedto the projectors �R induces a different system dynamics,each one represented by the states �R�t�. Each subspace canbe seen as a subreservoir. On the other hand, notice that thestandard projector P�T�t�=TrB��T�t�� B=�S�t� � �B �11� isrecuperated when all the states �R�t� have the same dynam-ics. Therefore, it is evident that the definition of the projector�11� implies the introduction of a generalized Born approxi-mation �22�, where instead of a uncorrelated form for thetotal system-environment density matrix, it is assumed aclassical correlated state.

By using that �R�R= IB, the system density matrix can bewritten as

�S�t� = �R

TrB��R�T�t��R�TrB��R�TrB��R�

�14a�

=TrB�P�T�t�� = �R

�R�t� . �14b�

This equation defines the system state as a sum over thestates �R�t�. Notice that the second line follows from thedefinition of the objects that define the projector �11�.

By writing the evolution Eq. �10� in an interaction repre-sentation with respect to HS+HB, and splitting the full dy-namics in the contributions P�T�t� and Q�T�t�, where Q=1−P, up to second order in the interaction Hamiltonian itfollows that �11�

dP�T�t�dt

= �0

t

dt�PLT�t�LT�t��P�T�t�� , �15�

where LT�t� is the total Liouville superoperator in a interac-tion representation. For writing the previous equation, wehave assumed Q�T�0�=0, which implies the absence of any

initial correlation between the system and the bath �T�0�=�S�0� � �B. Then, the initial condition of each state �R�t�can be written as

�R�0� = PR�S�0� . �16�

The parameters PR are defined by the weight of each subres-ervoir in the full stationary bath state

PR = TrB��R� = TrB��R�B� = ��R�

��R�B�R� , �17�

which trivially satisfies �RPR=1.Now, we split the interaction Hamiltonian as

HI = �R,R�

HIRR� �

R,R�

�RHI�R�. �18�

We notice that when �RHI�R�=0 for R�R�, the interactionHamiltonian can be written as a direct sum HI=HI1� HI2

¯ � HIR� HIR+1

¯, with HIR=�RHI�R. This case re-

cover the assumptions made in Ref. �22�. In fact, withoutconsidering the nondiagonal terms in Eq. �5� �aRR�

�� =0�, aftera trivial change of notation �R�t�→PR�R�t� in Eq. �4�, thedynamics reduce to a random Lindblad equation.

In order to proceed with the present derivation, we intro-duce the superoperator identity �34�

�a,�b, · �� =1

2��a, b�, · � +

1

2��a, b�+, · �+ − �a · b + b · a� ,

�19�

valid for arbitrary operators a and b. By using this identityand the splitting Eq. �18� into Eq. �15�, after a straightfor-ward calculation the evolution of �R�t� in the Schrödingerrepresentation can be written as in Eq. �5�. The effectiveHamiltonians read

HReff = HS − i

�

2�

0

d�TrBR��HI,HI�− ��BR

� . �20�

The nondiagonal operators DR�R read

DR�R =1

2�

0

d�TrBR��HIRR�

HIR�R�− �� + H.c.��BR

� , �21�

while the corresponding superoperators FRR� can be writtenas

FRR��·� = �0

d�TrBR�HIRR�

�− ��·� � �BR�HIR�R

+ H.c.� . �22�

The diagonal contributions follows from the previous expres-sions as DR=DRR, and FR�·�=FRR�·�. Furthermore, we havedefined TrBR

�·�TrB��R ·�R� and

�BR �R/PR. �23�

Notice that these objects correspond to the stationary state ofeach subreservoir.

In obtaining Eqs. �20�–�22� we have introduced a standardMarkovian approximation �1,2�, which allows one to obtainlocal in time evolutions for the set ��R�t��, as well as to

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extend the time integrals to infinite. This approximation ap-plies when the diagonal and nondiagonal correlations of thedifferent subreservoirs define the small time scale of theproblem. In order to clarify the introduction of the Markovapproximation, we assume that the interaction Hamiltoniancan be written as

HI = ��

V� � B�, �24�

where the operators V� and B� act on the system and bathHilbert spaces, respectively. By using HI=HI

†, the previousexpressions �21� and �22� read

DR�R =1

2��

0

d��R�R�� − ��V�

†V��− �� + H.c.� �25�

and

FRR��·� = ��

0

�d��RR�� − ��V��− ��·�V�

† + H.c.� .

�26�

Here, we have defined the “projected bath correlations”

�RR�� − �� TrBR�

��BR�B�

†�RB��− �� . �27�

Without taking in account the indexes R and R�, this expres-sion reduces to the standard definition of bath correlation�1–3,34�. Here, the same structure arises with projected ele-ments. As the integrals that appears in Eqs. �25� and �26�have the same structure that in the standard Born-Markovapproximation �34�, the meaning of the previous calculationsteps becomes clear.

