Quantum master equation for the microcanonical ensemble

Quantum master equation for the microcanonical ensemble

Massimiliano Esposito and Pierre Gaspard

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

共Received 8 May 2007; published 25 October 2007兲

By using projection superoperators, we present a new derivation of the quantum master equation first obtained by the authors in Phys. Rev. E 68, 066112共2003兲. We show that this equation describes the dynamics of a subsystem weakly interacting with an environment of finite heat capacity and initially described by a microcanonical distribution. After applying the rotating wave approximation to the equation, we show that the subsystem dynamics preserves the energy of the total system共subsystem plus environment兲 and tends towards an equilibrium state which corresponds to equipartition inside the energy shell of the total system. For infinite heat capacity environments, this equation reduces to the Redfield master equation for a subsystem interacting with a thermostat. These results should be of particular interest to describe relaxation and decoherence in nanosystems where the environment can have a finite number of degrees of freedom and the equivalence between the microcanonical and the canonical ensembles is thus not always guaranteed.

DOI:10.1103/PhysRevE.76.041134 PACS number共s兲: 05.30.⫺d, 03.65.Yz, 76.20.⫹q

I. INTRODUCTION

Irreversible processes do not occur in isolated quantum systems with a finite number of quantum levels. In order to understand relaxation toward equilibrium in these systems, one needs to take into account the effect of their interaction with their environment. The usual way to proceed is to con-sider a total Hamiltonian system composed of a subsystem coupled to an environment. The subsystem dynamics is then described by the reduced density matrix of the subsystem which is obtained by tracing out the degrees of freedom of the environment from the density matrix of the total system. In order to drive the subsystem into an effective irreversible process over reasonably long-time scales共of the order of the Heisenberg time scale of the environment兲 on need to as-sume a quasi-continuous environment spectrum. This is typi-cally valid when the energy spacing between the unperturbed quantum levels of the total system which are coupled to-gether by the subsystem-environment interaction is suffi-ciently small to make the interaction between these levels effective enough to “mix” them关1兴. Once this condition is satisfied, the generic way to obtain a closed master equation for the reduced density matrix of the subsystem is to use perturbation theory and the Born-Markov approximation which implicitly rely on the assumption that the environment has an infinite heat capacity and cannot be affected by the system dynamics. The environment thus plays the role of a thermostat for the subsystem and the descriptions in the ca-nonical and microcaca-nonical ensembles are equivalent 关2兴. The resulting quantum master equation is called the Redfield equation 关3–7兴. Since the Redfield equation can break the positivity of the subsystem density matrix it is sometimes simplified further by time averaging the equation in the in-teraction representation共rotating wave approximation兲 in or-der to get a master equation of Lindblad form which pre-serves positivity关8–12兴.

In the present paper, we consider situations where the environment has a quasicontinuous spectrum but a finite heat capacity. This means that the energy quanta of the subsystem

may significantly affect the microcanonical temperature of the environment so that the equivalence between the micro-canonical and micro-canonical statistical ensembles is compro-mised and only the microcanonical ensemble can be used a priori. Such situations should be generic since the density of state of a system growth exponentially with the number of degrees of freedom whereas the heat capacity only growth linearly. This means that there exists a range in the number of degrees of freedom where the quasicontinuous assumption can be satisfied without necessarily implying an infinite heat capacity. This domain where kinetic processes can occur in the subsystem but for which the usual master equations fail should be of particular importance in nanosystems where the number of degrees of freedom constituting the environment is not always large enough to be supposed infinite. In this sense, the present work can be viewed as a contribution to the recent attempts to apply statistical physics to small sys-tems 关13–18兴. Our results should also be of relevance for systems which display negative heat capacities in the ther-modynamic limit and to which the microcanonical ensemble description applies. A well known example is provided by coupled spin systems because their spectra remain bounded and forms bands of finite extension 关19–22兴. Other known examples of such systems are reported in Refs.关23–26兴.

II. NEW DERIVATION

In this section, with the help of projection superoperators, we give a different derivation of the quantum master equa-tion first derived in Ref.关27兴.

We consider the dynamics of a system with a Hamiltonian of the form

H = H0+␭V, 共1兲

where␭ is a small dimensionless parameter. The density ma-trix␳共t兲 of this total system obeys the von Neumann equa-tion ␳˙共t兲 = L␳共t兲 = − i ប关H,␳共t兲兴 = 共L0+␭L

⬘

兲␳共t兲 = − i ប关H0,␳共t兲兴 − ␭ i ប关V,␳共t兲兴. 共2兲

The solution of the von Neumann equation reads

␳共t兲 = U共t兲␳共0兲 = eLt_{␳共0兲. 共3兲}

In the interaction representation where

␳I共t兲 = e−L0t␳共t兲 = e共i/ប兲H0t␳共t兲e−共i/ប兲H0t, 共4兲

LI

⬘

共t兲 = e−L0

t_L

_⬘

_共t兲eL₀t

, 共5兲

the von Neumann equation takes the simple form

␳˙I共t兲 = ␭LI

⬘

共t兲␳I共t兲 = − ␭

i

ប关VI共t兲,␳I共t兲兴. 共6兲

By integrating Eq.共6兲 and truncating it to order ␭2_{, we get} the perturbative expansion

␳I共t兲 = W共t兲␳共0兲 = e−L0teLt␳共0兲 =关W₀共t兲 + ␭W₁共t兲 + ␭2W₂共t兲 + O共␭3兲兴␳共0兲, 共7兲 where W0共t兲 = I, W1共t兲 =

