Quantum master equation for a system influencing its environment

Quantum master equation for a system influencing its environment

Massimiliano Esposito and Pierre Gaspard

Center for Nonlinear Phenomena and Complex Systems, Universite´ Libre de Bruxelles, Code Postal 231, Campus Plaine, B-1050 Brussels, Belgium

共Received 16 June 2003; published 24 December 2003兲

A perturbative quantum master equation is derived for a system interacting with its environment, which is more general than the ones derived before. Our master equation takes into account the effect of the energy exchanges between the system and the environment and the conservation of energy in the finite total system. This master equation describes relaxation mechanisms in isolated nanoscopic quantum systems. In its most general form, this equation is non-Markovian and a Markovian version of it rules the long-time relaxation. We show that our equation reduces to the Redfield equation in the limit where the energy of the system does not affect the density of state of its environment. This master equation and the Redfield one are applied to a spin-environment model defined in terms of random matrices and compared with the solutions of the exact von Neumann equation. The comparison proves the necessity to allow energy exchange between the subsystem and the environment in order to correctly describe the relaxation in an isolated nanoscopic total system.

DOI: 10.1103/PhysRevE.68.066112 PACS number共s兲: 05.50.⫹q, 03.65.Yz

I. INTRODUCTION

Studying the dynamics of a simple system interacting with its environment is a very important problem in physics. The theoretical description of this problem started a long time ago.

In the context of classical mechanics several master equa-tions, such as the Boltzmann equation, the Chapman-Kolmogorov master equation, or the Fokker-Planck equation, were derived in order to describe the time evolution of the probability density of the system variables.

In the context of quantum mechanics, which interests us in this paper, the time evolution of a system interacting with its environment is described in terms of a reduced density matrix that is obtained by tracing out the degrees of freedom of the environment from the total共system plus environment兲 density matrix. In this way, the first quantum master equation was obtained by Pauli关1–3兴 in 1928. This equation is called the Pauli equation and describes the evolution of the popu-lations 共i.e., the diagonal elements of the density matrix兲 when the system is weakly perturbed by an additional term in its Hamiltonian. The transition rates between populations are given by the Fermi golden rule. In 1957, Redfield 关4兴 derived the so-called Redfield equation in the context of NMR for a system such as a spin interacting with its envi-ronment. This equation has been widely used and applied to many systems where the dynamics of the environment is faster than the dynamics of the system. This equation is Mar-kovian and has the defect of breaking the positivity on short time scales of the order of the environment correlation time for initial conditions near the border of the space of physi-cally admissible density matrices. Many similar master equa-tions for a system interacting with an environment have been derived since then starting from the von Neumann equation and making several assumptions共weak coupling limit, Mark-ovianity, separation of time scale between system and envi-ronment兲 关5–9兴. In 1976 Lindblad 关10兴 derived the most gen-eral quantum master equation which is Markovian and which preserves positivity. The Redfield equation has a Lindblad

form in the case of ␦-correlated environments. More re-cently, a non-Markovian Redfield equation has been obtained that preserves positivity and reduces to the Redfield equation in the Markovian limit关11兴. It has also been shown 关11,12兴 that the Markovian Redfield equation can preserve positivity if one applies a slippage of initial conditions that takes into account the non-Markovian effects on the early dynamics. Similar considerations have been proposed for different mas-ter equations 关13–15兴. As far as one considers the weak-coupling regime, all the master equations derived till now in the literature at second order of perturbation theory can be deduced from the non-Markovian Redfield equation.

The problem is that the non-Markovian Redfield equation as well as the other aforementioned master equations exist-ing in the literature are based on the fundamental assumption that the environment does not feel the effect of the system. This assumption seems realistic for macroscopic environ-ments but not in the case of nanoscopic isolated total systems in which the density of states of the environment can vary on an energy scale of the order of the system energy scale. Be-cause nanoscopic physics is experimentally progressing very fast, we expect that such effects will become important and measurable in future applications. Already, quantum dissipa-tion is being envisaged on the nanoscale for applicadissipa-tions such as spin dynamics in quantum dots 关16兴 or isomeriza-tions in atomic or molecular clusters in microcanonical sta-tistical ensembles 关17兴. Another possible application is the intramolecular energy relaxation in polyatomic molecules 关18兴.

The aim of the present paper is to systematically derive from the von Neumann equation a master equation which takes into account the fact that the energy of the total system 共system plus environment兲 is finite and constant and, there-fore, that the energy distribution of the environment is af-fected by energy exchanges with the system. The aforemen-tioned equations can be derived from our master equation, which thus appears to be very general.

non-Markovian Redfield equation from the von Neumann equa-tion, by performing a second-order perturbative expansion in the coupling parameter共under the assumption of weak cou-pling兲 for general environments. Thereafter, we consider the Markovian limit in both cases. We also show how, in this limit and neglecting the coupling between the populations and the quantum coherences, our master equation reduces to a simple equation of Pauli type for the total system, taking into account the modifications of the energy distribution of the environment due to the energy exchanges with the sys-tem. Finally, we compare our master equation to the Redfield equation and discuss how the Redfield equation can be seen as a particular case of our master equation. In Sec. III we apply our master equation and the Redfield equation to the case where the system is a two-level system interacting with a general environment. In Sec. IV we apply the master equa-tions to the case where the system is a two-level system interacting with a complex environment 共such as a classi-cally chaotic or many-body environment兲 that is modeled by random matrices from a Gaussian orthogonal ensemble, which we call Gaussian orthogonal random matrices 共GORM兲. In Sec. V, we compare the solutions of the non-Markovian and non-Markovian master equations to the exact so-lutions of the complete von Neumann equation in the case of our spin-GORM model. Conclusions are finally drawn in Sec. VI.

II. DERIVATION OF THE FUNDAMENTAL EQUATIONS The Hamiltonian of the total systems that we consider here is made of the sum of the system Hamiltonian HˆS and

the environment Hamiltonian HˆB plus a coupling term that

has the form of the product of a system operator Sˆ and a environment operator Bˆ . The generalization to a coupling term of the form␭兺iSˆiBˆiis easy. The amplitude of the

cou-pling term is determined by the coucou-pling parameter␭:

Hˆ_tot⫽Hˆ₀⫹␭Vˆ⫽Hˆ_S⫹Hˆ_B⫹␭SˆBˆ. 共1兲

The eigenstates of HˆS, respectively of HˆB, will be denoted

by兩s

典

, respectively兩b

典

. The eigenvalues of HˆS, respectively

of HˆB, will be denoted by Es, respectively Eb. Finally, the

eigenstates of Hˆ_totwill be denoted by兩␣典 and its eigenvalues by E_␣.

The evolution of the total density matrix is described by the von Neumann equation:

␳ˆ˙共t兲⫽⫺i关Hˆtot,␳ˆ共t兲兴⬅Ltot␳ˆ共t兲, 共2兲

whereLtotis the so-called quantum Liouvillian or von

Neu-mann operator of the total system. The interaction represen-tations of the operators are given by

␳ˆI共t兲⫽eiH ˆ

0t␳_ˆ共t兲e⫺iHˆ0t_,

Vˆ共t兲⫽eiHˆ₀t_{Vˆ e}⫺iHˆ₀t_, Bˆ共t兲⫽eiHˆBt_Bˆ e⫺iHˆBt_,

Sˆ共t兲⫽eiHˆSt_Sˆe⫺iHˆSt_. 共3兲

In the interaction representation, the von Neumann equation becomes

␳ˆ˙I共t兲⫽⫺i关␭Vˆ共t兲,␳ˆI共t兲兴⬅LI共t兲␳ˆI共t兲, 共4兲

with the interaction Liouvillian L_I(t)⫽e⫺L0tL

IeL0t where eL0t_Aˆ⫽e⫺iHˆ0t_Aˆ eiHˆ0t_{, the free Liouvillian} L

0⫽LS⫹LB⫽

⫺i关HˆS,•兴⫺i关HˆB,•兴, and Aˆ is an arbitrary operator. The

perturbative expression of the von Neumann equation in the interaction representation is given to order ␭2 by

␳ˆI共t兲⫽␳ˆ共0兲⫹

冕

0 t dt1LI共t1兲␳ˆ共0兲 ⫹

冕

0 t dt1

冕

0 t₁ dt2LI共t1兲LI共t2兲␳ˆ共0兲⫹O共␭3兲 共5兲 ⫽␳ˆ共0兲⫹

冕

0 t dTe⫺L0TL IeL0T␳ˆ共0兲 ⫹

冕

0 t dT

冕

0 T d␶e⫺L0TL IeL0␶LIe⫺L0␶eL0T␳ˆ共0兲 ⫹O共␭3_{兲, 共6兲}

if we set T⫽t1 and␶⫽t1⫺t2. Equation 共6兲 is the starting

point of all the derivations of a master equation in the weak coupling limit for a total system made of a system and its environment in mutual interaction.