Finally, in order to obtain the explicit expressions for thematrixes aRR�

�� and aR��, we define a matrix C��−�� from

V��− �� = e−i�HSV�e+i�HS = ��

C��− ��V�. �28�

By introducing these coefficients in Eqs. �25� and �26�, it ispossible to write the operators DR�R and FRR��·� as in Eq. �8�.The matrix aRR�

�� is defined by

aRR�� = �

��

0

d��RR�� − ��C��− ��

+ ��

0

d��RR�� *�− ��C��

* �− �� , �29�

while the diagonal matrix elements follows as aR��=aRR

��.Consistently, without taking in account the indexes R and R�,this matrix structure reduce to that of the standard Born-Markov approximation �34�.

Quantum master equation for a system influencing its en-vironment. In Ref. �10�, Esposito and Gaspard deduced aquantum master equation intended to describe physical situ-ations where the density of states of a reservoir is affected bythe changes of energy of an open system. While this physicalmotivation is different than that of the GBMA �22� �or ingeneral, than the correlated projector techniques �12��, herewe show that both formalisms can be deduced by using the

same calculations steps. Therefore, the evolution of Ref. �10�can also be written as a Lindblad rate equation.

In Ref. �10�, the system evolution is derived by taking inaccount the effect of the energy exchanges between the sys-tem and the environment and the conservation of energy bythe total �closed� system-reservoir dynamics. These condi-tions are preserved by tracing-out the bath coherences andmaintaining all the information with respect to the bathpopulations. Therefore, the system density matrix is writtenin terms of an auxiliary state that depends parametrically onthe energy of the environment, which is assumed in a micro-canonical state. By noting that in the GBMA there not existany coherence between the different sub-reservoirs �see Eq.�11��, we realize that the dynamics obtained in Ref. �10� canbe recovered with the previous results by associating the dis-crete index R with a continuos parameter , which label theeigenvalues of the reservoir, together with the replacements

�R�t� → �� ;t�, �R

→� d n� � , �30�

where n� � is the spectral density function of the reservoir.Consistently, the system state �Eq. �14�� is written as

�S�t� =� d n� �� ;t� � d �� ;t� . �31�

As in the GBMA, the evolution of �� ; t� can be written as aLindblad rate equation defined in terms of the matrix struc-ture �29� with the replacement �RR�

�� −��→� �� −��, where

� �� − �� = � �B�

† �� B� ��exp�− i� − �� . �32�

This last definition follows from the microcanonical state ofthe reservoir ��B→1�. Finally, by introducing the matrix el-ements

Pss�� ;t� �s�� ;t�s�� , �33�

where �s�� are the eigenstates of the system HamiltonianHSs�= ss�, the master equation of Ref. �10� is explicitlyrecovered. Due to the energy preservation condition, in gen-eral, the evolution involves a continuos parametric couplingbetween the matrix elements Pss�� ; t� and Pss�� ±� ; t�,where � is a energy scale that characterize the natural tran-sition frequencies of the system �10�.

We remark that the difference between both approachesrelies on the assumed properties of the environment. In thecontext of the GBMA, the index R labels a set of Hilbertsubspaces defined in terms of a manifold of bath eigenstatesable to induce, by itself, a Markovian system dynamics.Therefore, by hypothesis, the complete environment does notfeel the effects of the system energy changes. On the otherhand, the approach of Esposito and Gaspard applies to theopposite situation where, by hypothesis, the density of statesof the environment vary on a scale comparable to the systemenergy transitions. The stretched similarity between both ap-proaches follows from the absence of coherences betweenthe different �discrete or continuous� bath subspaces. In bothcases the system evolution can be written as a Lindblad rateequation.


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B. Composite environments

The previous analysis relies in a bipartite system-environment interaction described in a GBMA. Here, we ar-rive to a Lindblad rate equation by considering compositeenvironments, where extra degrees of freedom U modulatethe interaction �the entanglement� between a system S and aMarkovian reservoir B �21�. This formulation allows to findthe conditions under which Eq. �5� defines a completelypositive evolution.

The total Hamiltonian reads

HT = HS + HU + HSU + HB + HI. �34�

As before, HS represent the system Hamiltonian. Here, HB isthe Hamiltonian of the Markovian environment. On the otherhand, HU is the Hamiltonian of the extra degrees of freedomthat modulate the system-environment interaction. The inter-action Hamiltonian HI couples the three involved parts. Wealso consider the possibility of a direct interaction between Sand U, denoted by HSU.