冕

0 t dTLI

⬘

共T兲, W2共t兲 =

冕

0 t dT

冕

0 T d␶LI

⬘

共T兲LI

⬘

共T −␶兲. 共8兲

The inverse ofW共t兲 reads W−1_{共t兲 = W}

0共t兲 − ␭W1共t兲 + ␭2关W12共t兲 − W2共t兲兴 + O共␭3兲. 共9兲 Indeed, one can check thatW共t兲W−1共t兲=I+O共␭3兲. For later purpose, we also notice that

W˙ 共t兲AW−1_{共t兲 = ␭W˙}

1共t兲A + ␭2关W˙2共t兲A − W˙1共t兲AW1共t兲兴

+ O共␭3_{兲. 共10兲}

Now, we consider a subsystem S interacting with its en-vironment B. The Hamiltonian of this system is given by Eq. 共1兲 where

H0= HS+ HB, V =

兺

␬ S ␬_B

␬, 共11兲

S␬共B_␬兲 is a coupling operator of system S 共B兲. We will use the index s共b兲 to label the eigenstates of the Hamiltonian of system S共B兲.

We will use Liouville space where operators are mapped into vectors and superoperators into matrices关28兴. We recall some basic definitions

scalar product:具具A兩B典典 ⬅ trA†B, 共12兲 identity:I ⬅

兺

n,n⬘

兩nn

⬘

典典具具nn

⬘

兩, 共13兲 兩nn

⬘

典典 ↔ 兩n典具n

⬘

兩, 具具nn

⬘

兩 ↔ 兩n

⬘

典具n兩. 共14兲 Useful consequences of these definitions are

具具nn

⬘

兩n¯n¯

⬘

典典 =␦nn¯␦n⬘¯n⬘ 共15兲

具具nn

⬘

兩A典典 = 具n兩A兩n

⬘

典. 共16兲 An operator A in the total space which can be written as a product of a system and reservoir operator A = AS丢AB will

read in Liouville space兩A典典=兩AS典典丢兩AB典典.

We now define the following projection superoperators which act on the Liouville space of system B

P =

兺

b 兩bb典典具具bb兩, 共17兲 Q =

兺

b,b⬘ 兩bb

⬘

典典具具bb

⬘

兩共1 −␦bb⬘兲, 共18兲

and which satisfy the usual properties of projection superop-eratorsP+Q=IB,P2=P, Q2= Q, andPQ=QP=0. P, when

acting on the density matrix of the total system, eliminates the environment coherences but keeps the environment populations and leaves unaffected the subsystem degrees of freedom. Similar projection superoperators have been re-cently used in Refs.关30–33兴. For comparison, the projection superoperator used to derive the Redfield equation reads

PRed=

兺

b

兩␳eq_{典典具具bb兩, 共19兲}

where␳eq _{is the equilibrium density matrix of the} environ-ment. The two projection superoperators act differently on the density matrix ␳共t兲. In the Liouville space of the total system, we get, respectively,

P兩␳共t兲典典 =

兺

b

具具bb兩␳共t兲典典丢兩bb典典, 共20兲 PRed兩␳共t兲典典 = 兩␳S共t兲典典丢兩␳eq典典. 共21兲

the system state with the environment state. On the contrary, PRed assumes that the system reduced density matrix ␳S共t兲

⬅兺b具具bb兩␳共t兲典典=trB␳共t兲 is independent from the

environ-ment state which always remain at equilibrium.

We now letP and Q act on the density matrix of the total system in the interaction form共7兲 and find

P兩␳I共t兲典典 = PW共t兲共P + Q兲兩␳I共0兲典典 共22兲

Q兩␳I共t兲典典 = QW共t兲共P + Q兲兩␳I共0兲典典. 共23兲

From now on, we will consider initial conditions such that Q兩␳共0兲典典=0. This means that the environment part of the initial condition is diagonal in the environment eigenbasis and is thus invariant under the evolution when␭=0. Taking the time derivative of Eq. 共22兲 and Eq. 共23兲 and using 兩␳I共0兲典典=W−1共t兲兩␳I共t兲典典, we get

P兩␳˙I共t兲典典 = PW˙ 共t兲PW−1共t兲P兩␳I共t兲典典

+PW˙ 共t兲PW−1共t兲Q兩␳I共t兲典典, 共24兲

Q兩␳˙I共t兲典典 = QW˙ 共t兲PW−1共t兲P兩␳I共t兲典典

+QW˙ 共t兲PW−1共t兲Q兩␳I共t兲典典. 共25兲

These equations are still exact. If we restrict ourselves to second-order perturbation theory, we can obtain the impor-tant result that the P projected density matrix evolution is decoupled from theQ projected part. Indeed, with the help of Eq.共10兲, we have

PW˙ 共t兲PW−1_{共t兲Q = ␭PW˙}

1共t兲PQ + ␭2PW˙ 2共t兲PQ −␭2PW˙1共t兲PW1共t兲Q + O共␭3兲.

共26兲 The two first terms of the right-hand side are zero because PQ=0 and the third one also because

PW˙ 1共t兲P =

兺

b,b⬘ 兩bb典典具具bb兩LI

⬘

共t兲兩b

⬘

b

⬘

典典具具b

⬘

b

⬘

兩 = − i ប

兺

_␬ S␬共t兲_b,b

兺

_⬘兩bb典典具b兩关B␬共t兲,兩b

⬘

典具b

⬘

兩兴兩b典具具b

⬘

b

⬘

兩, 共27兲 where具b兩关B_␬共t兲,兩b

⬘

典具b

⬘

兩兴兩b典=0.

