Fluctuation Theorems for Quantum Master Equations

PHYSICAL REVIEW E 73, 046129 ͑2006͒

Fluctuation theorems for quantum master equations

Massimiliano Esposito* and Shaul Mukamel Department of Chemistry, University of California, Irvine, California 92697, USA ͑Received 17 November 2005; published 24 April 2006͒

A quantum fluctuation theorem for a driven quantum subsystem interacting with its environment is derived based solely on the assumption that its reduced density matrix obeys a closed evolution equation—i.e., a quantum master equation ͑QME͒. Quantum trajectories and their associated entropy, heat, and work appear naturally by transforming the QME to a time-dependent Liouville space basis that diagonalizes the instantaneous reduced density matrix of the subsystem. A quantum integral fluctuation theorem, a steady-state fluctuation theorem, and the Jarzynski relation are derived in a similar way as for classical stochastic dynamics.

DOI: 10.1103/PhysRevE.73.046129 PACS number͑s͒: 05.30.Ch, 05.70.Ln, 03.65.Yz

I. INTRODUCTION restricted situations ͓24–27͔. A quantum exchange fluctuation theorem has also been considered in ͓28͔. Some inter- The fluctuation theorems and the Jarzynski relation are esting considerations of the quantum definition of work in some of a handful of powerful results of nonequilibrium sta- the previous studies have been made in ͓29͔. tistical mechanics that hold far from thermodynamic equilib- It should be noted that the dynamics of an isolated rium. Originally derived in the context of classical mechan- ͑whether driven or not͒ quantum system is unitary and its ics ͓1͔, the Jarzynski relation has been subsequently von Neumann entropy is time independent. Therefore, fluc- extended to stochastic dynamics ͓2͔. It relates the distribu- tuation theorems for such closed systems are useful only pro- tion of the work done by a driving force of arbitrary speed on vided one defines some reduced macrovariable dynamics or a system initially at equilibrium ͑nonequilibrium property͒ to some measurement process on the system ͓30͔. the free energy difference between the initial and final equi- The purpose of this paper is to provide a unified deriva- librium states of the system ͑equilibrium property͒. This re- tion for the different quantum fluctuation relations ͑an inte- markable relation has recently been shown to hold for arbi- gral fluctuation theorem, a steady-state fluctuation theorem, ͒ trary coupling strength between the system and environment and the Jarzynski relation . We build upon the unification of ͑see Jarzynski’s reply ͓3͔ to criticism from Cohen and Mau- the different fluctuation relations recently accomplished by ͓ ͔͒ Seifert ͓19͔ for classical stochastic dynamics described by a zerall 4 . The fluctuation theorems are based on a funda- ͑ ͒ mental relation connecting the entropy production of a single birth and death master equation BDME . Quantum evolution involves coherences which make its interpretation in system trajectory to the logarithm of the ratio of the prob- term of trajectories not obvious. Nevertheless, we show that ability of forward and backward trajectories ͓5͔. The en- it is possible to formally develop a trajectory picture of quan- semble average of the trajectory entropy production is the tum dynamics which allows one to uniquely represent en- macroscopic entropy production of the system whereas its tropy, heat, and work distributions. This relies on the single distribution gives rise to various kinds of fluctuation theo- assumption that the reduced dynamics of a driven quantum rems. The first has been derived for classical mechanics and subsystem interacting with its environment is described by a initially for deterministic ͑but non-Hamiltonian͒ thermostat- ͓ ͔ closed evolution equation for the density matrix of the ted systems 6–8 . Some interesting studies of fluctuation subsystem—i.e., a quantum master equation ͑QME͓͒31–34͔. relations valid for far from equilibrium classical Hamiltonian ͓ ͔ However, while the physical quantities defined along classi- systems were made even earlier 9–11 . Fluctuation theorems cal trajectories are conceptually clear and experimentally for systems with stochastic dynamics have also been devel- ͓ ͔ measurable, how to measure the physical quantities associ- oped 12–19 . For classical stochastic dynamics, the connec- ated with quantum trajectories remains an open issue inti- tion between the fluctuation theorem and the Jarzynski rela- ͓ ͔ mately connected to quantum measurement. tion has been established by Crooks 17 . Seifert has recently The plan of the paper is as follows: We start in Sec. II by provided a unified description of the different fluctuation re- defining quantum heat and quantum work for a driven sub- lations and of the Jarzynski relation for classical stochastic ͓ ͔ system interacting with its environment, consistent with ther- processes described by master equations 19 . modynamics. We then discuss the consequences of defining The understanding of these two fundamental relations in heat and work in terms of the time-dependent basis which quantum mechanics is still not fully established. Quantum ͓ ͔ diagonalizes the subsystem density matrix in Sec. III. In Sec. Jarzynski relations have been investigated in 20–23 . Quan- IV, we show that by assuming a QME for the subsystem tum fluctuation theorems have been developed only in a few reduced density matrix we can recast its solution in a representation which takes the form of a BDME with time- dependent rates. In Sec. V, we show that the BDME repre- *Also at Center for Nonlinear Phenomena and Complex Systems, sentation allows us to split the entropy evolution into two Université Libre de Bruxelles, Code Postal 231, Campus Plaine, parts: the entropy flow associated with exchange processes B-1050 Brussels, Belgium. with the environment and the entropy production associated