A. Our quantum master equation

We now derive our master equation which is the central result of this paper. The main idea is to describe the time evolution in terms of quantities which are distributed over the energy of the environment. We thus define the following quantities in terms of which we intend to describe the prop-erties of the system:

Pss⬘共⑀;t兲⬅Tr␳ˆ共t兲兩s

⬘典具

s兩␦共⑀⫺HˆB兲. 共7兲

The diagonal element Pss(⑀;t) is the probability density to

find the system in the state s while the environment has the energy ⑀. The off-diagonal element Pss⬘(⑀;t) characterizes

the density of the quantum coherence between the states s and s

⬘

, density which is distributed over the energy⑀of the environment.

The matrix composed of the elements Pss⬘(⑀;t) is

Her-mitian

P_ss⬘共⑀;t兲⫽P_s ⬘s

Moreover, the normalization Tr␳ˆ (t)⫽1 of the total density

matrix implies that

兺

s

冕

d⑀Pss共⑀;t兲⫽1. 共9兲

In order to obtain a closed description in terms of the quantities共7兲, we suppose that the total density matrix can be described at all times by a density matrix of the following form: ␳ˆ共t兲⫽

兺

s,s⬘ 兩s

典具

s

⬘

兩Pss⬘共HˆB;t兲 n共HˆB兲 , 共10兲

where we have defined the energy density

n共⑀兲⫽TrB␦共⑀⫺HˆB兲, 共11兲

which is supposed to be smoothened on the energy scale of the mean level spacing. The assumption共10兲 has the effect of neglecting the contributions from the environment coher-ences to the system dynamics共albeit the system coherences are kept in the description兲. We remark that the form 共10兲 is not supposed to strictly hold at all times but is an assumption in order to obtain a closed set of equations for the quantities

Pss⬘(⑀;t).

In order to better understand the meaning of the above definitions, we notice that the reduced density matrix of the system takes the form

␳ˆ_S共t兲⫽Tr_B␳ˆ共t兲⫽

冕

d⑀Tr_B␦共⑀⫺Hˆ_B兲␳ˆ共t兲

⫽

冕

d⑀

兺

s,s⬘

兩s

典

Pss⬘共⑀;t兲

具

s

⬘

兩, 共12兲

which can be represented in the basis of the eigenstates of the system Hamiltonian as

␳ˆS共t兲⫽

冕

d⑀

冉

P₁₁共⑀;t兲 P₁₂共⑀;t兲 . . . P1N_S共⑀;t兲 P21共⑀;t兲 P22共⑀;t兲 . . . P2N_S共⑀;t兲 ⯗ ⯗  ⯗ PN_S1共⑀;t兲 PN_S2共⑀;t兲 . . . PN_SN_S共⑀;t兲

冊

. 共13兲 The goal of the precedent choice for the form of␳ˆ (t) is thus

to obtain a description in which the state s of the system is correlated with the energy ⑀ of the environment. In other words, the density matrix␳ˆ_Sof the system is decomposed as a distribution over the energy⑀of the environment.

We now proceed to the derivation of the equations of motion for our quantities Pss⬘(⑀;t) in the weak-coupling

limit. We start from the perturbative expansion 共6兲 of the total density matrix in the interaction representation共3兲. We first define the interaction representation of our quantities 共7兲:

PIss⬘共⑀;t兲⫽ei(Es⫺Es⬘)tPss⬘共⑀;t兲. 共14兲

We now have that

PIss⬘共⑀;t兲⫽Tr␳ˆI共t兲兩s

⬘典具

s兩␦共⑀⫺HˆB兲. 共15兲

Inserting the perturbative expansion 共6兲, we get

PIss⬘共⑀;t兲⫽Tr Xˆ␳ˆ共0兲⫹

冕

0 t dT Tr Xˆ e⫺L0TL IeL0T␳ˆ共0兲 ⫹

冕

0 t dT

冕

0 T d␶Tr Xˆ e⫺L0T ⫻LIeL0␶LIe⫺L0␶eL0T␳ˆ共0兲⫹O共␭3兲, 共16兲

where Xˆ⫽兩s

⬘典具

s兩␦(⑀⫺HˆB). Differentiating with respect to

time, we obtain the equation

P˙Iss⬘共⑀;t兲⫽Tr Xˆe⫺L0tLIeL0t␳ˆ共0兲 ⫹

冕

0 t d␶Tr Xˆ e⫺L0tL IeL0␶LIe⫺L0␶eL0t␳ˆ共0兲 ⫹O共␭3_{兲, 共17兲}

where the initial density matrix takes the assumed form共10兲 with t⫽0.

The first term is thus explicitly given by

Tr Xˆ e⫺L0tL

IeL0t␳ˆ共0兲⫽⫺i␭ Tr兩s

⬘典具

s兩␦共⑀⫺HˆB兲eiH ˆ

0t ⫻关Vˆ,e⫺iHˆ0t␳_ˆ共0兲eiHˆ0t兴e⫺iHˆ0t ⫽⫺i␭

兺

s ¯ ei(E_s⫺E_¯s)t

_具

_{s兩Sˆ兩s¯}

_典

⫻P¯ss⬘共⑀;0兲n共⑀兲

具

Bˆ

典

⑀ ⫹i␭

兺

s ¯ e⫺i(Es⬘⫺E¯s)t

具

_¯s兩Sˆ兩s

⬘典

⫻Pss¯共⑀;0兲n共⑀兲

具

Bˆ

典

⑀, 共18兲

with the environment coupling operator Bˆ averaged over the microcanonical state of the environment

具

Bˆ

典

_⑀⬅Tr␦共⑀⫺HˆB兲Bˆ

n共⑀兲 . 共19兲

We now assume that this average vanishes,

具

B

典

_⑀⫽0.

Oth-erwise, the nonvanishing average is absorbed in the system Hamiltonian by the following substitutions:

HˆS→HˆS⫹␭

具

Bˆ

典

⑀Sˆ, 共20兲

Vˆ→Vˆ⫺

具

Bˆ

典

_⑀Sˆ, 共21兲

leaving unchanged the total Hamiltonian. Thanks to this sim-plification, the first-order term of the perturbative expansion vanishes

Tr Xˆ e⫺L0tL

IeL0t␳ˆ共0兲⫽0. 共23兲

As a consequence, the time evolution of the total density matrix is given by the uncoupled Hamiltonian Hˆ0⫽HˆS

⫹HˆB up to correction of the order of␭2:

␳ˆ共t兲⫽eL0t␳_ˆ共0兲⫹O共␭2兲. 共24兲 According to our closure assumption that the total density matrix keeps the form 共10兲 during its time evolution, we have that eL0t␳_ˆ共0兲⫽␳_ˆ共t兲⫹O共␭2兲⫽

兺

s ¯,s¯⬘ 兩s¯

典具

¯s

⬘

兩P¯ss¯⬘共HˆB;t兲 n共HˆB兲 ⫹O共␭2_兲, 共25兲

which we can substitute in Eq. 共17兲 to get

P˙Iss⬘共⑀;t兲⫽

冕

0 t d␶Tr Xˆ e⫺L0tL IeL0␶LIe⫺L0␶eL0t␳ˆ共0兲 ⫹O共␭3_兲 ⫽⫺␭2_ei(E_s⫺E_s⬘)t

兺

s ¯,s¯⬘

冕

0 t d␶Tr

再

兩s

⬘典具

s兩␦共⑀⫺HˆB兲 ⫻

冋

SˆBˆ ,

冋

Sˆ共⫺␶兲Bˆ共⫺␶兲,兩s¯

典具

¯s

⬘

兩P¯ss¯⬘共HˆB;t兲 n共HˆB兲

册册冎

⫹O共␭3_{兲. 共26兲}

Going back to the original representation with Eq. 共14兲 and expanding the two commutators, we obtain

P˙ss⬘共⑀;t兲⫽⫺i共Es⫺Es⬘兲Pss⬘共⑀;t兲⫺␭2

兺

s ¯,s¯⬘

冕

0 t d␶

再

具

s兩SˆSˆ共⫺␶兲兩s¯

典具

¯s

⬘

兩s

⬘典

TrB␦共⑀⫺HˆB兲BˆBˆ共⫺␶兲 P¯ss¯⬘共HˆB;t兲 n共HˆB兲 ⫺

具

s兩Sˆ兩s¯

典具

¯s

⬘

兩Sˆ共⫺␶兲兩s

⬘典

TrB␦共⑀⫺HˆB兲Bˆ P¯ss¯⬘共HˆB;t兲 n共HˆB兲 Bˆ共⫺␶兲⫺

具

s兩Sˆ共⫺␶兲兩s¯

典具

¯s

⬘

兩Sˆ兩s

⬘典

⫻TrB␦共⑀⫺HˆB兲Bˆ共⫺␶兲 P¯ss¯⬘共HˆB;t兲 n共HˆB兲 Bˆ⫹

具

s兩s¯

典具

¯s

⬘

兩Sˆ共⫺␶兲Sˆ兩s

⬘典

TrB␦共⑀⫺HˆB兲 P¯ss¯⬘共HˆB;t兲 n共HˆB兲 Bˆ共⫺␶兲Bˆ

冎

⫹O共␭3兲. 共27兲

In order to evaluate the four last terms, we notice that, for a quasicontinuous energy spectrum, we can write