As B is a Markovian reservoir, we can trace out its de-grees of freedom in a standard way �1–3�. Therefore, weassume the completely positive Lindblad evolution

d�C�t�dt

=− i

��HC,�C�t�� − �DC,�C�t��+ + FC��C�t�� , �35�

with the definitions

DC =1

2�i,j

bijAj†Ai, FC�·� = �

i,jbijAi · Aj

†. �36�

The matrix �C�t� corresponds to the state of the “composesystem” S-U with Hilbert space HC=HS � HU. The sum in-dexes i and j run from one to 1 to �dim HC�2, withdim HC=dim HS dim HU. Consistently, the set �Ai� is a baseof operators in HC, and bij is an arbitrary Hermitian semi-positive matrix.

In order to get the system state it is also necessary to traceout the degrees of freedom U. In fact, �S�t�=TrU��C�t��,which deliver

�S�t� = TrU��C�t�� = �R

�R�C�t�R� ,

�R

�R�t� . �37�

where �R�� is a base of vector states in HU. We notice thathere, the sum structure Eq. �4� have a trivial interpretation interms of a trace operation.

By assuming an uncorrelated initial condition �C�0�=�S�0� � �U�0�, where �S�0� and �U�0� are arbitrary initialstates for the systems S and U, from Eq. �37� it follows theinitial conditions �R�0�= PR�S�0�, where

PR = �R�U�0�R� . �38�

Therefore, here the weights PR corresponding to Eq. �6� aredefined by the diagonal matrix elements of the initial state ofthe system U. From now on, we will assume that the set ofstates �R�� correspond to the eigenvectors basis of HU, i.e.,

HUR� = RR� . �39�

The evolution of the states �R�t�= �R�C�t�R� can be ob-tained from Eq. �35� after tracing over system U. Under spe-cial symmetry conditions, the resulting evolution can be castin the form of a Lindblad rate equation �5�. In fact, in ageneral case, there will be extra contributions proportional tothe components �R�C�t�R��. By noting that

TrS��R�C�t�R�� = �R�U�t�R�� , �40�

where �U�t�=TrS��C�t�� is the density matrix of the degreesof freedom U, we realize that the evolution of �R�t� can bewritten as a Lindblad rate equation only when the evolutionof �U�t� does not involves coupling between the populations�R�U�t�R� and coherences �R�U�t�R��, R�R�, of systemU. As is well known �1–3�, this property is satisfied when thedissipative evolution of �U�t� can be written in terms of theeigenoperators Lu of the unitary dynamic, i.e., �HU ,Lu�=�uLu. In what follows, we show explicitly that this propertyis sufficient to obtain a Lindblad rate equation for the set ofmatrixes ��R�t��.

First, we notice that the Hamiltonian HC in Eq. �35� mustto have the structure

HC = HS + HU + ��

V� � L0�, �41�

where L0� are the eigenoperators with a null eigenvalue, i.e.,

�HU ,L0��=0. With this structure, the populations and coher-

ences corresponding to U do not couple between them.Therefore, the effective Hamiltonian HR

eff in Eq. �5� reads

HReff = HS + �

�

�RL0�R�V�. �42�

After taking the operator base in HC=HS � HU as

�Ai� → �V� � Lu� , �43�

the superoperators Eq. �36� can be written as

DC =1

2 ��,�

u,v

buv��V�

†Lv†V�Lu, �44a�

FC�·� = ��,�

u,v

buv��V�Lu · V�

†Lv†. �44b�

With these definitions, by taking the trace operation over thesystem U in the evolution Eq. �35�, we notice that the evo-lution of the set ��R�t�� can be cast in the form of a Lindbladrate equation if the conditions

�u,v

buv��R�Lv

†R��RLuR�� = R�,R�aRR�� 45�

are satisfied. The factor R�,R� guarantees that the evolutionof the set ��R�t�� do not involve the terms �R�C�t�R��, R�R�, and in turn implies that the populations and coherencesof U do not couple between them. On the other hand, aRR�

��

defines the matrix elements corresponding to the structure


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�5�. The diagonal contributions follows from Eq. �45� bytaking R=R�.

The set of conditions Eq. �45� can be simplified by takingthe base

Lu → R��R , �46�

which from Eq. �39� satisfy �HU ,Lu�= � R− R��Lu. Thus,Eq. �45� can be consistently satisfied if we impose

buv�� = 0 for u � v . �47�

After changing �u→�R,R� in Eq. �45�, we get

aRR�� = b�R,R��R,R��

�� , aR�� = b�R,R��R,R�

�� , �48�

where we have used that R and R� are dumb indexes. Thisresult demonstrate that the evolution induced by the compos-ite environment can in fact be written as a Lindblad rateevolution Eq. �5� with the matrix elements defined by Eq.�48�.

From our previous considerations we deduce that Lind-blad rate equation arise from microscopic tripartite interac-tions having the structure

HI = L0 � HSB + �u

Lu � HSBu + Lu

†� �HSB

u �†, �49�

where �HU ,L0�=0 and Lu→ R��R� with R�R�. On theother hand, HSB

u are arbitrary interaction terms between thesystem S and the Markovian environment B. In fact, thestructure �49� guarantees that the populations and coherencesof U do not couple between them, which in turn implies thatthe evolutions of the system S is given by a Lindblad rateequation.