1539-3755/2006/73͑4͒/046129͑11͒/$23.00046129-1 ©2006 The American Physical Society M. ESPOSITO AND S. MUKAMEL PHYSICAL REVIEW E 73, 046129 ͑2006͒ with subsystem internal irreversible processes. In Sec. VI, t ͑ ͒ϵ͵ ␶ ˆ ͑ ͒␳˙͑ ͒ ͑ ͒ we show that the BDME representation naturally allows one QT t d TrHT t ˆ t . 6 to define quantum trajectories as well as their associated en- 0 tropy flow and production. We then derive the fundamental Using the von Neumann equation ͑2͒ and the invariance of relation of this paper ͓Eq. ͑67͔͒ which will allow us to de- the trace under cyclic permutation ͓35͔, we find that no heat rive, in Sec. VII, a quantum integral fluctuation theorem and, is generated in the isolated total system in Sec. VIII, a quantum steady-state fluctuation theorem. t Having identified in Sec. IX the heat and work associated ͑ ͒ ͵ ␶ ˆ ͑ ͓͒ ˆ ͑ ͒ ␳͑ ͔͒ ͑ ͒ QT t =−i d TrHT t HT t , ˆ t =0. 7 with the quantum trajectories, we show in Sec. X that the 0 fundamental relation of Sec. VI also allows one to derive a quantum Jarzynski relation. We finally draw conclusions in We next turn to the subsystem. Its reduced density matrix is ␴͑ ͒ϵ ␳͑ ͒ Sec. XI. defined as ˆ t TrB ˆ t , and its energy is given by ͗ ˆ ͘ ϵ ˆ ͑ ͒␳͑ ͒ ˆ ͑ ͒␴͑ ͒ ͑ ͒ HS t TrHS t ˆ t =TrSHS t ˆ t . 8 II. AVERAGE HEAT AND WORK The change in this energy between time 0 and t is given by We start by defining the average quantum heat and work t ˆ for a driven subsystem interacting with its environment and d͗H ͘␶ ⌬E ͑t͒ϵ͵ d␶ S = W ͑t͒ + Q ͑t͒, ͑9͒ show the consistency of these definitions with thermodynam- S ␶ S S 0 d ics. Heat and work can be rigorously expressed in terms of the reduced density matrix of the subsystem without having where the work and heat are defined as to refer explicitly to the environment. t t ˆ ͑ ͒ ͑ ͒ϵ͵ ␶ ˆ˙ ͑␶͒␳ ͑␶͒ ͵ ␶ ˆ˙ ͑␶͒␴͑␶͒ ͑ ͒ We consider a driven subsystem with Hamiltonian HS t . WS t d TrHS ˆ T = d TrSHS ˆ , 10 Everywhere in this paper we denote operators with a caret 0 0 ͑and superoperators with two carets͒ and use the Schrödinger picture where the time dependence of the observables is ex- t t ͑ ͒ϵ ␶ ˆ ͑␶͒␳˙ ͑␶͒ ␶ ˆ ͑␶͒␴˙ ͑␶͒ ͑ ͒ plicit and comes exclusively from external driving. We could QS t ͵ d TrHS ˆ T = ͵ d TrSHS ˆ . 11 0 0 ˆ ␭͑ ͒ ␭͑ ͒ also have written HS( t ), where t is the external-time- dependent driving. This subsystem is interacting with its en- Since the time dependence of the total system Hamiltonian ˆ ˆ˙ ˆ˙ vironment whose Hamiltonian is HB. The interaction energy comes solely from the subsystem Hamiltonian, HB =HI =0, between the subsystem and environment is described by Hˆ . ˆ˙ ˆ˙ I HT =HS, and the work done by the driving force on the sub- The Hamiltonian of the total system reads therefore system is the same as the work done by this force on the total system, Hˆ ͑t͒ = Hˆ ͑t͒ + Hˆ + Hˆ . ͑1͒ T S B I ͑ ͒ ͑ ͒ϵ ͑ ͒ ͑ ͒ WT t = WS t W t . 12 We have assumed that the driving acts exclusively on the This also means that the energy increase in the subsystem ˆ ˆ subsystem and does not affect HB and HI. minus the amount of heat which went to the environment is The state of the total system is described by the density equal to the energy increase in the total system: matrix ␳ˆ͑t͒ which obeys the von Neumann equation ͑ ͒ ⌬ ͑ ͒ ⌬ ͑ ͒ ͑ ͒ ͑ ͒ W t = ET t = ES t − QS t . 13 ˆ ␳ˆ˙͑ ͒ ͓ ˆ ͑ ͒ ␳ˆ͑ ͔͒ Lˆ ͑ ͒␳ˆ͑ ͒ ͑ ͒ It should be noticed that due to the absence of heat flux in the t =−i HT t , t = t t . 2 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ total system QT t =0. Using Eq. 6 with Eqs. 1 and 11 , The energy of the total system is given by we can also express the heat going from the subsystem to the environment as

͗ ˆ ͘ ϵ ˆ ͑ ͒␳͑ ͒ ͑ ͒ t t HT t TrHT t ˆ t . 3 d͗Hˆ ͘␶ d͗Hˆ ͘␶ Q ͑t͒ =−͵ d␶ B − ͵ d␶ I . ͑14͒ S d␶ d␶ The change in the total energy between time 0 and t due to 0 0 the time-dependent driving is therefore given by

t III. CALCULATING HEAT AND WORK d͗Hˆ ͘␶ ⌬E ͑t͒ϵ͵ d␶ T = W ͑t͒ + Q ͑t͒, ͑4͒ IN A TIME-DEPENDENT BASIS T ␶ T T 0 d As will become clear in Sec. IV, in order to associate where the work and heat have, respectively, been deﬁned as trajectories with the quantum dynamics, one needs to represent the dynamics in a time-dependent basis. This is a fun- t damental difference from classical thermodynamics where ͑ ͒ϵ͵ ␶ ˆ˙ ͑ ͒␳͑ ͒ ͑ ͒ the basis set ͑coordinate system͒ is ﬁxed. In order to associ- WT t d TrHT t ˆ t , 5 0 ate heat and work with single trajectories they must be de-