Tr_B␦共⑀⫺Hˆ_B兲Bˆ␦共⑀

⬘

⫺Hˆ_B兲Bˆ ⫽

兺

b,b⬘

␦共⑀⫺Eb兲␦共⑀

⬘

⫺Eb⬘兲円

具

b兩Bˆ兩b

⬘典

円2

⫽n共⑀兲n共⑀

⬘

兲F共⑀,⑀

⬘

兲, 共28兲 where the function F(⑀,⑀

⬘

) stands for

F共⑀,⑀

⬘

兲⬅‘‘円

具

⑀兩Bˆ兩⑀

⬘典

円2_{’’, 共29兲}

where 兩⑀典 denotes the eigenstate 兩b

典

of the environment Hamiltonian HˆB corresponding to the energy eigenvalue Eb

⫽⑀. Equation 共29兲 supposes some smoothening of the squares円

具

⑀兩Bˆ兩⑀

⬘典

円2of the matrix elements of Bˆ over a dense spectrum of eigenvalues around the energies ⑀and ⑀

⬘

. The function共29兲 has the symmetry

F共⑀,⑀

⬘

兲⫽F共⑀

⬘

,⑀兲. 共30兲

With the definition共29兲 and the identity

冕

d⑀

⬘

␦共⑀

⬘

⫺HˆB兲⫽Iˆ, 共31兲

we can now write that

TrB␦共⑀⫺HˆB兲

P¯ss¯⬘共HˆB;t兲 n共HˆB兲

Bˆ共⫺␶兲Bˆ

⫽P¯ss¯⬘共⑀;t兲

冕

d⑀

⬘

n共⑀

⬘

兲F共⑀,⑀

⬘

兲e⫺i(⑀⫺⑀⬘)t. 共35兲

Accordingly, our quantum master equation finally takes the closed form P˙ss⬘共⑀;t兲⫽⫺i共Es⫺Es⬘兲Pss⬘共⑀;t兲⫺␭2

兺

s ¯,s¯⬘

冕

d⑀

⬘

F共⑀,⑀

⬘

兲 ⫻

冕

0 t d␶兵

具

s兩Sˆ兩s¯

⬘典具

¯s

⬘

兩Sˆ兩s¯

典

P¯ss ⬘共⑀;t兲n共⑀

⬘

兲

⫻e⫹i(⑀⫺⑀⬘⫹E¯s⫺E¯s⬘)␶⫺

具

_s兩Sˆ兩s¯

典具

_¯s

⬘

兩Sˆ兩s

⬘典

⫻P¯ss¯⬘共⑀

⬘

;t兲n共⑀兲e⫹i(⑀⫺⑀⬘⫹Es⬘⫺E¯s⬘)␶⫺

具

s兩Sˆ兩s¯

典

⫻

具

¯s

⬘

兩Sˆ兩s

⬘典

P¯ss¯⬘共⑀

⬘

;t兲n共⑀兲e⫺i(⑀⫺⑀⬘⫹Es⫺E¯s)␶

⫹

具

¯s

⬘

兩Sˆ兩s¯

典具

¯s兩Sˆ兩s

⬘典

Pss¯⬘共⑀;t兲n共⑀

⬘

兲

⫻e⫺i(⑀⫺⑀⬘⫹E¯s⬘⫺E¯s)␶其⫹O共␭3兲. 共36兲

Equation 共36兲 determines the time evolution of the distribu-tion funcdistribu-tions Pss⬘(⑀;t) describing the populations and

quan-tum coherences of a system influencing its environment and is the central result of this paper. It is a non-Markovian equa-tion because of the presence of the time integral in the right-hand side.

In Eq. 共36兲 the function n(⑀)F(⑀,⑀

⬘

) determines the properties of the coupling to the environment and, in particu-lar, the time scale of the environment. If this time scale is supposed to be shorter then the system time scales 兵2␲/(Es⫺Es⬘)其, we can perform a Markovian approxima-tion in Eq. 共36兲. Such an approximation is justified for a

process evolving on time scales larger than the environment time scale. The Markovian approximation consists in taking the limit where the upper bound of the time integral goes to infinity and using the following relations:

冕

0

⬁

d␶e⫾i␻␶⫽⫾iP1

␻⫹␲␦共␻兲, 共37兲 where P denotes the principal part.

We finally obtain the Markovian version of our quantum master equation 共36兲 as

P˙_ss⬘共⑀;t兲⫽⫺i共E_s⫺E_s⬘兲P_ss⬘共⑀;t兲⫺i␭2

兺

s ¯,s¯⬘

再冋

冕

d⑀

⬘

n共⑀

⬘

兲F共⑀,⑀

⬘

兲P 1 ⑀⫺⑀

⬘

⫹E¯s⫺E¯s⬘

册

⫻关

具

s兩Sˆ兩s¯

⬘典具

¯s

⬘

兩Sˆ兩s¯

典

P_{¯ss ⬘}共⑀;t兲⫺

具

¯s兩Sˆ兩s¯

⬘典具

¯s

⬘

兩Sˆ兩s

⬘典

P_ss¯共⑀;t兲兴⫺

具

s兩Sˆ兩s¯

典具

¯s

⬘

兩Sˆ兩s

⬘典

n共⑀兲

冕

d⑀

⬘

F共⑀,⑀

⬘

兲P_¯ss¯⬘共⑀

⬘

;t兲 ⫻

冋

P 1 ⑀⫺⑀

⬘

⫹Es⬘⫺E¯s⬘ ⫺P 1 ⑀⫺⑀

⬘

⫹Es⫺E¯s

册冎

⫺␲␭2

兺

s ¯,s¯⬘

兵n共⑀⫹E¯s⫺E¯s⬘兲F共⑀,⑀⫹E¯s⫺E¯s⬘兲

⫻关

具

s兩Sˆ兩s¯

⬘典具

¯s

⬘

兩Sˆ兩s¯

典

P¯ss ⬘共⑀;t兲⫹

具

¯s兩Sˆ兩s¯

⬘典具

¯s

⬘

兩Sˆ兩s

⬘典

Pss¯共⑀;t兲兴⫺

具

s兩Sˆ兩s¯

典具

¯s

⬘

兩Sˆ兩s

⬘典

n共⑀兲

⫻关F共⑀,⑀⫹Es⬘⫺E¯s⬘兲P¯ss¯⬘共⑀⫹Es⬘⫺E¯s⬘;t兲⫹F共⑀,⑀⫹Es⫺E¯s兲P¯ss¯⬘共⑀⫹Es⫺E¯s;t兲兴其⫹O共␭3兲. 共38兲

We notice that the use of this Markovian equation may re-quire a slippage of initial conditions as shown in Refs. 关11,12兴. In Eq. 共38兲, the last terms in␲␭2 _{typically describe}

the relaxation to a stationary solution. The terms in i␭2 modify the frequencies of oscillations and include the so-called Lamb shifts of the zeroth-order energy eigenvalues. Indeed, if we consider only the evolution of the off-diagonal matrix element Pss⬘(⑀;t) by neglecting its coupling to all the

other matrix elements, we obtain the equation

P˙ss⬘共⑀;t兲⯝兵⫺i关E˜s共⑀兲⫺E˜s⬘共⑀兲兴⫺⌫ss⬘共⑀兲其Pss⬘共⑀;t兲,

共39兲

with the energies modified by the Lamb shifts

E ˜_{s共⑀兲⫽Es⫹␭}2

兺

s ¯ 円

具

s兩Sˆ兩s¯

典

円2

冕

d⑀

⬘

n共⑀

⬘

兲F共⑀,⑀

⬘

兲 ⫻P 1 ⑀⫺⑀

⬘

⫹Es⫺E¯s ⫹O共␭3_{兲 共40兲}

and the damping rates

⌫ss⬘共⑀兲⫽␲␭2

兺

s ¯(⫽s) 关円

具

s兩Sˆ兩s¯

典

円2n共⑀⫹Es⫺E¯s兲

⫻F共⑀,⑀⫹E_s⫺E_¯s兲⫹円

具

s

⬘

兩Sˆ兩s¯

典

円2n共⑀⫹E_s⬘⫺E_¯s兲

⫻F共⑀,⑀⫹Es⬘⫺E¯s兲兴⫹␲␭2共

具

s兩Sˆ兩s

典

in agreement with the results of Ref.关5兴. We notice that the complete equations for the off-diagonal matrix elements couple in general different energies because of the integrals over the environment energy⑀

⬘

.