Completely positive condition. We have presented two dif-ferent microscopic interactions that lead to a Lindblad rateequation. In order to use these equations as a valid tool formodeling open quantum system dynamics it is necessary toestablish the conditions under which the solution map�S�0�→�S�t� is a completely positive one. For an arbitraryLindblad rate equation this condition must to be defined interms of the matrixes aRR�

�� and aR��.

In order to find the allowed matrix structures, we noticethat the evolution Eq. �35� is a completely positive one whenbij→b�R,R��R,R��

�� is a semipositive defined matrix. Therefore,

by using Eq. �48� we arrive to the conditions

aRR�� 0, aR

�� 0, ∀ R,R�, �50�

i.e., for any value of R and R� both kind of matrixes must tobe semipositive defined in the system indexes �, �. The con-dition aR

�� 0 has a trivial interpretation. In fact, whenaRR�

�� =0, there not exist any dynamical coupling between thestates �R�t�. Thus, their evolutions are defined by a Lindbladstructure that under the constraint aR

��0 define a com-pletely positive evolution.

C. Quantum random walk

By using the similarity of Eq. �5� with a classical rateequation �33�, here we present a third derivation by con-

structing a stochastic dynamics that develops in the systemHilbert space and whose average evolution is given by aLindblad rate equation.

First, we assume that the system is endowed with a clas-sical internal degree of freedom characterized by a set �R� ofpossible states. The corresponding populations PR�t� obeythe classical evolution

dPR�t�dt

− �R�

R��R

�R�RPR�t� + �R�

R��R

�RR�PR��t� , �51�

with initial conditions PR�0�= PR, and where the coefficients��R�R� define the hopping rates between the different classi-cal states R.

To each state R we associate a different Markovian systemdynamics, whose evolution is generated by the superoperator

LR = LH + LR, �52�

with LH�·�= �−i /��HS , · � and a standard Lindblad contribu-tion LR�·�=−�DR , · �++FR�·�. Therefore, each state R definesa propagation channel with a different self-dynamic. The sys-tem state follows by tracing out any information about theinternal state. Thus, we write

�S�t� = �R

�R�t� , �53�

where each state �R�t� defines the system state given that theinternal degree of freedom is in the state R. Consistently, theinitial condition of the auxiliary states reads �R�0�= PR�S�0�.

Finally, we assume that in each transition R→R� of theinternal degree of freedom, it is applied a completely posi-tive superoperator ER �2–4�, which produces a disruptivetransformation in the system state.

The stochastic dynamics is completely defined after pro-

viding the self-channel dynamics, defined by �LR�, the set ofrates ��R�R� and the superoperators �ER�. By construction thisdynamics is completely positive. The explicit construction ofthe corresponding stochastic realizations, which develop inthe system Hilbert space, is as follows. When the system iseffectively in channel R, it is transferred to channel R� withrate �R�R. Therefore, the probability of staying in channel Rduring a sojourn interval t is given by

P0�R��t� = exp − t �

R�R��R

�R�R� . �54�

This function completely defines the statistics of the timeintervals between the successive disruptive events. As instandard classical rate equations, when the system “jumpoutside” of channel R, each subsequent channel R� is se-lected with probability


053815-6

tR�R =�R�R

�R�

R��R

�R�R

, �55�

in such a way that �R�tR�R=1. Furthermore, each transfer-ence R→R�, is attended by the application of the superop-erator ER, which produces the disruptive transformation�R�t�→ER��R�t��. This transformed state is the subsequentinitial condition for channel R�.

The average over realizations of the previous quantumstochastic process, for each state �R�t�, reads

�R�t� = P0�R��t�eR

tLR��0� + �0

t

d�P0�R��t − ��e�t−��LR

� �R�

R��R

�RR�ER��R�� . �56�

The structure of this equation has a clear interpretation. Thefirst contribution represents the realization where the systemremains in channel R without happening any scatteringevent. Clearly this term must be weighted by the probabilityof not having any event in the time interval �t ,0�, i.e., withthe probability P0

�R��t�. On the other hand, the terms insidethe integral correspond to the rest of the realizations. Theytake in account the contributions that come from any otherchannel R�, arriving at time � and surviving up to time t inchannel R. During this interval it is applied the self-channel

propagator exp��t−��LR�. As before, this evolution isweighted by the survival probability P0

�R��t−��.By working Eq. �56� in the Laplace domain, after a simple

calculation, it is possible at arrive at the evolution

d

dt�R�t� =

− i

��HS, �R�t�� − �DR, �R�t��+ + FR��R�t��

− �R�

R��R

�R�R�R�t� + �R�

R��R

�RR�ER��R��t�� .