After having showed that the relevant projected density matrix evolves in an autonomous way, we will now evaluate the generator of its evolution using second-order perturbation theory. Again using Eq.共10兲, we find that

PW˙ 共t兲PW−1_{共t兲P = ␭PW}˙

1共t兲P + ␭2PW˙2共t兲P −␭2_PW˙

1共t兲PW1共t兲P + O共␭3兲. 共28兲 The only term of right-hand side which is not zero is the second one关see Eq. 共27兲兴 whereupon we get

P兩␳˙I共t兲典典 = ␭2P

冕

0

t

d␶LI

⬘

共t兲LI

⬘

共t −␶兲P兩␳I共t兲典典 + O共␭3兲.

共29兲 Now leaving the interaction representation and using the fact thatPe−L0t_{= e}−LStP, we obtain P兩␳˙共t兲典典 = LSP兩␳共t兲典典 + ␭2eLStP

冕

0 t d␶LI

⬘

共t兲LI

⬘

共t −␶兲 ⫻e−LStP兩␳共t兲典典 + O共␭3兲. 共30兲

Now, we define the quantity P共Eb, t兲 that will become the

fundamental quantity of our theory P共Eb,t兲

n共Eb兲

⬅ 具具bb兩P兩␳共t兲典典, 共31兲 where n共Eb兲=trB␦共Eb− HB兲 is the density of state of the

en-vironment. Equation共30兲 can thus be rewritten as P˙ 共Eb,t兲 n共Eb兲 =LS P共Eb,t兲 n共Eb兲 +␭2

兺

b⬘

冕

0 t d␶ ⫻ eLSt具具bb兩L I

⬘

共t兲LI

⬘

共t −␶兲兩b

⬘

b

⬘

典典e−LS tP共Eb⬘,t兲 n共Eb⬘兲 , 共32兲 where we stop explicitly writing +O共␭3_{兲 from now on.} Equa-tion共32兲 can be evaluated further and we get

P˙ 共Eb,t兲 n共Eb兲 =LS P共Eb,t兲 n共Eb兲 −␭ 2 ប2

兺

␬␬⬘

兺

b⬘

冕

0 t d␶e−共i/ប兲HSt ⫻具b兩

冋

S␬共t兲B_␬共t兲,

冋

S␬⬘共t −␶兲B_␬⬘共t −␶兲, e共i/ប兲HStP共Eb⬘,t兲 n共Eb⬘兲

e−共i/ប兲HSt兩b

⬘

典具b

⬘

兩

册册

兩b典e共i/ប兲HSt_.

+ e−i␻bb⬘␶具b兩B

␬_⬘兩b

⬘

典具b

⬘

兩B␬兩b典P共E_n共Eb,t兲

b兲

S␬⬘共−␶兲S␬

冎

, 共34兲 where ␻bb⬘=共Eb− Eb⬘兲/ប are the Bohr frequencies of the

environment.

We assume now that the spectra of the environment is dense enough to be treated as quasicontinuum so that we can use the following equivalences

P共Eb,t兲 = n共Eb兲trB兩b典具b兩␳共t兲 → P共⑀,t兲 = trB␦共⑀− HB兲␳共t兲,

兺

b⬘

→

冕

d⑀

⬘

n共⑀

⬘

兲, n共Eb兲 → n共⑀兲, 具b兩B兩b

⬘

典 → 具⑀兩B兩⑀

⬘

典.

共35兲 The non-Markovian version of our master equation for envi-ronments with a quasicontinuous spectrum is therefore

P˙ 共⑀,t兲 = LSP共⑀,t兲 − ␭2 ប2

兺

␬␬⬘

冕

d⑀

⬘

冕

0 t d␶ ⫻兵+ ei共⑀−⑀⬘兲␶/ប_n共⑀

_⬘

_{兲具⑀兩B} ␬兩⑀

⬘

典具⑀

⬘

兩B␬_⬘兩⑀典S␬ ⫻S␬_{⬘共−␶兲P共⑀,t兲} − ei共⑀−⑀⬘兲␶/បn共⑀兲具⑀兩B_␬兩⑀

⬘

典具⑀

⬘

兩B_␬⬘兩⑀典S␬P共⑀

⬘

,t兲S␬⬘共−␶兲 − e−i共⑀−⑀⬘兲␶/បn共⑀兲具⑀兩B_␬⬘兩⑀

⬘

典具⑀

⬘

兩B_␬兩⑀典S␬⬘共−␶兲P共⑀

⬘

,t兲S␬ + e−i共⑀−⑀⬘兲␶/បn共⑀

⬘

兲具⑀兩B_␬⬘兩⑀

⬘

典具⑀

⬘

兩B_␬兩⑀典P共⑀,t兲 ⫻S␬⬘_{共−␶兲S}␬_{其. 共36兲}

This equation was first obtained in Ref.关27兴. Notice that the reduced density matrix of the subsystem is obtained from the quantity P共⑀, t兲 using

␳S共t兲 = trB␳共t兲 =

冕

d⑀P共⑀,t兲. 共37兲

The quantity P共⑀, t兲 can be seen as an environment energy distributed subsystem density matrix. One should also realize that Eq.共36兲 is a closed equation for the P共⑀, t兲’s but cannot be converted without approximations into a closed equation for␳S共t兲 using Eq. 共37兲. This is a first indication that Eq. 共36兲

contains more information than what can be contained in a closed equation for␳S共t兲.