The evolution equations for the populations of the states 兩s

典

of the system can be obtained by neglecting the contri-butions from the quantum coherences, i.e., by neglecting the terms involving off-diagonal elements of P_ss⬘(⑀;t). This is justified in the weak-coupling limit as long as the coherences vanish or are negligible in the initial conditions, i.e.,

Pss⬘(⑀;0)⫽0 for s⫽s

⬘

. Accordingly, we obtain the

follow-ing evolution equations for the populations:

P˙_ss共⑀;t兲⯝2␲␭2

兺

s⬘ 円

具

s兩Sˆ兩s

⬘典

円2_{F共⑀,⑀⫹E} s⫺Es⬘兲 ⫻关n共⑀兲Ps⬘s⬘共⑀⫹Es⫺Es⬘;t兲 ⫺n共⑀⫹Es⫺Es⬘兲Pss共⑀;t兲兴. 共42兲

This equation is a kind of Pauli equation established with the Fermi golden rule and the conversation of energy in the tran-sitions. Indeed, if a transition happens from a state in which the energy of the system is Es and the one of the

environ-ment ⑀ to a state in which the system has energy Es⬘, the

final energy of the environment should be ⑀

⬘

⫽⑀⫹Es

⫺Es⬘, which is well expressed by Eq. 共42兲. Nevertheless,

Eq. 共42兲 rules the populations of the states s of the system with the extra information given by the distribution over the environment energy⑀, which is not a feature of the standard Pauli equation and which turns out to be of importance to understand the relaxation inside a nanoscopic isolated sys-tem.

Our Markovian master equation共38兲 is more general than an equation for the populations because it also describes the time evolution of the distributions of the quantum coher-ences over the energy of the environment, as shown in the following sections.

B. Comparison with the Redfield master equation We now discuss the conceptual differences between our quantum master equation and another one known as the Red-field master equation. This equation is well known in the context of nuclear magnetic resonance 共NMR兲 where it de-scribes the time evolution of nuclear spins interacting with their environment.

The Redfield master equation describes the time evolution of the system density matrix obtained tracing out from the total density matrix the degrees of freedom of the environ-ment

␳ˆS共t兲⫽TrB␳ˆ共t兲. 共43兲

The Redfield equation is derived by using the closure ap-proximation that the total density matrix keeps the form

␳ˆ共t兲⫽␳ˆS共t兲丢␳ˆB 共44兲

during the whole time evolution, where ␳ˆB does not depend

on time. The Redfield master equation is derived in the weak-coupling limit by a method similar to the one of the previous section to get

␳ˆ˙S共t兲⫽⫺i关HˆS,␳ˆS共t兲兴⫺␭2Sˆ

冕

0 t d␶ ␣共␶兲Sˆ共⫺␶兲␳ˆS共t兲 ⫹␭2_Sˆ␳ˆ S共t兲

冕

0 t d␶␣*共␶兲Sˆ共⫺␶兲 ⫹␭2

冕

0 t d␶ ␣共␶兲Sˆ共⫺␶兲␳ˆ_S共t兲Sˆ ⫺␭2_␳ˆ S共t兲

冕

0 t d␶ ␣*共␶兲Sˆ共⫺␶兲Sˆ⫹O共␭3兲, 共45兲

with the correlation function of the environment operators ␣共t兲⫽

具

Bˆ共t兲Bˆ共0兲

典

⫽TrB␳ˆBBˆ共t兲Bˆ共0兲. 共46兲

Equation 共45兲 is a non-Markovian Redfield equation. The non-Markovianity comes from the fact that the integrals over expressions containing the correlation function depend on time. The density matrix of the system can be represented in the basis of the system eigenstates as

␩ss⬘共t兲⬅

具

s兩␳ˆS共t兲兩s

⬘典

. 共47兲

In this representation, the non-Markovian Redfield equation has the following form:

␩˙_ss ⬘共t兲⫽⫺i共Es⫺Es⬘兲␩ss⬘共t兲 ⫺␭2

兺

s ¯,s¯⬘

冕

d⑀

⬘

n共⑀

⬘

兲F共⑀,⑀

⬘

兲

冕

0 t d␶兵

具

s兩Sˆ兩s¯

⬘典

⫻

具

¯s

⬘

兩Sˆ兩s¯

典

␩¯ss ⬘共t兲e⫹i(⑀⫺⑀⬘⫹E¯s⫺E¯s⬘)␶⫺

具

s兩Sˆ兩s¯

典

⫻

具

¯s

⬘

兩Sˆ兩s

⬘典

␩¯ss¯⬘共t兲e⫹i(⑀⫺⑀⬘⫹E¯s⫺Es)␶⫺

具

s兩Sˆ兩s¯

典

⫻

具

¯s

⬘

兩Sˆ兩s

⬘典

␩_¯ss¯⬘共t兲e⫺i(⑀⫺⑀⬘⫹E¯s⬘⫺Es⬘)␶⫹

具

_¯s

⬘

兩Sˆ兩s¯

典

⫻

具

¯s兩Sˆ兩s

⬘典

␩_ss¯⬘共t兲e⫺i(⑀⫺⑀⬘⫹E¯s⬘⫺E¯s)␶其⫹O共␭3兲.

共48兲 If the environment is large enough, the correlation func-tion in Eq.共45兲 goes to zero after a certain time. This time, called the environment correlation time␶corr, determines the

time scale of the environment dynamics. If we perform the

Markovian approximation that consists of putting the upper

bound of the time integral in the non-Markovian Redfield equation to infinity, one gets the standard Redfield equation. We notice that, in doing so, the time evolution may be spoiled on a time scale of order␶corrunless some use is made

of some slipped initial conditions 关11,12兴. Performing this Markovian approximation, one gets the standard 共Markov-ian兲 Redfield equation given by Eq. 共45兲 with 兰₀t replaced by 兰0⬁. As shown in Refs.关11,12兴, the use of this Redfield

initial conditions.

In order to compare the Redfield equation with our master equation derived in the previous section, we consider the case where the environment is initially in the microcanonical state: ␳ˆB⫽ ␦共⑀⫺HˆB兲 TrB␦共⑀⫺HˆB兲 ⫽␦共⑀⫺HˆB兲 n共⑀兲 . 共49兲

Having chosen the microcanonical density matrix 共49兲 for the environment, the correlation function共46兲 takes the form

␣共␶,⑀兲⫽ 1

n共⑀兲TrB␦共⑀⫺HˆB兲Bˆ共␶兲Bˆ

⫽_n共1_⑀兲

冕

d⑀

⬘

TrB␦共⑀⫺HˆB兲Bˆ共␶兲␦共⑀

⬘

⫺HˆB兲Bˆ

⫽

冕

d⑀

⬘

n共⑀

⬘

兲F共⑀,⑀

⬘

兲ei(⑀⫺⑀⬘)␶. 共50兲 In the basis of the system eigenstates, the Redfield equation takes the form

␩˙_ss

⬘共t兲⫽⫺i共Es⫺Es⬘兲␩ss⬘共t兲⫺i␭2

兺

s ¯,s¯⬘

再冋

冕

d⑀

⬘

n共⑀

⬘

兲F共⑀,⑀

⬘

兲P 1 ⑀⫺⑀

⬘

⫹E¯s⫺E¯s⬘

册

关

具

s兩Sˆ兩s¯

⬘典具

¯s

⬘

兩Sˆ兩s¯

典

␩¯ss ⬘共t兲⫺

具

¯s兩Sˆ兩s¯

⬘典

⫻

具

¯s

⬘

兩Sˆ兩s

⬘典

␩ss¯共t兲兴⫹

冕

d⑀

⬘

n共⑀

⬘

兲F共⑀,⑀

⬘

兲

冋

P 1 ⑀⫺⑀

⬘

⫹E¯s⬘⫺Es⬘ ⫺P 1 ⑀⫺⑀

⬘

⫹E¯s⫺Es

册

具

s兩Sˆ兩s¯

典具

¯s

⬘

兩Sˆ兩s

⬘典

␩¯ss¯⬘共t兲

冎

⫺␲␭2

兺

s ¯,s¯⬘

兵n共⑀⫹E¯s⫺E¯s⬘兲F共⑀,⑀⫹E¯s⫺E¯s⬘兲关

具

s兩Sˆ兩s¯

⬘典具

¯s

⬘

兩Sˆ兩s¯

典

␩¯ss ⬘共t兲⫹

具

¯s兩Sˆ兩s¯

⬘典具

¯s

⬘

兩Sˆ兩s

⬘典

␩ss¯共t兲兴

⫺关n共⑀⫹E¯s⬘⫺Es⬘兲F共⑀,⑀⫹E¯s⬘⫺Es⬘兲⫹n共⑀⫹E¯s⫺Es兲F共⑀,⑀⫹E¯s⫺Es兲兴

具

s兩Sˆ兩s¯

典具

¯s

⬘

兩Sˆ兩s

⬘典

␩¯ss¯⬘共t兲其⫹O共␭ 3_兲.