046129-2 FLUCTUATION THEOREMS FOR QUANTUM MASTER EQUATIONS PHYSICAL REVIEW E 73, 046129 ͑2006͒

fined with respect to a time-dependent basis set. The en- ͗m ͉␴ˆ˙ ͑t͉͒mЈ͘ = P˙ ͑m͒␦ − ͗m˙ ͉mЈ͘P ͑mЈ͒ − ͗m ͉m˙ Ј͘P ͑m͒. semble average of the quantities defined for the trajectories t t t mmЈ t t t t t t will therefore also depend on this basis set. For this reason, ͑25͒ we introduce a modified definition of heat and work. The Notice also that for m=mЈ, we have effect of the basis time dependence on heat and work is given in Appendix A. ͗m ͉␴ˆ˙ ͑t͉͒m ͘ = P˙ ͑m͒͑26͒ The energy of the total system ͑3͒ can also be written as t t t ͗ ͉ ͘ ͗ ͉ ͘ d ͑͗ ͉ ͒͘ because m˙ t mt + mt m˙ t = mt mt =0. ͗ ˆ ͘ T͑␣͒͗␣ ͉ ˆ ͑ ͉͒␣ ͘ ͑ ͒ dt HT t = ͚ Pt t HT t t , 15 Using Eq. ͑22͒, the change in the subsystem energy be- ␣ tween time 0 and t can be rewritten as where we have introduced the time-dependent basis ͕͉␣ ͖͘ t ⌬ ͑ ͒ ˜ ͑ ͒ ˜ ͑ ͒ ͑ ͒ which diagonalizes the instantaneous density matrix at all ES t = WS t + QS t , 27 times: where the work and heat are defined in analogy to Eqs. ͑18͒ ͑ ͒ ͗␣ ͉␳͑ ͉͒␣Ј͘ ͗␣ ͉␳͑ ͉͒␣ ͘␦ ϵ T͑␣͒␦ ͑ ͒ and 19 as t ˆ t t = t ˆ t t ␣␣Ј Pt ␣␣Ј. 16 t d ˜ ͑ ͒ϵ͵ ␶ ͑ ͒ ͓͗ ͉ ˆ ͑␶͉͒ ͔͘ ͑ ͒ The change in this energy between time 0 and t due to the W t d ͚ P␶ m m␶ H m␶ , 28 S ␶ S time-dependent driving can therefore be rewritten as 0 m d

⌬E ͑t͒ = W˜ ͑t͒ + Q˜ ͑t͒, ͑17͒ t T T T ˜ ͑ ͒ϵ͵ ␶ ˙ ͑ ͒͗ ͉ ˆ ͑␶͉͒ ͘ ͑ ͒ QS t d ͚ P␶ m m␶ HS m␶ . 29 where the modified work and heat have, respectively, been 0 m defined as It is shown in Appendix A that the work and heat defined in t ͕͉ ͖͘ d the time-dependent basis mt are related to the original ˜ ͑ ͒ϵ͵ ␶ T͑␣͒ ͓͗␣ ͉ ˆ ͑␶͉͒␣ ͔͘ ͑ ͒ W t d ͚ P ␶ H ␶ , 18 T ␶ ␶ T work and heat defined in any time-independent basis by 0 ␣ d ˜ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ WS t = WS t + AS t , 30 t ˜ ͑ ͒ϵ͵ ␶ ˙ T͑␣͒͗␣ ͉ ˆ ͑␶͉͒␣ ͘ ͑ ͒ QT t d ͚ P␶ ␶ HT ␶ . 19 ␣ ˜ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ 0 QS t = QS t − AS t , 31 Because the total system is driven but otherwise isolated, its where ͑ ͒ evolution is unitary and we have see Appendix A t ͑ ͒ϵ͵ ␶ ͑ ͓͒͗ ˙ ͉ ˆ ͑␶͉͒ ͘ ͗ ͉ ˆ ͑␶͉͒ ˙ ͔͘ ˜ ͑ ͒ ͑ ͒ ⌬ ͑ ͒ ͑ ͒ AS t d ͚ P␶ m m␶ HS m␶ + m␶ HS m␶ . WT t = WT t = ET t , 20 0 m ͑32͒ Q˜ ͑t͒ = Q ͑t͒ =0. ͑21͒ T T It should be emphasized that both the original and modified This means that defining heat and work on the time- work and heat of the subsystem can be defined exclusively in dependent basis which diagonalizes the instantaneous den- terms of subsystem quantities, without referring explicitly to sity matrix for unitary evolution is equivalent to the original the environment. definition of heat and work in a time-independent basis. Using Eqs. ͑12͒ and ͑13͒ with Eqs. ͑30͒ and ͑31͒,weget The energy of the subsystem ͑9͒ can also be written in ⌬ ͑ ͒ ͑ ͒ ˜ ͑ ͒ ͑ ͒ ⌬ ͑ ͒ ˜ ͑ ͒ ͑ ͒ analogy to Eq. ͑15͒ as ET t = W t = WS t − AS t = ES t − QS t − AS t . ͑33͒ ͗ ˆ ͘ ͑ ͒͗ ͉ ˆ ͑ ͉͒ ͘ ͑ ͒ HS t = ͚ Pt m mt HS t mt , 22 m IV. BDME REPRESENTATION ͕͉ ͖͘ where we have introduced the time-dependent basis mt OF THE QME SOLUTION which diagonalizes the instantaneous subsystem reduced density matrix: In this section we show that if we assume a closed evolution equation for the subsystem reduced density matrix, we ͗ ͉␴͑ ͉͒ Ј͘ ͗ ͉␴͑ ͉͒ ͘␦ ϵ ͑ ͒␦ ͑ ͒ mt ˆ t mt = mt ˆ t mt mmЈ Pt m mmЈ. 23 can recast its solution in a BDME form with time-dependent rates. Let us note for future reference that We assume that the reduced subsystem density matrix ␴ˆ ͑t͒ obeys a closed QME ͓31–34͔. This QME can be derived d ͑͗ ͉␴͑ ͉͒ Ј͒͘ ͗ ͉␴˙ ͑ ͉͒ Ј͘ ͗ ͉␴͑ ͉͒ Ј͘ mt ˆ t mt = mt ˆ t mt + m˙ t ˆ t mt microscopically by perturbation theory like in the Redfield dt theory or using a quantum dynamical semigroup approach ͗ ͉␴͑ ͉͒ Ј͘ ͑ ͒ leading to Lindblad-type master equations. In Liouville + mt ˆ t m˙ . 24 t space ͓36͔, the QME of the externally driven subsystem in- Equations ͑23͒ and ͑24͒ give teracting with its environment reads