In the environment space, the equilibrium correlation function between two coupling operators B_␬and B_␬⬘is

␣␬␬_⬘共t兲 = trB␳eqe共i/ប兲HBtB␬e−共i/ប兲HBtB␬⬘, 共38兲

where␳eq_{is the equilibrium density matrix of the} environ-ment. If the environment is described by a microcanonical distribution at energy⑀

␳mic

eq _{共⑀兲 =␦共⑀}

− HB兲/n共⑀兲, 共39兲

the correlation function reads

␣␬␬_⬘共⑀,t兲 =

冕

d⑀

⬘

n共⑀

⬘

兲ei共⑀−⑀⬘兲t/ប具⑀兩B␬兩⑀

⬘

典具⑀

⬘

兩B␬_⬘兩⑀典. 共40兲 Its Fourier transform关␣˜共␻兲=兰⬁_−⬁共dt/2␲兲ei␻t_{␣共t兲兴 is}

␣

˜_␬␬⬘共⑀,␻兲 = បn共⑀+ប␻兲具⑀兩B_␬兩⑀+ប␻典具⑀+ប␻兩B_␬⬘兩⑀典. 共41兲 This quantity has a useful physical interpretation.兺_␬␬⬘␣˜_␬␬⬘ is, up to a factorប2/共␲␭2兲, the Fermi golden rule transition rate for the environment in a microcanonical distribution at energy ⑀ to absorb 共emit兲 a quantum of energy ប␻ when submitted to a periodic perturbation兺_␬B_␬cos␻t. Using Eq. 共41兲, we can easily verify the important property

␣

˜_␬␬⬘共⑀,␻兲n共⑀兲 =␣˜_␬⬘␬共⑀+ប␻,−␻兲n共⑀+ប␻兲. 共42兲 We can now rewrite Eq.共36兲 as

P˙ 共⑀,t兲 = LSP共⑀,t兲 + ␭2 ប2

兺

␬␬⬘

冕

0 t d␶

冕

−⬁ ⬁ d␻ ⫻兵− e−i␻␶_␣˜ ␬␬_⬘共⑀,␻兲S␬S␬⬘共−␶兲P共⑀,t兲 − ei␻␶_␣˜ ␬_⬘␬共⑀,␻兲P共⑀,t兲S␬⬘共−␶兲S␬ + e−i␻␶␣˜_␬⬘␬共⑀+ប␻,−␻兲S␬P共⑀+ប␻,t兲S␬⬘共−␶兲 + ei␻␶␣˜_␬␬⬘共⑀+ប␻,−␻兲S␬⬘共−␶兲P共⑀+ប␻,t兲S␬其. 共43兲 This equation is the same non-Markovian master equation as Eq.共36兲 but written in a more compact and intuitive form. It is explicit now that the effect of the environment on the subsystem dynamics only enters the description via the en-vironment microcanonical correlation function. In standard master equations, the same is true but with canonical instead of microcanonical correlation functions. The Markovian ver-sion of this equation is obtained by taking the upper bound of the integral to infinity 兰₀t→兰₀⬁. This approximation is well known in the literature and is justified when the environment correlation function decays on time scales ␶c much shorter

then the fastest time scale of the free subsystem evolution␶S

共typically given by the inverse of the largest subsystem Bohr frequency兲.

A remark concerning the terminology is required at this point. In our terminology, the generator of time evolution is said non-Markovian if it explicitly depends on time whether or not it acts on ␳S or P共⑀兲. An alternative definition 共see

关33兴兲 defines the system dynamics as non-Markovian if it cannot be described by a time-independent generator acting on␳S. With this definition, Eq.共43兲 with the upper bound of

the time integral taken to infinity would describe a non-Markovian dynamics.

III. ROTATING WAVE APPROXIMATION

version of the master equation in the interaction form. When applying this approximation to Eq.共43兲, the equation takes a simple form which allows to prove important results.

The RWA is most commonly used in quantum optic 关7,11,12兴 because the free subsystem dynamics generally evolves on times scales ␶S which are much faster than the

relaxation time scales␶rinduced by the coupling to the

en-vironment. The master equation in the interacting represen-tation evolves then very slowly compared to the Bohr fre-quencies of the subsystem which can therefore be averaged out. In other words, the RWA is justified if the time scale separation ␶cⰆ␶SⰆ␶r exist. In the mathematical physics,

peoples usually refer to this averaging procedure共which is performed after a rescaling of time t

⬘

=␭2_{t兲 as the weak}

cou-pling limit关9,10兴 since the smaller the coupling the longer␶r.

Their motivation is to impose a Lindblad form to the master equation generator in order to ensure the positivity of the subsystem density matrix 关7,10,29兴. The same situation oc-curs in our case. In Ref. 关33兴, Breuer has generalized the Lindblad theory to generators which act on projected total density matrices of the type共20兲 which correlate the system-reservoir dynamics. Equation共43兲 is not of the generalized Lindblad form and could in principle lead to positivity break-down similarly as the Redfield equation关6,34–36兴. The RWA can be used to guaranty that our equation preserves the posi-tivity of the subsystem density matrix. By writing Eq.共43兲 in the interaction representation and by projecting the resulting equation in the subsystem eigenbasis, we get