共51兲

The off-diagonal elements of the system density matrix individually obey the equations

␩˙_ss

⬘共t兲⯝兵⫺i关E˜s共⑀兲⫺E˜s⬘共⑀兲兴⫺⌫ss⬘共⑀兲其␩ss⬘共t兲, 共52兲

with the same Lamb shifts共40兲 and damping rates 共41兲 as in our master equation and as expected from Ref.关5兴. There is no difference between our quantum master equation and the Redfield one at this stage.

On the other hand, the Redfield equation predicts an evo-lution of the populations ruled by the following equation obtained by neglecting all the contributions coming from the coherences in Eq. 共51兲:

␩˙

ss共t兲⫽2␲␭2

兺

s⬘

円

具

s兩Sˆ兩s

⬘典

円2关F共⑀,⑀⫹Es⬘⫺Es兲

⫻n共⑀⫹Es⬘⫺Es兲␩s⬘s⬘共t兲⫺F共⑀,⑀⫹Es⫺Es⬘兲

⫻n共⑀⫹Es⫺Es⬘兲␩ss共t兲兴. 共53兲

This equation is the same as the master equation for the populations derived by Cohen-Tannoudji and co-workers in Ref. 关5兴.

We notice that important differences now exists between the population equation 共53兲 obtained from the Redfield equation and the other population equation 共42兲 obtained from our master equation. Both equations describe the evo-lution of the populations as a random walk process in the spectrum. However, these processes are significantly

differ-ent for Eqs. 共42兲 and 共53兲. Let us focus on the evolution of the probability to be on a system state corresponding to the system energy E_s. In both equations we see that for the loss contributions to the evolution coming from the jumps from an energy Es to an energy Es⬘, the density of states of the

environment is modified by the energy Es⫺Es⬘. This is

con-sistent with the Fermi golden rule applied to the total system and, thus, keeps the total energy constant. We care now on the gain contributions to the evolution. In our equation共42兲, we see that for these contributions due to jumps from the system energy Es⬘ to Es, the density of states of the

envi-ronment is modified by the energy Es⬘⫺Es. This is also

consistent with the Fermi golden rule applied to the total system and, thus, keeps the total energy constant. However, for the Redfield equation we see that for the jumps from an energy Es⬘ to an energy Es, the density of states of the

environment is not modified by the energy Es⬘⫺Es, which

is not consistent with the Fermi golden rule applied to the total system and does not keep the total energy constant.

condition. The Redfield equation describes transitions that occur along a vertical line at constant environment energy and is therefore not consistent with energy conservation in the total system. This is acceptable if the environment is sufficiently large and has an arbitrarily large energy. How-ever, this is inadequate if the total energy of the system and the environment is finite as in nanoscopic systems, in which case our master should replace the Redfield equation.

We can summarize as follows the differences between our quantum master equation 共38兲 and the Redfield equation 共51兲. The derivation of both equations is based on the per-turbative expansion of the total density matrix, but a specific form is imposed in each equation to the total density matrix 关see Eqs. 共10兲 and 共44兲兴. The consequence of this choice can be seen on the reduced density matrix of the system. In the Redfield theory, we have

␳ˆ_S共t兲⫽

兺

s,s⬘

兩s

典

␩ss⬘共t兲

具

s

⬘

兩, 共54兲

while, for our master equation, using Eq. 共12兲 we have

␳ˆS共t兲⫽

兺

s,s⬘

冕

d⑀兩s

典

Pss⬘共⑀;t兲

具

s

⬘

兩. 共55兲

The system density matrix is related to the distribution func-tions according to

具

s兩␳ˆS共t兲兩s

⬘典

⫽␩ss⬘共t兲⫽

冕

d⑀Pss⬘共⑀;t兲. 共56兲

We see that, in our master equation, the matrix elements of the system density matrix are decomposed on the energy of the environment. This is not the case for the Redfield equa-tion. The decomposition allows us to correlate the states of the system with the states of the environment. This is the main point of our master equation. The density matrix adopted for the Redfield equation cannot describe such cor-relations. In the Redfield equation, during the evolution, the environment is always in the same state while, in our master equation, the state of the environment is determined by the state of the system. As a consequence, we obtain a descrip-tion which is consistent with energy conservadescrip-tion thanks to our master equation.

III. APPLICATION TO THE SPIN-ENVIRONMENT MODEL

In this section we consider a specific class of two-level systems interacting with an environment. The two-level sys-tem may be supposed to be a spin. An example is the spin-boson model in which the environment is a set of harmonic oscillators behaving as phonons关19兴.

The Hamiltonian of the spin-environment model we con-sider here is the following:

Hˆtot⫽

⌬

2␴ˆz⫹HˆB⫹␭␴ˆxBˆ . 共57兲

The eigenvalue equation of the system is

FIG. 2. Generalization of the previous Fig. 1 for the case where the system has more than two levels共here four levels兲. One can see that the system levels 共horizontal lines兲 that do not intersect the total energy diagonal line E⫽ES⫹EBwithin the environment en-ergy spectrum delimited by the sparse-dotted vertical lines do not participate in the dynamics.

FIG. 1. Schematic representation of the energy exchanges de-scribed, respectively, by the Redfield and by our master equation in the Markovian limit for a two-level system model, in the plane of the system energy ESversus the environment energy EB. The en-ergy splitting between the two levels of the system is denoted by⌬. The energy spectrum of the system is discrete共two levels兲 while the one of the environment is a quasicontinuum represented by the density of states given by the Wigner semicircular law共88兲 of width equal to unity. The total energy of the system is given by E⫽ES ⫹EB, which corresponds to the diagonal line. The initial condition is denoted by two empty superposed circles. We see that transitions preserving the total energy have to occur along the diagonal line

HˆS兩s

典

⫽

⌬

2␴ˆz兩s

典

⫽s ⌬

2兩s

典

, 共58兲 where s⫽⫾1. Like in Sec. II, we first derive our master equation and then the Redfield equation in order to compare both equations.

A. Using our master equation

Let us now apply our master equation to the spin-environment model. In our theory and for a two-level sys-tem, the total density matrix becomes

␳ˆ共t兲⫽ 1 n共HˆB兲

关P⫹⫹共HˆB;t兲兩⫹

典具

⫹兩⫹P⫹⫺共HˆB;t兲兩⫹

典具

⫺兩

⫹P⫺⫹共HˆB;t兲兩⫺

典具

⫹兩⫹P⫺⫺共HˆB;t兲兩⫺

典具

⫺兩兴. 共59兲

For the spin-environment model, our non-Markovian master equation共36兲 is given by

P˙_⫹⫹共⑀;t兲⫽⫺␭2P_⫹⫹共⑀;t兲

冕

d⑀

⬘

F共⑀,⑀

⬘

兲n共⑀

⬘

兲 ⫻

冕

0 t

d␶关ei(⑀⫺⑀⬘⫹⌬)␶_⫹e⫺i(⑀⫺⑀⬘⫹⌬)␶_兴

⫹␭2_n共⑀兲

冕

_d⑀

_⬘

_F共⑀,⑀

_⬘

_兲P

⫺⫺共⑀

⬘

;t兲 ⫻

冕

0 t

d␶关ei(⑀⫺⑀⬘⫹⌬)␶_⫹e⫺i(⑀⫺⑀⬘⫹⌬)␶_{兴, 共60兲}

P˙_⫺⫺共⑀;t兲⫽⫺␭2_P

⫺⫺共⑀;t兲

冕

d⑀

⬘

F共⑀,⑀

⬘

兲n共⑀

⬘

兲 ⫻

冕

0 t

d␶关ei(⑀⫺⑀⬘⫺⌬)␶⫹e⫺i(⑀⫺⑀⬘⫺⌬)␶兴

⫹␭2_n共⑀兲

冕

_d⑀

_⬘

_F共⑀,⑀

_⬘

_兲P

⫹⫹共⑀

⬘

;t兲 ⫻

冕

0 t

d␶关ei(⑀⫺⑀⬘⫺⌬)␶⫹e⫺i(⑀⫺⑀⬘⫺⌬)␶兴, 共61兲

P˙_⫹⫺共⑀;t兲⫽⫺i⌬P_⫹⫺共⑀;t兲⫺␭2P_⫹⫺共⑀;t兲 ⫻

冕

d⑀

⬘

F共⑀,⑀

⬘

兲n共⑀

⬘

兲

冕

0 t d␶关ei(⑀⫺⑀⬘⫹⌬)␶ ⫹e⫺i(⑀⫺⑀⬘⫺⌬)␶_兴⫹␭2_n共⑀兲 ⫻

冕

d⑀

⬘

F共⑀,⑀

⬘

兲P_⫺⫹共⑀

⬘

;t兲 ⫻

冕

0 t

d␶关ei(⑀⫺⑀⬘⫺⌬)␶⫹e⫺i(⑀⫺⑀⬘⫹⌬)␶兴, 共62兲

and a further equation for P˙_⫺⫹(⑀;t) given by the complex conjugate of Eq.共62兲.