046129-3 M. ESPOSITO AND S. MUKAMEL PHYSICAL REVIEW E 73, 046129 ͑2006͒

ˆ environment, the generator is anti-Hermitian and the evolu- ͉␴ˆ˙ ͑t͒͘͘ = Kˆ ͑t͉͒␴ˆ ͑t͒͘͘. ͑34͒ ͉␴͑ ͒͘͘ Mˆ ͉␴͑ ͒͘͘ tion superoperator unitary. In this case ˆ t = t ˆ 0 If the interaction with the environment vanishes, the genera- ͗͗ ͉ ͗͗ ͉Mˆ −1 ˆ ˆ and mtmt = m0m0 t , so that tor Kˆ ͑t͒ becomes the anti-Hermitian superoperator Kˆ ͑t͒ ͑ ͒ ͗ ͉␴͑ ͉͒ ͘ ͑ ͒ ͑ ͒ Lˆ ͑ ͒ ͓ ˆ ͑ ͒ ͔ Mˆ Pt m = m0 ˆ 0 m0 = P0 m . 41 = S t =−i HS t ,· and the evolution superoperator t de- ˆ ͑ ͒ ͉␴͑ ͒͘͘ Mˆ ͉␴͑ ͒͘͘ This shows that the Pt m ’s evolve only if the dynamics is ﬁned by ˆ t = t ˆ 0 becomes the unitary superopera- nonunitary. Mˆ ͕͐t ␶Lˆ ͑␶͖͒ tor t =exp+ 0d S . However, for nonvanishing cou- When there is no driving and the subsystem does interact pling this generator is not anti-Hermitian and leads to a with its environment, the dynamics is nonunitary and the nonunitary evolution. subsystem will reach equilibrium ␴ˆ eq on long time scales. The QME in some given ͑possibly time-dependent͒ basis For an inﬁnite isothermal environment this equilibrium state reads will correspond to the canonical subsystem reduced density ␤ ˆ ␤ ˆ ␴ˆ eq − HS − HS ␤ ͑ ˆ matrix =e /ZS where ZS =Tre and =1/T kB ͗͗iiЈ͉␴ˆ˙ ͑t͒͘͘ = ͚ ͗͗iiЈ͉Kˆ ͑t͉͒jjЈ͗͗͘͘jjЈ͉␴ˆ ͑t͒͘͘, ͑35͒ ϵ1͒. In this case the basis diagonalizing ␴ˆ eq becomes time jjЈ independent and will also diagonalize the subsystem Hamil- ␤ tonian so that Peq͑m͒=e− Em /Z where E are the eigenval- where ͗͗jjЈ͉␴ˆ ͑t͒͘͘ is the superoperator representation of S m ues of the subsystem Hamiltonian. ͗j͉␴ˆ ͑t͉͒jЈ͘. Let us now use the time-dependent basis ͕͉m ͖͘ t For a subsystem with nonequilibrium boundary conditions ͑ ͒ ␴͑ ͒ introduced in Eq. 23 . Since the QME keeps ˆ t Hermitian, and interacting with its environment, the subsystem can this diagonalization is always possible: reach a steady state ␴ˆ st at long times. In this case the matrix ͗͗ Ј͉␴͑ ͒͘͘ ͑ ͒␦ ͑ ͒ diagonalizing the density matrix is again time independent mtmt ˆ t = Pt m m,mЈ. 36 ͑ ͒ st͑ ͒ and both the probabilities Pt m = P m and rates ͑ Ј ͒ st͑ Ј ͒ A crucial property of this basis is that ͓see Eq. ͑26͔͒ Wt m ,m =W m ,m become time independent.

˙ ͑ ͒ ͗͗ ͉␴˙ ͑ ͒͘͘ ͑ ͒ Pt m = mtmt ˆ t . 37 V. ENTROPY FOR QUANTUM ENSEMBLES The consequence of this property is that by defining In this section we define the von Neumann entropy associated with the subsystem and separate its evolution into two ͑ Ј ͒ϵ͗͗ ͉Kˆ ͑ ͉͒ Ј Ј͘͘ ͑ ͒ parts: the entropy flow associated with the heat going from Wt m ,m mtmt t mt mt 38 the subsystem to the environment and the always positive entropy production associated with the internal entropy and by projecting the QME ͑34͒ onto the time-dependent growth of the subsystem. superbra ͗͗m͑t͒m͑t͉͒ we get The von Neumann entropy of the subsystem is defined by ˙ ͑ ͒ ͑ Ј ͒ ͑ Ј͒ ͑ ͒ Pt m = ͚ Wt m ,m Pt m . 39 ͑ ͒ϵ ␴͑ ͒ ␴͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ S t −Trˆ t ln ˆ t =−͚ Pt m ln Pt m . 42 mЈ m Since the QME ͑34͒ preserves probability, we have ͑ ͒ ͚ ͑ ͒ ͑ ͒ Using Eq. 40 , we can write its time derivative as mWt mЈ,m =0 and Wt mЈ,m real. Therefore, we can re- ͑ ͒ write Eq. 39 as ˙͑ ͒ ˙ ͑ ͒ ͑ ͒͑͒ S t =−͚ Pt m ln Pt m 43 m ˙ ͑ ͒ ͕ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͖͒ Pt m = ͚ Wt mЈ,m Pt mЈ − Wt m,mЈ Pt m . mЈm P ͑mЈ͒ ͑40͒ =− ͚ P ͑m͒W ͑m,mЈ͒ln t . ͑44͒ t t P ͑m͒ m,mЈ t Even though this equation appears like a BDME, it should not be viewed as an equation of motion. It is merely a way of In analogy with ͓13,15͔ for classical systems, this can be recasting the solution of the QME ͑34͒ in a diagonal basis. In partitioned as ͑ ͒ ͑ ͒ fact, in order to get the Pt m ’s and Wt mЈ,m ’s, we need to ˙͑ ͒ ˙ ͑ ͒ ˙ ͑ ͒ ͑ ͒ solve the QME first and find the time-dependent unitary S t = Se t + Si t , 45 transformation diagonalizing the solution ␴ˆ ͑t͒ at any time. Equation ͑39͒ should therefore be viewed as a formal defi- where nition of the rate matrix W ͑mЈ,m͒. We will show that t ͑ ͒ ͑ Ј ͒ Wt m,mЈ Wt m ,m defined in this way can be used to derive quantum S˙ ͑t͒ϵ− ͚ P ͑m͒W ͑m,mЈ͒ln ͑46͒ ͑ Ј ͒ e t t W ͑mЈ,m͒ fluctuation relations. Note that Wt m ,m depends on the m,mЈ t subsystem initial condition ␴ˆ ͑0͒. If the subsystem ͑driven or not͒ does not interact with the and