P˙_ssI_⬘共⑀,t兲 =␭ 2 ប2

兺

␬␬⬘

兺

¯ss¯⬘

冕

0 ⬁ d␶

冕

−⬁ ⬁ d␻

兵

− e−i共␻+␻¯ss¯_⬘兲␶_ei␻ss¯_⬘t␣_˜ ␬␬_⬘共⑀,␻兲Sss␬¯S¯s␬s¯⬘⬘P¯s⬘s⬘ I _共 ⑀,t兲 − ei共␻−␻_¯ss¯_⬘兲␶_ei␻¯ss_⬘t␣_˜ ␬_⬘␬共⑀,␻兲Pss¯ I _共 ⑀,t兲S¯ss␬¯⬘⬘S¯s⬘s⬘ ␬ + e−i共␻+␻¯s⬘s⬘兲␶_ei共␻ss¯+␻¯s⬘s⬘兲t␣_˜ ␬_⬘␬共⑀+ប␻,−␻兲Sss␬¯P¯ss¯⬘ I 共⑀+ប␻,t兲S¯_s⬘s⬘ ␬_⬘ + ei共␻−␻ss¯兲␶_ei共␻ss¯+␻¯s⬘s⬘兲t␣_˜ ␬␬_⬘共⑀+ប␻,−␻兲Sss␬¯⬘ ⫻P¯ss¯_⬘ I _共 ⑀+ប␻,t兲S¯_s⬘s⬘ ␬

其

_{. 共44兲}

The RWA consist in time averaging limT→⬁共1/2T兲兰−T

T

dt the right-hand side of Eq.共44兲 so that P˙ss⬘ I 共⑀,t兲 =␭2 ប2

兺

␬␬_⬘

兺

¯ss¯⬘

冕

0 ⬁ d␶

冕

−⬁ ⬁ d␻

兵

−␦ss¯⬘e−i共␻+␻¯ss兲␶␣˜_␬␬⬘共⑀,␻兲Sss␬¯S¯ss␬⬘Pss⬘ I 共⑀,t兲 −␦¯ss⬘ei共␻−␻s⬘¯s⬘兲␶␣˜␬_⬘␬共⑀,␻兲Pss⬘ I 共⑀,t兲Ss⬘¯s⬘ ␬⬘ _S s ¯⬘s⬘ ␬ +关共1 −␦ss⬘兲␦ss¯␦¯s⬘s⬘+␦ss⬘␦¯s⬘¯s兴e−i共␻+␻¯s⬘s⬘兲␶␣˜_␬⬘_␬共⑀+ប␻,−␻兲Sss␬¯P¯ss¯⬘ I 共⑀+ប␻,t兲S_¯s␬_⬘⬘_s⬘ +关共1 −␦ss⬘兲␦ss¯␦¯s⬘s⬘+␦ss⬘␦¯s⬘¯s兴ei共␻−␻ss¯兲␶␣˜_␬␬⬘共⑀+ប␻,−␻兲Sss␬¯⬘P¯ss¯⬘ I 共⑀+ប␻,t兲S¯_s⬘s⬘ ␬

其

. 共45兲

Using兰₀⬁d␶e±i␻␶=␲␦共␻兲±iP共1/␻兲, we finally get the Markovian version of our master equation in the RWA P˙_ssI _⬘共⑀,t兲 =␭ 2 ប2

兺

␬␬⬘

兺

¯ss¯⬘

再

−␲␦ss¯⬘␣˜_␬␬⬘共⑀,−␻¯ss兲Sss␬¯S¯ss␬⬘Pss⬘ I 共⑀,t兲 −␲␦¯ss⬘␣˜␬_⬘␬共⑀,␻s⬘¯s⬘兲Pss⬘ I 共⑀,t兲S_s␬_⬘⬘¯_s⬘S¯s_⬘s⬘ ␬ +␲关共1 −␦ss⬘兲␦ss¯␦¯s⬘s⬘+␦ss⬘␦¯s⬘¯s兴␣˜␬⬘␬共⑀−ប␻¯s⬘s⬘,␻¯s⬘s⬘兲Sss␬¯P¯ss¯_⬘ I _共 ⑀−ប␻¯s⬘s⬘,t兲S¯s_⬘s⬘ ␬⬘ +␲关共1 −␦ss⬘兲␦ss¯␦¯s_⬘s⬘+␦ss⬘␦¯s_⬘¯s兴␣˜_␬␬⬘共⑀+ប␻ss¯,−␻ss¯兲Sss␬¯⬘P¯ss¯⬘ I 共⑀+ប␻ss¯,t兲S¯_s⬘s⬘ ␬ + i␦ss¯⬘

冕

−⬁ ⬁ d␻P␣˜␬␬⬘共⑀,␻兲 ␻+␻¯ss Sss␬¯S¯ss␬⬘Pss⬘ I _共 ⑀,t兲 − i␦¯ss⬘

冕

−⬁ ⬁ d␻P␣˜␬⬘␬共⑀,␻兲 ␻−␻s⬘¯s_⬘ PI_ss⬘共⑀,t兲S␬_s⬘⬘¯_s⬘S¯s⬘s⬘ ␬ − i关共1 −␦ss⬘兲␦ss¯␦¯s⬘s⬘+␦ss⬘␦¯s⬘¯s兴Sss␬¯

冕

−⬁ ⬁ d␻P␣ ˜_␬⬘␬共⑀+ប␻,−␻兲P¯ss¯⬘ I 共⑀+ប␻,t兲 ␻+␻¯ss⬘ S¯_s⬘s⬘ ␬⬘ + i关共1 −␦ss⬘兲␦ss¯␦¯s⬘s⬘+␦ss⬘␦¯s⬘¯s兴Sss␬¯⬘