We observe that the diagonal and off-diagonal elements of

Pss⬘(⑀;t) obey decoupled equations in the case of the

spin-environment model. Therefore, the time evolution of the populations is independent of the time evolution of the quan-tum coherences. We now perform the Markovian

approxima-tion that consists of putting the upper bound of the time

integral to infinity. Using Eq. 共37兲, we find

P˙_⫹⫹共⑀;t兲⫽2␲␭2F共⑀,⑀⫹⌬兲关n共⑀兲P_⫺⫺共⑀⫹⌬;t兲 ⫺n共⑀⫹⌬兲P⫹⫹共⑀;t兲兴, 共63兲 P˙_⫺⫺共⑀;t兲⫽2␲␭2F共⑀,⑀⫺⌬兲关n共⑀兲P_⫹⫹共⑀⫺⌬;t兲 ⫺n共⑀⫺⌬兲P⫺⫺共⑀;t兲兴, 共64兲 P˙_⫹⫺共⑀;t兲⫽⫺i⌬P_⫹⫺共⑀;t兲⫹i␭2

冕

d⑀

⬘

F共⑀,⑀

⬘

兲 ⫻P 2⌬ 共⑀⫺⑀

⬘

兲2_⫺⌬2关n共⑀

⬘

兲P⫹⫺共⑀;t兲 ⫹n共⑀兲P⫺⫹共⑀

⬘

;t兲兴⫺␲␭2关n共⑀⫹⌬兲F共⑀,⑀⫹⌬兲 ⫹n共⑀⫺⌬兲F共⑀,⑀⫺⌬兲兴P_⫹⫺共⑀;t兲⫹␲␭2n共⑀兲 ⫻关F共⑀,⑀⫹⌬兲P_⫺⫹共⑀⫹⌬;t兲 ⫹F共⑀,⑀⫺⌬兲P_⫺⫹共⑀⫺⌬;t兲兴. 共65兲

We notice that during the time evolution of the populations, the following quantity remains a constant of motion:

P共⑀;t兲⬅P_⫹⫹共⑀;t兲⫹P_⫺⫺共⑀⫹⌬;t兲⫽P共⑀;0兲. 共66兲 Accordingly, the difference of the populations defined as

Z共⑀;t兲⬅P_⫹⫹共⑀;t兲⫺P_⫺⫺共⑀⫹⌬;t兲 共67兲 obeys the differential equation

Z˙共⑀;t兲⫽2␲␭2关n共⑀兲⫺n共⑀⫹⌬兲兴F共⑀,⑀⫹⌬兲P共⑀;0兲 ⫺2␲␭2_{关n共⑀兲⫹n共⑀⫹⌬兲兴F共⑀,⑀⫹⌬兲Z共⑀;t兲,}

共68兲 the solution of which is given by

Z共⑀;t兲⫽Z共⑀;⬁兲⫹关Z共⑀;0兲⫺Z共⑀;⬁兲兴e⫺␥Paulit 共69兲 with the asymptotic equilibrium value

Z共⑀;⬁兲⫽n共⑀兲⫺n共⑀⫹⌬兲

n共⑀兲⫹n共⑀⫹⌬兲P共⑀;0兲 共70兲

and the relaxation rate

Therefore, the populations relax to their asymptotic equilib-rium values for each pair of energies ⑀and⑀⫹⌬ of the en-vironment, keeping constant the initial distribution of the quantity P共⑀;0兲.

The time evolution of the distribution functions P_⫾⫿(⑀;t) of the quantum coherences is more complicated because there is now a coupling between a continuum of values of the environment energy instead of only two values. Accordingly, the distributions of quantum coherence is ruled by a couple of two integro-differential equations, instead of an ordinary differential equation.

B. Using the Redfield equation

For the spin-environment model, the non-Markovian and Markovian Redfield equations can be derived from Eqs.共48兲 and共51兲. Using Eq. 共48兲, the non-Markovian Redfield equa-tions here write

␩˙

⫹⫹共t兲⫽⫺␭2␩⫹⫹共t兲

冕

d⑀

⬘

n共⑀

⬘

兲F共⑀,⑀

⬘

兲 ⫻

冕

0 t

d␶关ei(⑀⫺⑀⬘⫹⌬)␶⫹e⫺i(⑀⫺⑀⬘⫹⌬)␶兴

⫹␭2_{␩⫺⫺共t兲}

冕

_d⑀

_⬘

_n共⑀

_⬘

_{兲F共⑀,⑀}

_⬘

_兲

⫻

冕

0 t

d␶关ei(⑀⫺⑀⬘⫺⌬)␶⫹e⫺i(⑀⫺⑀⬘⫺⌬)␶兴, 共72兲

␩˙_{⫺⫺共t兲⫽⫺␭}2_{␩⫺⫺共t兲}

冕

_d⑀

_⬘

_n共⑀

_⬘

_{兲F共⑀,⑀}

_⬘

_兲

⫻

冕

0 t

d␶关ei(⑀⫺⑀⬘⫺⌬)␶⫹e⫺i(⑀⫺⑀⬘⫺⌬)␶兴

⫹␭2_{␩⫹⫹共t兲}

冕

_d⑀

_⬘

_n共⑀

_⬘

_{兲F共⑀,⑀}

_⬘

_兲

⫻

冕

0 t

d␶关ei(⑀⫺⑀⬘⫹⌬)␶⫹e⫺i(⑀⫺⑀⬘⫹⌬)␶兴, 共73兲

␩˙

⫹⫺共t兲⫽⫺i⌬␩⫹⫺共t兲⫺␭2␩⫹⫺共t兲

冕

d⑀

⬘

n共⑀

⬘

兲F共⑀,⑀

⬘

兲 ⫻

冕

0 t

d␶关ei(⑀⫺⑀⬘⫹⌬)␶⫹e⫺i(⑀⫺⑀⬘⫺⌬)␶兴

⫹␭2_{␩⫺⫹共t兲}

冕

_d⑀

_⬘

_n共⑀

_⬘

_{兲F共⑀,⑀}

_⬘

_兲

⫻

冕

0 t

d␶关ei(⑀⫺⑀⬘⫺⌬)␶⫹e⫺i(⑀⫺⑀⬘⫹⌬)␶兴, 共74兲

and a further equation for ␩_⫺⫹(t) given by the complex conjugate of Eq.共74兲.

Here again, there is a decoupling between the time evo-lutions of the populations and of the quantum coherences.

Taking the Markovian approximation by replacing兰₀t into 兰0⬁and using Eq.共37兲, we get the Markovian Redfield

equa-tions for the spin-environment model:

␩˙ ⫹⫹共t兲⫽2␲␭2关n共⑀⫺⌬兲F共⑀,⑀⫺⌬兲␩⫺⫺共t兲 ⫺n共⑀⫹⌬兲F共⑀,⑀⫹⌬兲␩_⫹⫹共t兲兴, 共75兲 ␩˙ ⫺⫺共t兲⫽2␲␭2关n共⑀⫹⌬兲F共⑀,⑀⫹⌬兲␩⫹⫹共t兲 ⫺n共⑀⫺⌬兲F共⑀,⑀⫺⌬兲␩_⫺⫺共t兲兴, 共76兲 ␩˙_{⫹⫺共t兲⫽⫺i⌬␩⫹⫺共t兲⫹i␭}2

冕

_d⑀

_⬘

_n共⑀

_⬘

_{兲F共⑀,⑀}

_⬘

_兲 ⫻P 2⌬ 共⑀⫺⑀

⬘

兲2_⫺⌬2关␩⫹⫺共t兲⫹␩⫺⫹共t兲兴⫺␲␭ 2 ⫻关n共⑀⫹⌬兲F共⑀,⑀⫹⌬兲⫹n共⑀⫺⌬兲F共⑀,⑀⫺⌬兲兴 ⫻关␩⫹⫺共t兲⫺␩⫺⫹共t兲兴. 共77兲 The populations of the two-level system are controlled by the

z component of the spin defined as the difference

z_Redfield共t兲⫽␩_⫹⫹共t兲⫺␩_⫺⫺共t兲. 共78兲

According to the Markovian Redfield equations 共75兲 and 共76兲, the z component obeys the differential equation

z˙Redfield⫽2␲␭2关n共⑀⫺⌬兲F共⑀,⑀⫺⌬兲⫺n共⑀⫹⌬兲F共⑀,⑀⫹⌬兲兴

⫺2␲␭2_{关n共⑀⫺⌬兲F共⑀,⑀⫺⌬兲}

⫹n共⑀⫹⌬兲F共⑀,⑀⫹⌬兲兴zRedfield. 共79兲

Its solution is given by

z_Redfield共t兲⫽z_Redfield共⬁兲

⫹关zRedfield共0兲⫺zRedfield共⬁兲兴e⫺␥Redfieldt, 共80兲

with the equilibrium value

z_Redfield共⬁兲⫽n共⑀⫺⌬兲F共⑀,⑀⫺⌬兲⫺n共⑀⫹⌬兲F共⑀,⑀⫹⌬兲 n共⑀⫺⌬兲F共⑀,⑀⫺⌬兲⫹n共⑀⫹⌬兲F共⑀,⑀⫹⌬兲,

共81兲 and the relaxation rate

␥Redfield⫽2␲␭2关n共⑀⫺⌬兲F共⑀,⑀⫺⌬兲

We notice that the rate predicted by the Redfield equation coincides with the one predicted by our master equation only in the limit⌬⫽0. A more important difference appears in the asymptotic equilibrium values for the z component of the spin predicted by both equations. These differences find their origin in the problem of conservation of energy with the Redfield equation, as explained above. Comparison with nu-merical data will confirm this explanation in a following sec-tion in the case of the spin-GORM model.