046129-4 FLUCTUATION THEOREMS FOR QUANTUM MASTER EQUATIONS PHYSICAL REVIEW E 73, 046129 ͑2006͒

P ͑m͒W ͑m,mЈ͒ formations the probability distribution remains Gibbsian ˙ ͑ ͒ϵ ͑ ͒ ͑ Ј͒ t t ͑ ͒ Si t ͚ Pt m Wt m,m ln . 47 ˙ ͑ ͒ P ͑mЈ͒W ͑mЈ,m͒ along the adiabatic levels. Because Si t =0, we also have m,mЈ t t S˙͑t͒=S˙ ͑t͒. Using Eq. ͑48͒, this means that for a reversible ͑ ͒ ͑ ͒ജ e As a consequence of the inequality R1 −R2 ln R1 /R2 0, transformation the change in the entropy of the subsystem ˙ ͑ ͒ജ we notice that Si t 0 is always a positive quantity. We will results exclusively from the heat flow to the environment, therefore identify it with the entropy production. The remain- ˙ S˙͑t͒=Q˜ /T, consistent with thermodynamics. ˙ ͑ ͒ S ing part of the entropy Se t is thus associated with the en- When there is no driving, Eq. ͑51͒ with Eq. ͑48͒ defines tropy flow to the environment since in thermodynamics the equilibrium. At equilibrium we have S˙͑t͒=S˙ ͑t͒=S˙ ͑t͒=0. entropy evolution is partitioned in the ͑reversible͒ entropy i e flow to the environment and the ͑irreversible͒ entropy production ͓37,38͔. To further rationalize this identification, let VI. ENTROPY FOR QUANTUM TRAJECTORIES ͑ ͒ us assume that Wt m,mЈ satisfy the detailed balance condition ͓17,38,39͔. For isothermal environments at temperature In this section we introduce quantum trajectories and dis- T, the detailed balance condition with respect to Hˆ ͑t͒, which tributions. We will associate an entropy with these trajecto- s ries and identify the entropy flow and production of these means that the nondriven subsystem tends to thermal equi- trajectories, whose ensemble averages recover the entropies librium at long time, reads discussed in Sec. V. This will allow us to derive a fundamen- ͑ ͒ Wt m,mЈ ␤͑͗ ͉ ˆ ͑ ͉͒ ͘ ͗ Ј͉ ˆ ͑ ͉͒ Ј͒͘ tal quantum relation similar to the classical relation obtained = e mt HS t mt − mt HS t mt . ͑48͒ ͑ ͒ by Crooks ͓17͔ and Seifert ͓19͔ connecting the ratio of the Wt mЈ,m probability of a forward trajectory and the backward one Noticing that the heat ͑29͒ can be rewritten as with the trajectory entropy production. ͑ ͒ ˙ From Eq. 40 it seems natural to unravel the evolution ˜ ͑ ͒ ͚ ˙ ͑ ͒͗ ͉ ˆ ͑ ͉͒ ͘ ͑ ͒ Q t = Pt m mt HS t mt equation for the probability Pt m in the same way as is done m for classical stochastic processes ͓19͔. Let us consider a sto- =− ͚ P ͑m͒W ͑m,mЈ͓͒͗m ͉Hˆ ͑t͉͒m ͘ − ͗mЈ͉Hˆ ͑t͉͒mЈ͔͘ chastic trajectory of duration t which contains N jumps. t t t S t t S t Different trajectories can of course have a different number m,mЈ of jumps N. ␶=͓0,t͔ labels time during the process. ␤͗m ͉Hˆ ͑t͉͒m ͘ e t S t j=1,...,N labels the jumps. The trajectory n͑␶͒ ͑see Fig. 1͒ is ͑ ͒ ͑ ͒ ͑ ͒ =−T ͚ Pt m Wt m,mЈ ln 49 ␤͗mЈ͉Hˆ ͑t͉͒mЈ͘ made by the successive states taken by the system in time: m,mЈ e t S t ͑ ͒ ͑ ͒ → → → → ͑ ͒ and using Eq. 46 , the immediate consequence of Eq. 48 is n͑␶͒ = n0 n1 n2 ¯ nN. 53 that the entropy flow is equal to the modified heat going from the subsystem to the environment divided by the environ- ␶ The system starts in n0, jumps at time j from nj−1 to nj, and ment temperature as expected from thermodynamics: ␶ ␶ ends up at time t in nN. We will denote 0 =0 and N+1=t. ˙ The entropy associated with the trajectory n͑␶͒ reads Q˜ ˙ ͑ ͒ S ͑ ͒ Se t = . 50 T s͑␶͒ϵ−lnP␶͑n͑␶͒͒, ͑54͒ This motivates our partition of the entropy ͑45͒ and the definition of the modified heat in Sec. III. where P␶͑n͑␶͒͒ is the solution of Eq. ͑40͒ for an initial con- ͑ ͒ We can further show that the entropy flow is associated dition P0 n0 , evaluated along the trajectory n͑␶͒. with reversible entropy variations. In a thermodynamic The time derivative of this trajectory entropy, sense, a reversible transformation is a one during which the ˙ ͑ ͒ N ͒ ␶P␶͑n͒ P␶͑nץ -entropy production is zero, Si t =0. This property holds pro s˙͑␶͒ =−ͯ ͯ − ͚ ␦͑␶ − ␶ ͒ln j , ͑55͒ vided the following condition is satisfied ͓see Eq. ͑47͔͒: ͑ ͒ j ͑ ͒ P␶ n n͑␶͒ j=1 P␶ nj−1 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ Pt m Wt m,mЈ = Pt mЈ Wt mЈ,m . 51 Using Eq. ͑48͒, we find that for a reversible transformation will be partitioned as the subsystem has to be in the time-dependent state ˙͑␶͒ ˙ ͑␶͒ ˙ ͑␶͒ ͑ ͒ ␤͗ ͉ ˆ ͑ ͉͒ ͘ s = se + si , 56 e− mt HS t mt P ͑m͒ = . ͑52͒ t ␤͗ ͉ ˆ ͑ ͉͒ ͘ ͚ e− mt HS t mt where the trajectory entropy flux reads m N W␶͑n ,n ͒ This state correspond to the instantaneous Gibbs state of the s˙ ͑␶͒ϵ− ͚ ␦͑␶ − ␶ ͒ln j−1 j ͑57͒ ␤ ˆ ͑ ͒ e j ␴͑ ͒ − HS t ͕͉ ͘ ͖ ͑ ͒ W␶͑n ,n ͒ subsystem ˆ t =e /ZS. In this case m t in Eq. 51 j=1 j j−1 becomes the adiabatic basis ͑basis diagonalizing the subsystem Hamiltonian͒. We thus show that for reversible trans- and the trajectory entropy production reads