冕

−⬁ ⬁ d␻P␣ ˜_␬␬⬘共⑀+ប␻,−␻兲P_¯ss¯⬘ I 共⑀+ប␻,t兲 ␻−␻ss¯ S¯_s⬘s⬘ ␬

冎

_{. 共46兲}

This equation is a central result of this paper. It might look a little complicated but is in fact relatively simple. The first four terms are responsible for the damping of the subsystem and the four last term are small shifts in the Bohr frequencies of the subsystem. The populations共s=s

⬘

兲 evolve independently from the coherences 共s⫽s

⬘

兲 and the coherences evolve indepen-dently for each other following exponentially damped oscillations

P˙ss⬘共⑀,t兲 = 共− ⌫ss⬘− i⍀ss⬘兲Pss⬘共⑀,t兲, 共47兲

⌫ss⬘= ␭2 ប2

兺

␬␬⬘

再

− 2␲␣˜_␬␬⬘共⑀,0兲S_ss␬⬘S_s␬_⬘s⬘+␲

兺

s ¯ 关˜␣_␬␬⬘共⑀,␻ss¯兲S␬ss¯S¯ss␬⬘+␣˜␬⬘␬共⑀,␻s⬘¯s兲Ss␬⬘⬘¯sS¯ss␬_⬘兴

冎

共48兲

and where the modified Bohr frequencies are given by

⍀ss⬘=␻ss⬘− ␭2 ប2

兺

␬␬⬘

兺

¯s

冋

冕

−⬁ ⬁ d␻P␣˜␬␬⬘共⑀,␻兲 ␻+␻¯ss Sss␬¯S¯ss␬⬘ −

冕

−⬁ ⬁ d␻P␣˜␬␬⬘共⑀,␻兲 ␻+␻¯ss_⬘ S_s⬘¯_s ␬ S_¯ss␬⬘_⬘

册

. 共49兲 This equation is local in the energy of the environment. This is not the case of the equation which rules the population dynamics P˙ss共⑀,t兲 = ␭2 ប2

兺

␬␬⬘

兺

¯s

兵

− 2␲␣˜_␬␬⬘共⑀,−␻¯ss兲S␬ss¯S¯ss␬⬘Pss共⑀,t兲 + 2␲␣˜_␬⬘␬共⑀−ប␻¯ss,␻¯ss兲S␬ss¯S¯ss␬⬘P¯ss¯共⑀−ប␻¯ss,t兲

其

. 共50兲 This equation is a kind of Pauli equation for the total system which preserves the unperturbed energy of the total system

具H0典t=

冕

d⑀

兺

s 共Es+⑀兲Pss共⑀,t兲 =

冕

d␧ ␧f共␧兲, 共51兲 where f共␧兲 =

兺

s Pss共␧ − Es,t兲 共52兲

represents the probability distribution inside a given energy shell of energy ␧ of the unperturbed total system. Indeed, using Eq.共50兲, we find

f˙共␧兲 =

兺

s

P˙ss共␧ − Es,t兲 = 0. 共53兲

This shows that the energy of the total system is conserved by the dynamics because the probability is preserved inside each energy shell of the total system. If the initial condition of the total system is a product of a subsystem pure state of energy Eswith a microcanonical distribution at energy⑀0for the environment关P共⑀, 0兲=兩s典具s兩␦共⑀−⑀0兲兴 the energy distribu-tion of the total system corresponds to f共␧兲=␦共␧−Es−⑀0兲. If the subsystem is initially described by a density matrix␳S共0兲,

we get f共␧兲=兺s␳S,ss共0兲␦共␧−Es−⑀0兲. The population dynam-ics independently evolves in each energy shell of the total system. The equilibrium distribution of Eq.共50兲 is

P¯ss¯共⑀−ប␻¯ss,⬁兲 Pss共⑀,⬁兲 = ␣˜␬␬⬘共⑀,−␻¯ss兲 ␣ ˜_␬⬘␬共⑀−ប␻¯ss,␻¯ss兲 . 共54兲

Using Eq.共42兲 and ␧=⑀+ Es, Eq. 共54兲 becomes

P¯ss¯共␧ − E¯s,⬁兲

Pss共␧ − Es,⬁兲

=n共␧ − E¯s兲 n共␧ − Es兲

, 共55兲

which expresses detailed balance at equilibrium. Because f共␧兲 is invariant under the dynamics, we finally get

Pss共␧ − Es,⬁兲 = n共␧ − Es兲

兺

s ¯ n共␧ − E¯s兲 f共␧兲. 共56兲 At equilibrium, each quantum level inside a given energy shell of the total system has the same probability. This means that our equation describes how any initial distribution inside a given energy shell of the total system reaches equilibrium.

IV. EQUIVALENCE BETWEEN THE ENSEMBLES

In this section, we start by showing using a simple quali-tative argument that if the environment is initially described by a canonical distribution and is assumed large enough for not being affected by the subsystem dynamics, our equation reduces to the Redfield master equation. However, the most important part of this section is devoted to the problem of understanding in detail how our master equation共which rules the dynamics of a subsystem interacting with an environment described by a microcanonical distribution兲 effectively re-duces to a Redfield master equation共which rules the dynam-ics of a subsystem interacting with an environment described by a canonical distribution兲 if the equivalence between the microcanonical and the canonical ensemble is satisfied for the environment.