IV. APPLICATION TO THE SPIN-GORM MODEL In order to confront our master equation and the Redfield equation with numerical data and test their respective do-mains of validity, we now apply our theory to a specific class of two-level systems interacting with an environment, for which the environment operators are Gaussian orthogonal random matrices 共GORM兲. We call this model the spin-GORM model and its detailed properties will be described elsewhere关20兴.

The system is a two-level system, while the environment is supposed to be a system with a very complex dynamics. Here, the term complex is used in a generic way. The com-plexity can come, for example, from the fact that the corre-sponding classical system is chaoticlike in a quantum billiard or for the hydrogen atom in a strong magnetic field关21,22兴. It can also come from a large number of coupling between states in an interacting many-body system like those appear-ing in nuclear physics 关22兴 or in systems of interacting fermions like quantum computers 关22兴. A well-known method, developed by Wigner in the 1950s, for modeling the energy spectrum of a complex quantum system contain-ing many states interactcontain-ing with each other, consists of as-suming that their Hamiltonian is a random matrix关23–25兴. Here, we suppose that the Hamiltonian of the environment is a Gaussian orthogonal random matrix共GORM兲. The interac-tion between the spin and the environment is given by a coupling operator which is the product of a system and en-vironment operators. The latter is also represented by a GORM because of its complex interaction with the many degrees of freedom of the environment. Such random-matrix models have recently turned out to be of great relevance for the discussion of relaxation and dissipation in quantum sys-tems 关16,26–29兴.

The spin-GORM model can therefore be considered as a particular case of the spin-environment model in which the environment operators are GORM. The Hamiltonian of the spin-GORM model is thus given by

Hˆtot⫽

⌬

2␴ˆz⫹HˆB⫹␭␴ˆxBˆ 共83兲 where the Hamiltonian of the environment is

HˆB⫽

1

冑

8NXˆ , 共84兲 and the environment coupling operator by

Bˆ⫽ 1

冑

8NX

ˆ

_⬘

_{. 共85兲}

Xˆ and Xˆ

⬘

are two statistically independent N/2⫻N/2 random matrices belonging to the Gaussian orthogonal ensemble 共GOE兲 of probability density

p共Xˆ兲⫽C exp共⫺aXˆ Tr Xˆ2兲, 共86兲

with aXˆ⫽

1

2 and a normalization constantC. N/2 is the num-ber of states of the environment. The off-diagonal and diag-onal elements of Xˆ are independent Gaussian random num-bers with mean zero and standard deviations␴_off-diag.⫽1 and ␴diag⫽

冑

2, respectively.

In the limit N→⬁, the density of states of the environ-ment gets smooth and can be calculated by an average over the random-matrix ensemble

n共⑀兲⫽

兺

b_⫽1

N/2

␦共⑀⫺Eb兲. 共87兲

It is known that the GOE level density is given by the Wigner semicircular law

n共⑀兲⫽

再

4N ␲

冑

14⫺⑀ 2 _{if 兩⑀兩⬍}1 2 0 if 兩⑀兩⭓12 . 共88兲

The random matrices are normalized so that the level density of the environment has a width equal to unity. To simplify the notations in the following, we use the convention

冑

x⬅

再

冑

x

if 0⬍x

0 if x⭐0. 共89兲

For the following, we also need to evaluate the function

F(⑀,⑀

⬘

) for our random-matrix model. For this purpose, we need the random-matrix average of the quantity共28兲. Since, in the GOE, we have that

円

具

b兩Bˆ兩b

典

円2⫽ 1

4N, 共90兲

円

具

b兩Bˆ兩b

⬘典

円2_⫽ 1

8N, 共91兲

Tr_B␦共⑀⫺Hˆ_B兲Bˆ␦共⑀

⬘

⫺Hˆ_B兲Bˆ⫽

兺

b,b⬘ ␦共⑀⫺Eb兲␦共⑀

⬘

⫺Eb⬘兲円

具

b兩Bˆ兩b

⬘典

円2 ⫽

兺

b ␦共⑀⫺Eb兲␦共⑀

⬘

⫺Eb兲 円

具

b兩Bˆ兩b

典

円2⫹

兺

b⫽b⬘ ␦共⑀⫺Eb兲␦共⑀

⬘

⫺Eb⬘兲 円

具

b兩Bˆ兩b

⬘典

円2 ⫽_4N1

兺

b ␦共⑀⫺Eb兲␦共⑀

⬘

⫺Eb兲⫹ 1 8N _b

兺

_⫽b⬘␦共⑀⫺Eb兲␦共⑀

⬘

⫺Eb⬘兲 ⫽ 1 8N

兺

b ␦共⑀⫺Eb兲␦共⑀

⬘

⫺Eb兲⫹ 1 8N _b,b

兺

_⬘␦共⑀⫺Eb兲␦共⑀

⬘

⫺Eb⬘兲 ⯝_8N1 ␦共⑀⫺⑀

⬘

兲n共⑀兲⫹_8N1 n共⑀兲n共⑀

⬘

兲. 共92兲

In the limit N→⬁, the first term becomes negligible in front of the second term so that the comparison with Eq. 共28兲 shows that for the spin-GORM model,

F共⑀,⑀

⬘

兲⯝ 1

8N. 共93兲

The total system contains N states. The unperturbed den-sity of states of the total system is schematically depicted in Fig. 3, for␭⫽0. The model has different regimes whether the splitting ⌬ between the two levels of the spin is larger or smaller than the width of the environment level density. The spin-GORM model can describe a large variety of physical situations. In the present paper we focus on the perturbative regimes (␭ⰆHˆ_B). When ⌬ is larger than the width of the semicircular density of states of the environment, we are in a highly non-Markovian regime. The dynamics of the system is faster than that of the environment. On the other hand, when⌬ is smaller than unity, we are in a Markovian regime because the dynamics of the environment is much faster then the one of the system.

Now, we apply our master equation and the Redfield equation to the spin-GORM model in both their Markovian and non-Markovian versions.

A. Using our master equation

We now apply our master equation 共36兲 to the spin-GORM model. Using Eqs.共60兲–共62兲, 共88兲, and 共93兲, we get the non-Markovian equations

P˙_⫹⫹共⑀;t兲⫽⫺␭ 2 ␲ P⫹⫹共⑀;t兲

冕

d⑀

⬘

冑

1 4⫺⑀

⬘

2 ⫻sin共⑀⫺⑀

⬘

⫹⌬兲t 共⑀⫺⑀

⬘

⫹⌬兲 ⫹ ␭2 ␲

冑

1 4⫺⑀ 2 ⫻

冕

d⑀

⬘

P_⫺⫺共⑀

⬘

;t兲sin共⑀⫺⑀

⬘

⫹⌬兲t 共⑀⫺⑀

⬘

⫹⌬兲 , 共94兲 P˙_⫺⫺共⑀;t兲⫽⫺␭ 2 ␲ P⫺⫺共⑀;t兲

冕

d⑀

⬘

冑

1 4⫺⑀

⬘

2 ⫻sin共⑀⫺⑀

⬘

⫺⌬兲t 共⑀⫺⑀

⬘

⫺⌬兲 ⫹ ␭2 ␲

冑

1 4⫺⑀ 2 ⫻

冕

d⑀

⬘

P_⫹⫹共⑀

⬘

;t兲sin共⑀⫺⑀

⬘

⫺⌬兲t 共⑀⫺⑀

⬘

⫺⌬兲 , 共95兲 P˙_⫹⫺共⑀;t兲⫽⫺i⌬P_⫹⫺共⑀;t兲⫺ ␭ 2 2␲P⫹⫺共⑀;t兲

冕

d⑀

⬘

⫻

冑

1₄⫺⑀

⬘

2

冕

0 t

d␶关ei(⑀⫺⑀⬘⫹⌬)␶⫹e⫺i(⑀⫺⑀⬘⫺⌬)␶兴

⫹ ␭ 2 2␲

冑

1 4⫺⑀ 2

冕

_d⑀

_⬘

_P ⫺⫹共⑀

⬘

;t兲 ⫻

冕

0 t

d␶关ei(⑀⫺⑀⬘⫺⌬)␶⫹e⫺i(⑀⫺⑀⬘⫹⌬)␶兴. 共96兲

We notice that the equations for the populations are decou-pled from the ones for the quantum coherences.