046129-5 M. ESPOSITO AND S. MUKAMEL PHYSICAL REVIEW E 73, 046129 ͑2006͒

FIG. 1. Representation of a quantum forward trajectory n͑␶͒ and of the associated backward trajectory ˜n͑˜␶͒.

-P␶͑n͒ W˜ ␶͑m,mЈ͒=W ␶͑m,mЈ͒. The time dependence of the quan ץ s˙ ͑␶͒ϵ−ͯ t ͯ t− i ͑ ͒ tum transition matrix W ͑m,mЈ͒ does not come exclusively P␶ n n͑␶͒ t from the external driving force and is different for different N P␶͑n ͒W␶͑n ,n ͒ initial conditions of the subsystem ␴ˆ ͑0͒. Therefore, reversing ␦͑␶ ␶ ͒ j j j−1 ͑ ͒ − ͚ − j ln . 58 the driving protocol does not simply reverse the time depen- P␶͑n ͒W␶͑n ,n ͒ j=1 j−1 j−1 j dence of the transition matrix. In the quantum case, we de- The ensemble average over trajectories is carried out by us- fine the backward process by taking ␶→t–␶ everywhere in ͑ ͒ ͑ ͒ ͑ ing the probability P␶ nj−1 W␶ nj−1,nj that a transition oc- the QME the protocol is reversed and a minus sign appears ␶ ͑ ͒ ͒ curs at time j between nj−1 and nj. We get in the left hand side of Eq. 34 due to the time derivative . This has the effect of reversing the time dependence of the ˙͑␶͒ ͗ ͑␶͒͘ ͑ ͒ ͑ ͒ S = s˙ , 59 quantum transition matrix Wt m,mЈ . This definition will allow us to derive important fluctuation relations in next sec- S˙ ͑␶͒ = ͗s˙ ͑␶͒͘, ͑60͒ tions. e e Let us consider a new dynamics in the time interval ˜␶ =͓0,t͔ obeying ˙ ͑␶͒ ͗ ͑␶͒͘ ͑ ͒ Si = s˙i . 61 ˜˙ ˜ ˜ The probability of a forward trajectory n͑␶͒ starting at time 0 P˜␶͑m͒ = ͚ W˜␶͑m˜ Ј,m˜ ͒P˜␶͑m˜ Ј͒, ͑63͒ and ending at time t is given by m˜ Ј

N ␶ j where the rates are related to the previous rates in the fol- ␮ ͓ ͔ ͑ ͒ͫ͟ ͵ ␶Ј͚ ͑ ͒ n͑␶͒ = P n expͩ− d W␶ n ,m ͪ F 0 0 Ј j−1 lowing way: ␶ j=1 j−1 m t W˜ ␶͑m˜ Ј,m˜ ͒ = W ␶͑m˜ Ј,m˜ ͒. ͑64͒ ϫ ͑ ͒ͬ ͵ ␶Ј͚ ͑ ͒ ˜ t−˜ W␶ n ,n expͩ− d W␶ n ,m ͪ. j−1 j Ј N j ␶ N m The backward dynamics may have an arbitrary initial condi- ͑62͒ ˜ ͑ ͒ tion P0 ñ0 . We now define the following trajectory for this The exponentials represent the probabilities to stay in a given backward dynamics: state during the time interval between two successive jumps, → → → → and the transition rates evaluated at the jump times give the ñ͑˜␶͒ = ñ0 ñ1 ñ2 ¯ ñN probability for the jumps to occur at these times. = n → n → n → → n , ͑65͒ Defining the backward process as done for classical sto- N N−1 N−2 ¯ 0 chastic dynamics ͓19͔ is not possible. In the classical case, it ␶ where the jumps between ñj−1 and ñj occur at time ˜j =t is sufficient after the forward process to revert the driving ␶ − N−j+1 and where ñj =nN−j. Because this dynamics and the protocol ˜␭͑␶͒=␭͑t−␶͒ and to ask for the probability of a dynamics ͑40͒ both span the same configuration space, sum- backward trajectory ͑system taking the sequence of states of ming over all trajectories of the backward process is equiva- the forward trajectory but in the reversed order͒ to occur. The lent to summing over all the trajectories of the original pro- reversal of the driving protocol has the consequence of re- cess. The trajectory ͑65͒ is depicted in Fig. 1. The probability versing the time dependence of the transition matrix of this trajectory is evaluated in Appendix B and reads