�57�

We notice that this expression does not correspond to themore general structure of a Lindblad rate equation, �5�. Nev-ertheless, there exist different nontrivial situations that fall inthis category. As we demonstrate in the next section, theadvantage of this formulation is that it provides a simpleframework for understanding some nonusual characteristicsof the system dynamics.

III. NON-MARKOVIAN DYNAMICS

In this section we obtain the master equation that definethe evolution of the system state �S�t� associated to anarbitrary Lindblad rate equation �5�. In order to simplifythe notation, we define a column vector defined in R spaceand whose elements are the states �R, i.e., ��= ��1 , �2 , . . . , �R , . . . �T, where T denote a transposition opera-tion. Then, the evolution Eq. �5� can be written as

d��t�…dt

= LH��t�… + M��t�… , �58�

where LH�·�=−�i /��HS , · �, and the matrix elements of Mreads

MRR��·� = R,R��− i

��HR� , · � − �DR, · �+ + FR�·��

+ FRR��·� − R,R� �R�

R��R

�DR�R, · �+, �59�

where HR� =HReff−HS, is the shift Hamiltonian produced by

the interaction with the reservoir. The initial condition reads��0�)= P��S�0�, where we have introduced the vector P�= �P1 , P2 , . . . , PR , . . . �T. The system state Eq. �4� reads �S�t�= �1 ��t��, where 1� is the row vector with elements equal to1. Notice that due to the normalization of the statisticalweights it follows �1 P�=1.

From Eq. �58�, the system state can be trivially written inthe Laplace domain as

�S�u� = �11

u − �LH + M�P��S�0� , �60a�

�1G�u�P��S�0� , �60b�

where u is the conjugate variable. Multiplying the right term

by the identity operator written in the form 1/ �1G�u��u− �LH+M��P�, it is straightforward to arrive to the nonlocalevolution

d�S�t�dt

= LH��S�t�� + �0

t

d�L�t − ��S�� , �61�

where the superoperator L�t� is defined by the relation

�1G�u�MP��·� = �1G�u�P�L�u��·� . �62�

In general, depending on the underlying structure, the evolu-tion �61� involves many different memory kernels, each oneassociated to a Lindblad contribution.

We notice that a similar master equation was obtained inRefs. �21,22�. Nevertheless, here the dynamics may stronglydeparts with respect to the evolutions that arise from Lind-blad equations with a random rate �aRR�

�� =0�. In fact, the pre-vious calculation steps are valid only if

limu→0�1uG�u�P� = 0. �63�

By using that limt→f�t�=limu→0uf�u�, this condition is

equivalent to limt→�1G�t�P�=0. In the general case aRR��

�0, Eq. �63� is not always satisfied. In this situation, thedensity matrix evolution becomes nonhomogeneous and thestationary state may depends on the system initial condition.In general, this case may arises when the diagonal contribu-tions are null, i.e., aR

��=0 and aRR�� 0. We remark that these

matrix structures values are completely consistent with theconditions �50�. On the other hand, in the context of the


053815-7

GBMA, this case arise when the diagonal sub-bath correla-tions are null, �RR

��−��=0, which in turn implies that theinteraction Hamiltonian �18� satisfies �RHI�R�=0 if R=R�.

In order to characterize the dynamics when the condition�63� is not satisfied, we introduce the difference

�S�u� �S�u� −1

ulimu→0

�1uG�u�P��S�0� , �64a�

=�1G�u� −1

ulimu→0

uG�u�P��S�0� , �64b�

�1G�u�P��S�0� , �64c�

where now the pseudopropagator G�u� satisfies

limu→0�1uG�u�P�=0. Therefore, �S�t� satisfies an evolu-tion such as Eq. �61� where the kernel is defined by Eq. �62�with G�u�→G�u�. Notice that the system state, even inthe stationary regime, involves the contribution

limu→0�1uG�u�P��S�0�, that in fact depends on the systeminitial condition. In the next examples we show the meaningof this property, as well as its interpretation in the context ofthe stochastic approach.

A. Dephasing environment

Here we analyze the case of a qubit system interactingwith a dispersive reservoir �4,15� whose action can be writ-ten in terms of a dispersive Lindblad rate equation. We as-sume a complex reservoir with only two subspaces, R=a ,b,whose statistical weights �Eq. �17�� satisfy Pa+ Pb=1. Thus,the system state reads

�S�t� = �a�t� + �b�t� . �65�

A generalization to an arbitrary number of subreservoir isstraightforward.