The canonical density matrix of the environment is related to the microcanonical density matrix of the environment by

␳caneq共␤兲 = e−␤HB Z =

冕

d⑀W共⑀兲␳mic eq_{共⑀兲, 共57兲} where W共⑀兲 =e−␤⑀n共⑀兲 Z . 共58兲

We notice that the condition共42兲 found for the microcanoni-cal correlation functions is in fact at the origin of the KMS condition for the canonical correlation functions. Indeed, us-ing Eq.共42兲 and

␣

˜_␬␬⬘共␤,␻兲 =

冕

d⑀W共⑀兲␣˜_␬␬⬘共⑀,␻兲, 共59兲 we obtain the KMS condition

␣

ini-tially described by a canonical distribution and remains in it at any time during its interaction with the subsystem. This assumption can be expressed by the ansatz

P共⑀,t兲 ⬇␳S共t兲

e−␤⑀n共⑀兲

Z . 共61兲

This can be qualitatively justified by assuming that the envi-ronment is very large compared to the subsystem. By inte-grating Eq.共43兲 over the energy of the environment and with the help of Eq. 共37兲, Eq. 共59兲, and Eq. 共61兲, we get the Redfield equation关3–7兴 ␳˙S共t兲 = LS␳S共t兲 + ␭2 ប2

兺

␬␬⬘

冕

0 t d␶

兵

−␣_␬␬⬘共␤,␶兲S␬S␬⬘共−␶兲␳S共t兲 −␣_␬␬* _⬘共␤,␶兲␳S共t兲S␬⬘共−␶兲S␬ +␣_␬␬* _⬘共␤,␶兲S␬␳S共t兲S␬⬘共−␶兲 +␣_␬␬⬘共␤,␶兲S␬⬘共−␶兲␳S共t兲S␬

其

, 共62兲

with␣共␶兲=兰d␻e−i␻␶␣共␻兲. Exactly the same procedure can be used for Eq.共46兲 and we thus get the RWA version of the Redfield equation共also called the weak-coupling-limit mas-ter equation兲 关7,9–12兴. In this last case, one can also show that, by integrating the equilibrium distribution共54兲 over the energy of the environment and by using共61兲 and the KMS condition共60兲, we get

␳¯ss¯ S

␳ss

S = e−␤ប␻¯ss. 共63兲

As expected, using the normalization condition兺¯s␳¯ss¯ S

= 1, we find that the subsystem equilibrium distribution of the RWA form of the Redfield equation is a canonical distribution at the temperature of the environment

␳ss S =e −␤Es ZS , 共64兲 where ZS=兺se−␤Es.

After this qualitative discussion, we now show that the precise condition for the Redfield equation to provide an effective description of our master equation is the equiva-lence between the canonical and microcanonical ensembles. Integrating Eq.共43兲 over the energy of the environment and using Eq.共37兲, we get

␳˙S共t兲 = LS␳S共t兲 + ␭2 ប2

兺

␬␬⬘

冕

0 t d␶

冕

−⬁ ⬁ d␻ ⫻

再

− e−i␻␶

冕

d⑀ ␣˜_␬␬⬘共⑀,␻兲S␬S␬⬘共−␶兲P共⑀,t兲 − ei␻␶

冕

d⑀ ␣˜_␬⬘␬共⑀,␻兲P共⑀,t兲S␬⬘共−␶兲S␬ + e−i␻␶

冕

d⑀ ␣˜_␬⬘␬共⑀,−␻兲S␬P共⑀,t兲S␬⬘共−␶兲 + ei␻␶

冕

d⑀ ␣˜_␬␬⬘共⑀,−␻兲S␬⬘共−␶兲P共⑀,t兲S␬

冎

. 共65兲 In order to close the equation for the subsystem density ma-trix, we have to assume that the microcanonical correlation functions can be put out of the energy integral. This can only be justified if the environment is such that

␣

˜_␬␬⬘共⑀+ប␻S,␻兲 ⬇␣˜␬␬⬘共⑀,␻兲. 共66兲

By doing this, we obtain Eq.共62兲, but where the canonical correlation functions have to be replaced by the microca-nonical correlation function␣˜_␬␬⬘共␤,␻兲→␣˜_␬␬⬘共⑀,␻兲. There-fore, in order to obtain the Redfield equation, we need to further assume that we can identify the microcanonical with the canonical correlation functions of the environment

␣

˜_␬␬⬘共⑀,␻兲 ⬇˜␣_␬␬⬘共␤,␻兲. 共67兲 Let us now find the conditions under which the two assump-tions共66兲 and 共67兲 are valid. The entropy, associated with the microcanonical distribution in an energy shell of the environ-ment of width␦_⑀and corresponding to an energy⑀, is given by

S共⑀兲 ⬅ kBln w = kBln n共⑀兲␦⑀, 共68兲

where w is the complexion number共i.e., the number of avail-able states in the energy shell兲. The microcanonical tempera-ture of the environment associated with this microcanonical entropy is defined as ␤共⑀兲 = 1 kBT共⑀兲 ⬅ 1 kB dS共⑀兲 d⑀ = 1 n共⑀兲 dn共⑀兲 d⑀ . 共69兲 Suppose that the environment is in a microcanonical distri-bution at the energy⑀m. This environment can be effectively

described by a canonical distribution at the temperature 共kB␤兲−1 if W共⑀兲 is a sharp function with its maximum at the

energy ⑀m, which is therefore the most probable energy. In

this case, we can expand ln W共⑀兲 around⑀m. We get

ln W共⑀兲 = ln W共⑀m兲 +

冋

d2 d⑀2ln W共⑀兲

册

_⑀=⑀ m 共⑀−⑀m兲2 2! +

冋

d 3 d⑀3ln W共⑀兲

册

_⑀=⑀ m 共⑀−⑀m兲3 3! + ¯ , 共70兲

since d ln W共⑀m兲/d⑀= 0. Because the energy⑀m corresponds

to the maximum of ln W共⑀兲, the canonical temperature is equal to the microcanonical temperature at⑀m