Performing the Markovian approximation and using limt_→⬁(sin␻t/␻)⫽␲␦(␻) and Eq.共37兲, we get

P˙_⫹⫺共⑀;t兲⫽⫺i⌬P_⫹⫺共⑀;t兲⫹i␭ 2 ␲

冕

d⑀

⬘

P ⌬ 共⑀⫺⑀

⬘

兲2_⫺⌬2 ⫻

冋

冑

1₄⫺⑀

⬘

2_P ⫹⫺共⑀;t兲⫹

冑

1 4⫺⑀ 2_P ⫺⫹共⑀

⬘

;t兲

册

⫺␭ 2 2

冋

冑

1 4⫺共⑀⫹⌬兲 2_⫹

冑

1 4⫺共⑀⫺⌬兲 2

册

⫻P_⫹⫺共⑀;t兲⫹␭ 2 2

冑

1 4⫺⑀ 2_关P ⫺⫹共⑀⫹⌬;t兲 ⫹P_⫺⫹共⑀⫺⌬;t兲兴, 共99兲

where the expressions and integrals over energy extend over the interval of definition of the level density n(E) and of the distributions Pss⬘(E;t) which is always⫺1/2⬍E⬍⫹1/2, E

being the argument of these functions.

We now focus our attention on the evolution of the popu-lations. We see from Eqs. 共97兲 and 共98兲 that the transitions conserve the total energy of the system and environment so that the transitions occur between the only two energies ⑀ and⑀⫹⌬ of the environment. As a consequence, the quantity

P共⑀;t兲⬅P_⫹⫹共⑀;t兲⫹P_⫺⫺共⑀⫹⌬;t兲⫽P共⑀;0兲 共100兲 is a constant of the motion for each energy⑀of the environ-ment, as already noticed with Eq.共66兲. The difference 共67兲 of populations obeys the differential equation共68兲. If the initial distributions P(⑀

⬘

;0) and Z(⑀

⬘

;0) are Dirac delta distribu-tions centered on the initial energy ⑀:

P共⑀

⬘

;0兲⫽␦共⑀

⬘

⫺⑀兲, 共101兲

Z共⑀

⬘

;0兲⫽␦共⑀

⬘

⫺⑀兲zPauli共0兲. 共102兲

The z component of the spin defined as

z_Pauli共t兲⫽

冕

d⑀Z共⑀;t兲 共103兲

obeys the same differential equation as the distribution

Z(⑀;t), z˙_Pauli共t兲⫽␭2

冋

冑

1 4⫺⑀ 2_⫺

冑

1 4⫺共⑀⫹⌬兲 2

册

⫺␭2

冋

冑

1 4⫺⑀ 2_⫹

冑

1 4⫺共⑀⫹⌬兲 2

册

_z Pauli共t兲. 共104兲 The solution of Eq. 共104兲 is given by

zPauli共t兲⫽zPauli共⬁兲⫹关zPauli共0兲⫺zPauli共⬁兲兴e⫺␥Paulit, 共105兲

with the asymptotic equilibrium value

zPauli共⬁兲⫽

冑

1 4⫺⑀ 2_⫺

冑

1 4⫺共⑀⫹⌬兲 2

冑

1 4⫺⑀ 2_⫹

冑

1 4⫺共⑀⫹⌬兲 2 , 共106兲

and the relaxation rate

␥Pauli⫽␭2

冋

冑

1 4⫺⑀ 2_⫹

冑

1 4⫺共⑀⫹⌬兲 2

册

_{. 共107兲}

With the convention 共89兲, the expressions are nonvanishing only over the interval of definition of their argument. Figure 4 helps us to represent the different values that take Eqs. 共106兲 and 共107兲 in the space of the environment energy⑀and of the splitting energy ⌬ of the two-level system.

B. Using the Redfield equation

For the spin-GORM model, the correlation function 共46兲 can be calculated by performing a GOE average. Using the

level density 共88兲 and the value 共93兲, the microcanonical correlation function共50兲 becomes

␣共␶,⑀兲⫽

冕

d⑀

⬘

n共⑀

⬘

兲F共⑀,⑀

⬘

兲ei(⑀⫺⑀⬘)␶ ⯝

冕

⫺1/2 ⫹1/2 d⑀

⬘

4N ␲

冑

1 4⫺⑀

⬘

2 1 8Ne i(⑀⫺⑀⬘)␶ ⫽ J1

冉

␶ 2

冊

4␶ e i⑀␶_{, 共108兲}

in the limit N→⬁, where J1(t) is the Bessel function of the

first kind.

Therefore, using the Redfield equation共72兲–共74兲 and the microcanonical correlation function of the spin-GORM model, we get ␩˙_{⫹⫹共t兲⫽⫺␭}2_{␩⫹⫹共t兲}

冕

0 t d␶cos关共⑀⫹⌬兲␶兴 J1

冉

␶ 2

冊

2␶ ⫹␭2_␩ ⫺⫺共t兲

冕

0 t d␶cos关共⑀⫺⌬兲␶兴 J₁

冉

␶ 2

冊

2␶ , 共109兲 ␩˙_{⫺⫺共t兲⫽⫺␭}2_{␩⫺⫺共t兲}

冕

0 t d␶cos关共⑀⫺⌬兲␶兴 J1

冉

␶ 2

冊

2␶ ⫹␭2␩ ⫹⫹共t兲

冕

0 t d␶cos关共⑀⫹⌬兲␶兴 J1

冉

␶ 2

冊

2␶ , 共110兲 ␩˙ ⫹⫺共t兲⫽⫺i⌬␩⫹⫺共t兲⫺␭2␩⫹⫺共t兲

冕

0 t d␶ei⌬␶cos共⑀␶兲 J₁

冉

␶ 2

冊

2␶ ⫹␭2_{␩⫺⫹共t兲}

冕

0 t d␶e⫺i⌬␶cos共⑀␶兲 J1

冉

␶ 2

冊

2␶ . 共111兲

These are the non-Markovian Redfield equations for the spin-GORM model. The Markovian Redfield equations for the spin-GORM model take the following forms:

␩˙ ⫹⫹共t兲⫽⫺␭2

冑

1 4⫺共⑀⫹⌬兲 2_␩ ⫹⫹共t兲 ⫹␭2

冑

1 4⫺共⑀⫺⌬兲 2_{␩⫺⫺共t兲, 共112兲} ␩˙_{⫺⫺共t兲⫽⫺␭}2

冑

1 4⫺共⑀⫺⌬兲 2_{␩⫺⫺共t兲} ⫹␭2

冑

1 4⫺共⑀⫹⌬兲 2_{␩⫹⫹共t兲, 共113兲} ␩˙_{⫹⫺共t兲⫽⫺i⌬␩⫹⫺共t兲⫹i}␭ 2 ␲

冕

d⑀

⬘

冑

1 4⫺⑀

⬘

2 ⫻P ⌬ 共⑀⫺⑀

⬘

兲2_⫺⌬2关␩⫹⫺共t兲⫹␩⫺⫹共t兲兴 ⫺␭ 2 2

冋

冑

1 4⫺共⑀⫹⌬兲 2_⫹

冑

1 4⫺共⑀⫺⌬兲 2

册

⫻关␩⫹⫺共t兲⫺␩⫺⫹共t兲兴. 共114兲 We focus on the evolution of the populations. The popu-lation of the two-level system is controlled by the z compo-nent of the spin by Eq. 共78兲. According to the Markovian Redfield equations共112兲 and 共113兲, the time evolution of the

z component is given by Eq.共80兲 with the equilibrium value FIG. 4. Schematic representation of the different regimes of the

spin-GORM model for situations where the initial state of the spin is s⫽⫹1, in the plane of the environment energy⑀ versus the spin energy splitting ⌬. The different regions correspond to different values of the functions n(⑀), n(⑀⫹⌬), and n(⑀⫺⌬), where n(E) denotes the DOS defined by the semicircular law共88兲. One can take a value of the environment energy anywhere between ⑀⫽⫺1₂ and ⑀⫽1

2, where n(⑀)⫽0. In region 1: n(⑀⫹⌬)⫽0 and n(⑀⫺⌬)⫽0.