046129-6 FLUCTUATION THEOREMS FOR QUANTUM MASTER EQUATIONS PHYSICAL REVIEW E 73, 046129 ͑2006͒

N ␶ ͗x͘ j ജe this relation also means that on average the quantity ␮ ͓˜ ͔ ˜ ͑ ͒ͫ͟ ͵ ␶Ј͚ ͑ ͒ ˜ n͑␶͒ = P n expͩ− d W␶ n ,m ͪ B ˜ 0 N Ј j−1 r͑t͒ is always non-negative, ͗r͑t͒͘ജ0. Choosing P ͑n ͒ ␶ 0 N j=1 j−1 m ͑ ͒ ͑ ͒ ⌬ ͑ ͒ ͓ = Pt nN we have r t = si t and we show again see text t below Eq. ͑47͔͒ that the ensemble averaged trajectory en- ϫ ͑ ͒ͬ ͵ ␶Ј͚ ͑ ͒ W␶ n ,n expͩ− d W␶ n ,m ͪ. ͗⌬ ͑ ͒͘ജ j j j−1 Ј N tropy production is always non-negative, si t 0. ␶ N m ͑66͒ VIII. QUANTUM FLUCTUATION THEOREM We now calculate the ratio of the forward ͑62͒ and backward FOR A STEADY STATE ͑66͒ probabilities. Noticing that the exponentials cancel, we find the fundamental result of the paper: We consider a subsystem subjected to nonequilibrium constraints, whose dynamics is described by a QME of the ␮ ͓ ͔ F n͑␶͒ P ͑n ͒ ͑ ͒ r͑t͒ =ln =ln 0 0 − ⌬s ͑t͒, ͑67͒ form 34 . When the subsystem is in a steady state, its den- ␮ ͓ ͔ e ͑ ͒ B ñ͑˜␶͒ ˜P ͑n ͒ sity matrix does not evolve in time and the rates in Eq. 40 0 N are time independent. An example of such system could be a where the trajectory entropy flow is two-level atom driven by a coherent single-mode field on N resonance ͑in the dipole approximation and in the rotating t W␶ ͑n ,n ͒ ⌬ ͑ ͒ϵ ͑ ͒ ͑ ͒ ͵ ␶ ͑␶͒ j j j−1 wave approximation͒ described by Bloch equations ͑see p. se t se t − se 0 = d s˙e = ͚ ln . W␶ ͑n ,n ͒ 154 of Ref. ͓34͔͒. The basis set of the forward process is time 0 j=1 j j−1 j independent and is the same as the basis set of the backward ͑68͒ process. By definition, we have In analogy with the classical results of Seifert ͓19͔,wecan ͑ ͒ ͗␦ ͑ ͒ ͑ ͒ ͘ now derive the various fluctuation theorems by specific pF„R t … = „R t − rF t … F choices of initial conditions for the backward trajectories. By ͚ ␮ ͓ ͔␦ ͑ ͒ ͑ ͒ ͑ ͒ ˜ ͑ ͒ ͑ ͒ = F n͑␶͒ „R t − rF t …. 72 choosing P0 nN = Pt nN and using the trajectory entropy n͑␶͒ P ͑n ͒ ⌬s͑t͒ϵs͑t͒ − s͑0͒ =ln 0 0 , ͑69͒ Using Eq. ͑67͒, we can write ͑ ͒ Pt nN ͑ ͒ ͑ ͒ ͑ ͒ ␮ ͓ ͔ rF t ␦ ͑ ͒ ͑ ͒ Eq. 67 becomes pF„R t … = ͚ B n͑␶͒ e „R t − rF t … n͑␶͒ ␮ ͓ ͔ F n͑␶͒ r͑t͒ =ln = ⌬s͑t͒ − ⌬s ͑t͒ = ⌬s ͑t͒, ͑70͒ ␮ ͓ ͔ R͑t͒␦ ͑ ͒ ͑ ͒ ␮ ͓ ͔ e i = ͚ B n͑␶͒ e R t − rF t B ñ͑˜␶͒ „ … n͑␶͒ ⌬ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ where si t =si t −si 0 is the trajectory entropy production. ͗␦ ͑ ͒ ͑ ͒ ͘ R t = „R t + rB t … Be Equation ͑67͒ was first derived by Crooks ͓17͔ for classi- ͑ ͒ R͑t͒ ͑ ͒ cal stochastic processes and later generalized by others = pB„− R t …e , 73 ͓5,19͔. We have shown that this relation may be extended to quantum systems. In the classical case the time dependence where to go from the second to the third line, we used ͑ ͒ ͑ ͒ ͑ ͒ ˜ ͑ ͒ of the rates is exclusively due to the external driving. In the rF t =−rB t which comes from Eq. 67 . When P0 nN ͑ ͒ ͑ ͒ ͑ ͒ quantum case it is also due to the quantum evolution of the = Pt nN and therefore Eq. 70 holds, Eq. 73 becomes a density matrix itself. However, in both cases, the backward fluctuation theorem for the entropy production: dynamics is obtained by reversing the time dependence of ⌬ ͑ ͒ the rate matrix. ⌬ ͑ ͒ ⌬ ͑ ͒ Si t ͑ ͒ pF„ Si t … = pB„− Si t …e . 74 This relation shows that at steady state, the ratio of the prob- VII. QUANTUM INTEGRAL FLUCTUATION THEOREM ability to observe a given entropy production during a fro- Summing over all possible trajectories of the backward ward process and the probability to observe the same entropy process is equivalent to summing over all possible trajecto- production with a minus sign during the backward process is ries of the original process ͚ =͚ . By averaging Eq. ͑67͒ given by the exponential of the entropy production. This is ñ͑˜␶͒ n͑␶͒ over all possible trajectories, we find the most familiar form of the fluctuation theorem. In the infinite-time limit, if the subsystem has a finite number of ͑ 1=͚ ␮ ͓ñ͑␶͔͒ = ͚ ␮ ͓ñ͑␶͔͒ levels this condition is usually implicitly assumed in QME B ˜ B ˜ t→ϱ ñ͑˜␶͒ n͑␶͒ ͒ ⌬ ͑ ͒ ⌬ ͑ ͒ ⌬ ͑ ͒ theory , S t will be bounded and Si t = Se t so that ␮ ͓ ͔ −r͑t͒ ͗ −r͑t͒͘ ͑ ͒ Eq. ͑74͒ also becomes a fluctuation theorem for the entropy = ͚ F n͑␶͒ e = e . 71 n͑␶͒ flow and therefore also for the heat. For completeness, we give in Appendix C a different derivation of a fluctuation This integral fluctuation theorem ͓19͔ is valid for any choice theorem similar to Eq. ͑74͒ and which is not restricted to ͑ ͒ ˜ ͑ ͒ ͑ ͒ ͗ x͘ of P0 n0 and P0 nN in Eq. 67 . Using the fact that e steady states.