The evolution of the auxiliary states are taken as

d

dt�a�t� = − �a��a�t� − �z�a�t��z� − �ba�a�t� + �ab�z�b�t��z,

�66a�

d

dt�b�t� = − �b��b�t� − �z�b�t��z� − �ab�b�t� + �ba�z�a�t��z,

�66b�

where �z is the z Pauli matrix. The completely positive con-ditions Eq. �50� imply

�a � 0, �b � 0, �67a�

�ab � 0, �ba � 0. �67b�

By denoting the matrix elements by �R=a ,b�

�R�t� = ��R+�t� �R

+�t��R

−�t� �R−�t�

� , �68�

the evolution corresponding to the populations read

d

dt�a

±�t� = − �ba�a±�t� + �ab�b

±�t� , �69a�

d

dt�b

±�t� = − �ab�b±�t� + �ba�a

±�t� , �69b�

with �R±�0�= PR�S

±�0�, while for the coherences we obtain

d

dt�a

±�t� = − ��a + �ba��a±�t� − �ab�b

±�t� , �70a�

d

dt�b

±�t� = − ��b + �ab��b±�t� − �ba�a

±�t� , �70b�

with �R±�0�= PR�S

±�0�. For expressing the initial conditionswe have extended trivially the notation Eq. �68� to the matrixelements of �S�t�. We notice that all coherences and popula-tions evolve independently each of the others. From the evo-lution of the populations �69� it follows that

d

dtTr��a�t�� = − �baTr��a�t�� + �abTr��b�t�� , �71a�

d

dtTr��b�t�� = − �abTr��b�t�� + �baTr��a�t�� , �71b�

with Tr��a�0��+Tr��b�0��= Pa+ Pb=1, which implies that thetrace of the auxiliary states perform a classical random walk.

From Eqs. �65� and �69� it becomes evident that the popu-lations of the system remain unchanged during all the evo-lution. On the other hand, the dynamic of the coherences canbe obtained straightforwardly in the Laplace domain. FromEq. �70� we get

�a±�u� = hab�u��S

±�0�, �b±�u� = hba�u��S

±�0� , �72�

where we have introduced the auxiliary function

hab�u� =�Pa − Pb��ab + Pa�u + �b�

�ba�u + �a� + �ab�u + �b� + �u + �a��u + �b�. �73�

Therefore, from Eq. �65�, the matrix elements of �S�t� read

�S±�t� = �S

±�0�, �S±�t� = h�t��S

±�0� , �74�

where h�t�=hab�t�+hba�t�, gives the coherences decay. Fromthese solutions, it is straightforward to obtain the correspond-ing system evolution

d�S�t�dt

= �0

t

d�K�t − ��L��S�� , �75�

with L�·�= �−· +�z ·�z� and K�u�= �1−uh�u�� /h�u�.In order to check the completely positive condition, we

write the solution map as

�S�t� = g+�t��0� + g−�t��z��0��z �76�

with g±�t�= �1±h�t�� /2. This mapping is completely positiveat all times if g±�t��0 �2–4�, and in turn implies the con-straint


053815-8

h�t� � 1. �77�

In the upper curve of Fig. 1 we plot the normalized coher-ences �S

±�t� /�S±�0�=h�t� for the case in which the non-

diagonal rates are null, �ab=�ba=0. Then, the dynamics re-duce to a superposition of exponential decays, each oneparticipating with weights Pa and Pb.

In the lower curve of Fig. 1 the nondiagonal rates arenon-null, while the rest of the parameters remain the same asin the upper curve. In contrast to the previous case, here thecoherence decay develops an oscillatory behavior that attainnegative values. Clearly, this regime is unreachable by a su-perposition of exponential decays. In both cases, the condi-tion �77� is satisfied, guaranteeing the physical validity of therespective solutions.

Stochastic representation. The evolution �66� admits astochastic interpretation such as that proposed previously.The stochastic trajectories can be simulated with the follow-ing algorithms. First, for being consistent with the initialcondition, the system initialization must be realized as fol-lows: �i� Generate a random number r� �0,1�. �ii� If r� Pa �r� Pa� the dynamic initialize in channel a �b� with�a�0�=�S�0� ��b�0�=�S�0��. Trivially, with this procedure thechannel a �b� is initialized with probability Pa �Pb�.

By comparing Eqs. �66� and �57�, the scattering superop-erator results E�·�=�z ·�z, which does not depends on thechannel �a and b�. It action over an arbitrary state �Eq. �68��is �R=a ,b�

E��R�t�� = �z�R�t��z = � �R+�t� − �R

+�t�− �R

−�t� �R−�t�

� . �78�

Therefore, its application implies a change of sign for thecoherence components. On the other hand, the self-dynamics

�52� of each channel is defined by La/b�·�=�a/b�−· +�z ·�z�.

With the previous information, the single trajectories canbe constructed with the following algorithm.

�1� Given that the system has arrived at time ti to channela, generate a random number r� �0,1� and solve for �ti+1

− ti� from the equation P0�a��ti+1− ti�=r, where P0

�a��t�=exp�−�bat�.