␤=␤共⑀m兲. 共71兲

Now using the microcanonical heat capacity defined by 1

C共⑀兲⬅ dT共⑀兲

d⑀ , 共72兲

d2 d⑀2 ln W共⑀兲 = d␤共⑀兲 d⑀ = − 1 kBT2共⑀兲C共⑀兲 . 共73兲

We see now that the rational to truncate the series共70兲 is a large and positive heat capacity. This is true for most envi-ronments in the limit of a large number of degrees of free-dom N, since typically C共⑀兲⬃N. We can now rewrite Eq. 共70兲 as W共⑀兲 ⬇ W共⑀m兲exp

冋

− 共⑀−⑀m兲2 2␴2共⑀m兲

册

, 共74兲 where ␴2_共⑀ m兲 = kBT2共⑀m兲C共⑀m兲. 共75兲

This result confirms that ⑀m is the maximum of W共⑀兲, and

also shows that⑀mis its mean value if ␴共⑀m兲 is small

com-pared to the typical energies of the environment. This is true for large N, since⑀/␴共⑀m兲⬃

冑

N. Under these conditions, an

environment described by a microcanonical distribution at the energy ⑀m can be effectively described by a canonical

distribution at the temperature␤共⑀m兲 so that the second

as-sumption共67兲 becomes justified. However, in order to justify the first assumption共66兲, the environment effective canonical temperature has also to remain unchanged under energy shifts of the environment energy of the order of the typical subsystem quantaប␻S

␤共⑀±ប␻S兲 =␤共⑀兲 ±

d␤共⑀兲

d⑀ ប␻S+ ¯ . 共76兲 The condition to have such an isothermal environment is

冏

ប␻S ␤共⑀兲 d␤共⑀兲 d⑀

冏

=

冏

ប␻S T共⑀兲C共⑀兲

冏

Ⰶ 1. 共77兲 This condition is again satisfied when the number of degrees of freedom of the environment becomes large because ប␻S/关T共⑀兲C共⑀兲兴⬃1/N. Our conclusion is that, in the limit of

an infinitely large environment N→⬁, ␴共⑀兲 becomes infi-nitely small compared to typical environment energy scales 共so that we have the equivalence between the microcanonical and canonical ensembles兲 but also becomes infinitely large compared to typical subsystem energy scalesប␻S共so that the

environment is isothermal for the subsystem兲. It is in this limit that the Redfield master equation becomes valid and can provide an effective description of our master equation.

V. CONCLUSIONS

In this paper, we have considered a subsystem interacting with an environment which has a sufficiently large number of degrees of freedom so that its spectrum can be supposed quasicontinuous. As a consequence, this environment can

drive the subsystem into a relaxation process on time scales typically shorter than the Heisenberg time of the environ-ment关tH=បn共⑀兲, where n共⑀兲 is the average density of states

of the environment兴. However, the number of degrees of freedom of this environment may still be too small for the transitions to leave it isothermal. Indeed, the heat capacity is finite and the energy exchanges between the subsystem and the environment can modify the energy distribution of the environment. This is not an unrealistic assumption since the density of states of the environment grows exponentially with the number of degrees of freedom albeit the heat capac-ity only grows linearly. By using projection operators, we derived a master equation governing the relaxation dynamics of such a subsystem. This equation describes the evolution of the subsystem density matrix distributed over the energy of the environment. This allows us to take into account the changes of the environment energy distribution due to its interaction with the subsystem. By performing the rotating wave approximation 共RWA兲 on this master equation, we have been able to show that the subsystem populations evolve independently from the coherences. The coherences decay in the form of exponentially damped oscillations, while the populations obey a kind of Pauli equation for the total system. This equation conserves the energy of the total system and its equilibrium distribution corresponds to the uniform distribution of the probability over the energy shell of the unperturbed total system. Our equation provides a natural representation of the dynamics of a subsystem inter-acting with an environment described by a microcanonical distribution. The Redfield master equation is the usual way to represent the dynamics of a subsystem interacting with an environment described by a canonical distribution and is a closed equation for the density matrix of the subsystem. We have shown that, if the equivalence between the microca-nonical and camicroca-nonical ensembles is satisfied for the environ-ment 共containing infinitely many degrees of freedom兲, our equation reduces to the Redfield master equation. If this equivalence is not satisfied, our equation becomes of crucial importance to correctly describe the subsystem relaxation. In the same sense that the microcanonical ensemble is more fundamental than the canonical ensemble, our master equa-tion is more fundamental then the Redfield equaequa-tions. We believe that this work improves the understanding of kinetic processes in nanosystems where the thermodynamical limit cannot always be taken.

ACKNOWLEDGMENTS

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Equilib-rium Statistical Mechanics, 2nd ed.共Springer, Berlin, 1998兲.

关4兴 A. G. Redfield, IBM J. Res. Dev. 1, 19 共1957兲.

关5兴 N. G. van Kampen, Stochastic Processes in Physics and

Chemistry, 2nd ed.共North-Holland, Amsterdam, 1997兲.

关6兴 R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II:

Nonequilibrium Statistical Mechanics, 2nd ed.共Springer,

Ber-lin, 1998兲.

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