�2� For times satisfying t� �ti+1 , ti� the dynamics inchannel a is defined by its self-propagator �a�t�=exp��t− ti�La��a�ti�.

�3� At time ti+1 the system is transferred from channel a tob, implying the transformation �b�ti+1�→E��a�ti+1�� and theposterior resetting of channel a, defined by �a�ti+1�→0.

�4� Go to �1� with a↔b and i→ i+1.At this point, it is immediate to realize that the classical

rate equations �69� and �71� arise straightforwardly from the�transfer� jumps between both channels. The correspondingstationary traces read

Tr��a�� =�ab

�ab + �ba, Tr��b�� =

�ba

�ab + �ba, �79�

which do not depend on the system initial state.In contrast with the population evolution, some nonstand-

ard dynamical properties can be found in the coherences evo-lution when �a=�b=0. In Fig. 2 we show the normalizedcoherences �S

±�t� /�S±�0�=h�t� corresponding to this case. In

the inset, it is shown a typical stochastic realization of thecoherences of the auxiliary matrixes �a�t� and �b�t� obtainedwith the previous algorithm. As expected, in each applicationof E the coherences are transferred between both channelswith a change of sign. We also show an average over 500realizations. We checked that by increasing the number ofrealizations, the average behavior result indistinguishablewith the dynamics �74�.

In strong contrast with the previous figure, in Fig. 2 thestationary values of the coherences are “not null and depend

FIG. 1. Normalized coherences �S±�t� /�S

±�0�=h�t�, Eq. �74�. Inthe upper curve the parameters are �a=0.1,�b=1, and �ab=�ba=0.In the lower curve they are �a=0.1,�b=1,�ab=1, and �ba=0.1. Therates are expressed in arbitrary units �a.u.�. In both curves we takePa=0.1 and Pb=0.9.

FIG. 2. Normalized coherences �S±�t� /�S

±�0�=h�t�, Eq. �74�.The parameters are �a=�b=0,�ab=1,�ba=0.1, with the statisticalweights Pa=0.1 and Pb=0.9. The noisy curve correspond to anaverage over 500 realizations of the trajectories defined in the text.The inset show a particular realization for the coherences �a

±�t� and�b

±�t� of the auxiliary matrixes �a�t� and �b�t�, respectively.


053815-9

on the initial condition.” In fact, their normalized asymptoticvalue is limt→�S

±�t� /�S±�0��−0.654. This characteristic is

consistent with the breakdown of condition �63� and can beunderstood in terms of our previous analysis. By taking �a=�b=0 in Eq. �72� we get

�a±�u� =

Pa�u + �ab� − Pb�ab

u�u + �ab + �ba��S

±�0� , �80�

which implies the asymptotic value

limt→

�a±�t� = �Pa − Pb�

�ab

�ab + �ba�S

±�0� , �81a�

=�Pa − Pb�Tr��a��S±�0� . �81b�

This last expression can be easily interpreted in terms of therealizations of the proposed stochastic dynamics. From theinset of Fig. 2, it is clear that, in spite of a change of sign, thecoherence transferred between both channels does notchange along all the evolution. In fact, notice that due to theelection �a=�b=0, the self-propagators of both channels �seeprevious step �2�� are the identity operator. Therefore, allrealizations that begin in channel a �measured by Pa� that arefound in channel a in the stationary regime (measured byTr��a��), contribute to the stationary value of the coher-ence �a

±�t� with the value �S±�0�. This argument explain the

contribution proportional to PaTr��a��S±�0� in Eq. �81�.

On the other hand, a similar contribution is expected fromthe realizations that begin in channel b. Nevertheless, due tothe action of the superoperator E �Eq. �78�� they contributeswith the opposite sign.

By adding the contributions of both auxiliary matrixes,from Eq. �81� the stationary system coherences reads

limt→

�S±�t� = �Pa − Pb��ab − �ba

�ab + �ba��S

±�0� � 0. �82�

This expression fits the stationary value of Fig. 2.The stochastic realizations corresponding to the system

coherence �S±�t� can be trivially obtained from the the real-

izations of �a±�t� and �b

±�t�. By adding the upper and lowerrealizations in the inset of Fig. 2, we get a function thatfluctuates between the values ±�S

±�0�. By considering theinitial conditions and the superoperator action from theserealizations it is also possible to understand the four contri-bution terms of Eq. �82�. Finally, we remark that when any ofboth channels have a nontrivial self-dynamics, the coher-ences vanish in the stationary regime, losing any dependenceon the system initial condition �S�0� �See Fig. 1�.

B. Depolarizing reservoir

Another example that admits a stochastic representation isthe case of a depolarizing reservoir �4,15�, which is definedby the superoperator

E�·� = ��x · �x + �y · �y�/2, �83�

where �x and �y are the x and y Pauli matrixes, respectively.For simplifying the analysis we assume channels without

self-dynamics. Therefore, the evolution reads

d

dt�a�t� = − �ba�a�t� + �abE��b�t�� , �84a�

d

dt�b�t� = − �ab�b�t� + �baE��a�t�� . �84b�

The action of the superoperator E over the states �R�t� �Eq.�68�� is given by �R=a ,b�

E��R�t�� = ��R−�t� 0

0 �R+�t�

� . �85�

Therefore, its application destroy the coherences componentsand interchange the populations of the upper and lowerstates.

The populations of the auxiliary states evolve as

d

dt�a

+�t� = − �ba�a+�t� + �ab�b

−�t� , �86a�

d

dt�b

−�t� = − �ab�b−�t� + �ba�a

+�t� , �86b�

subject to the initials conditions �a+�0�= Pa�S

+�0� and�b

−�0�= Pb�S−�0�. The evolution of �b

+�t� and �a−�t� follows

after changing a↔b. Notice that this splitting of the popu-lation couplings follows from the superoperator action de-fined by Eq. �85�. On the other hand, the coherence evolu-tions read

d

dt�a

±�t� = − �ba�a±�t�,

d

dt�b

±�t� = − �ab�b±�t� . �87�

Therefore, in this case the stationary coherences are null.This fact also follows trivially from Eq. �85�. In contrast, thestationary populations reads

�a+�� = ��S

+�0�Pa + �S−�0�Pb�

�ab

�ab + �ba, �88a�

�b−�� = ��S

+�0�Pa + �S−�0�Pb�

�ba

�ab + �ba, �88b�

where �b+�� and �a

−�� follows after changing a↔b. Thisresult has an immediate interpretation in the context of thestochastic approach. In fact, the last fractional factors corre-spond to the “natural” stationary solutions of �86�. This so-lution is corrected by the terms in brackets, which in facttake in account the system initialization �notice that �a

+�0�+�b

−�0��1� and the transformations induced by the super-operator E Eq. �85�. Finally, the system stationary popula-tions �S

±��=�a±��+�b

±�� read


053815-10

�S±�� = �S

±�0�Pa�ab + Pb�ba

�ab + �ba+ �S

��0�Pa�ba + Pb�ab

�ab + �ba.

�89�

As in the previous case, the dependence of the stationarystate in the initial conditions is lost when the channels have aproper dissipative self-dynamics.

IV. SUMMARY AND CONCLUSIONS

We have presented a class of dynamical master equationsthat provide an alternative framework for the characteriza-tion of non-Markovian open quantum system dynamics. Inthis approach, the system state is written in terms of a set ofauxiliary matrixes whose evolutions involve Lindblad contri-butions with coupling between all of them, resembling thestructure of a classical rate equation.

We have derived the previous structure from different ap-proaches. In the context of the GBMA, a complex structuredreservoir is approximated in terms of a direct sum of Mar-kovian subreservoirs. Then, the Lindblad rate structure arisesby considering arbitrary interaction Hamiltonians that couplethe different subspaces associated to each subreservoir. Thematrix structures that define the system evolution are ex-pressed in terms of the projected bath correlations.

On the other hand, we have derived the same structurefrom composite environments, where the entanglement be-tween the system and a Markovian environment is modu-lated by extra unobserved degrees of freedom. The Lindbladrate structure arises straightforwardly when the tripartite in-

teraction Hamiltonian that involve the three parts does notcouple the coherences and populations of the extra degreesof freedom. This scheme also allows one to find the condi-tions under which an arbitrary Lindblad rate equation pro-vides a completely positive evolution.

Due to the apparent similarity of the evolution with aclassical rate equation, we have also formulated a quantumstochastic dynamics that on average is described by a Lind-blad rate equation. The stochastic dynamic consists of a setof transmission channels, each one endowed with a differentself-system evolution, and where the transitions betweenthem are attended by the application of a completely positivesuperoperator. This formalism allows one to understandsome amazing properties of the non-Markovian dynamics,such as the dependence of the stationary state in the initialconditions. This phenomenon arises from the interplay be-tween the initial channel occupations and the structure of thestochastic dynamics. We exemplified our results by analyz-ing the dynamical action of nontrivial complex dephasingand depolarizing reservoirs over a single qubit system. Inconclusion, we have presented a close formalism that definesan extra class of non-Markovian quantum processes that maybe of help for understanding different physical situationswhere the presence of nonlocal effects is relevant �24–32�.

ACKNOWLEDGMENTS

This work was partially supported by Secretaría de Estadode Universidades e Investigación, MCEyC, Spain. The au-thor also thanks financial support from CONICET, Argen-tine.

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Lindblad rate equations