046129-7 M. ESPOSITO AND S. MUKAMEL PHYSICAL REVIEW E 73, 046129 ͑2006͒

IX. HEAT AND WORK FOR QUANTUM TRAJECTORIES ␤͗ ͉ ˆ ͑ ͉͒ ͘ e− nN HS t nN ͑ ͒ ͑ ͒ Pt nN = , 78 If we use the relation ͑48͒ together with the definition of Zt the trajectory entropy flow ͑68͒, we find that the heat asso- ͚ ͓ ␤͗ ͉ ˆ ͑ ͉͒ ͔͘ ͚ ciated with a single trajectory is given by where Z0 = nexp − n0 HS 0 n0 and Zt = n ϫ ͓ ␤͗ ͉ ˆ ͑ ͉͒ ͔͘ ͕͉ ͖͘ ͓͕͉ ͖͔͘ exp − nN HS t nN . Notice that n0 nt is now the ˜q ͑t͒ϵ␤−1⌬s ͑t͒ ˆ ͑ ͓͒ˆ ͑ ͔͒ S e eigenbasis of HS 0 HS t . N The free energy difference between the initial and final ͓͗ ͉ ˆ ͑␶ ͉͒ ͘ ͗ ͉ ˆ ͑␶ ͉͒ ͔͘ ͑ ͒ = ͚ nj HS j nj − nj−1 HS j nj−1 . 75 states is given by j=1 Z ⌬F͑t͒ = F͑t͒ − F͑0͒ =−␤−1 ln t . ͑79͒ The interpretation of this result is that the heat flowing to the Z0 environment results from transitions between the subsystem Using Eq. ͑75͒ which defines the heat of a single subsystem states nj. trajectory, we can write Eq. ͑70͒ as The energy associated with a trajectory is a state function ⌬ ͑ ͒ ͑ ͒ ͑ ͒ ␤ ͑ ͒ ͑ ͒ and only depends on the initial and final states of the trajec- si t =−lnPt nN +lnP0 n0 − ˜qS t . 80 tory: Using now Eqs. ͑78͒, ͑79͒, ͑76͒, and ͑77͒, we can rewrite Eq. ͑80͒ as ⌬ ͑ ͒ ͗ ͉ ˆ ͑ ͉͒ ͘ ͗ ͉ ˆ ͑ ͉͒ ͘ eS t = nN HS t nN − n0 HS 0 n0 ⌬ ͑ ͒ ␤⌬ ͑ ͒ ␤ ͑ ͒ ͑ ͒ N si t =− F t + w˜ S t . 81 ͓͗ ͉ ˆ ͑␶ ͉͒ ͘ ͗ ͉ ˆ ͑␶ ͉͒ ͔͘ = ͚ nj HS j nj − nj−1 HS j nj−1 Finally, by inserting Eq. ͑81͒ into the integral fluctuation j=1 ͑ ͒ ͑ ͒ ⌬ ͑ ͒ theorem 71 where r t = si t , we find the quantum Jarzyn- ͑ ͒ ͑ ͒ ͑ ͒ ski relation = w˜ S t + ˜qS t . 76 ␤⌬ ͑ ͒ ␤ ͑ ͒ e− F t = ͗e− w˜ S t ͘. ͑82͒ The work is therefore given by

͑ ͒ ⌬ ͑ ͒ ͑ ͒ w˜ S t = eS t − ˜qS t XI. CONCLUSIONS N We have presented a unified derivation of a quantum in- ͓͗ ͉ ˆ ͑␶ ͉͒ ͘ ͗ ͉ ˆ ͑␶ ͉͒ ͔͘ = ͚ nj−1 HS j nj−1 − nj−1 HS j−1 nj−1 . tegral fluctuation theorem, a quantum steady-state fluctuation j=1 theorem, and the quantum Jarzynski relation for a driven ͑77͒ subsystem interacting with its environment and described by a QME. This generalizes earlier results obtained for quantum The work thus results from the time evolution of the Hamil- systems. By recasting the solution of the QME in a BDME ͑ ͒ tonian due to the driving force along the states nj of the form with time-dependent rates for the eigenvalues of the subsystem between the transitions. It is interesting to point subsystem density matrix, we naturally define quantum tra- out the parallel between our description of heat and work in jectories and their associated entropy, heat, and work and ͕͉ ͖͘ the mt basis set and the adiabatic basis description of Ref. study their fluctuation properties. The connection between ͓22͔. In the latter the work comes from the evolution along the trajectory quantities which naturally enter our formula- the adiabatic states and the heat comes from the transitions tion and measurable quantum trajectory quantities is still an between the adiabatic state. This can be understood by com- open issue. Deriving quantum fluctuation relations without paring Eqs. ͑A9͒ and ͑A10͒ with Eqs. ͑A12͒ and ͑A13͒. having to assume QME’s, which do not correctly account for strong subsystem-environment entanglement, is an exiting perspective. X. QUANTUM JARZYNSKI RELATION

We assume that the subsystem is initially at equilibrium ACKNOWLEDGMENTS ˆ ͑ ͒ ˆ ␭͑ ͒ with respect to the Hamiltonian HS 0 =HS( 0 ) and is The support of the National Science Foundation ͑Grant therefore described by a canonical distribution. The system is No. CHE-0446555͒, NIRT ͑Grant No. EEC 0303389͒, and then driven out of equilibrium by turning the driving force the National Institute of Health ͑Grant No. GM59230-05͒ is from ␭͑0͒ to ␭͑tЈ͒ at time tЈ. After tЈ the driving force stop gratefully acknowledged. M.E. is also collaborateur scienti- evolving. On long time scales after tЈ—say, t ͑tӷtЈ͒—the ﬁque at the FNRS Belgium. system is again at equilibrium in a canonical distribution but ˆ ͑ ͒ ˆ ␭͑ ͒ now with respect to HS t =HS( t ). We choose APPENDIX A: BASIS DEPENDENCE OF HEAT AND WORK ␤͗ ͉ ˆ ͑ ͉͒ ͘ e− n0 HS 0 n0 ͑ ͒ We consider a system with a time-dependent Hamiltonian P0 n0 = , ˆ Z0 H͑t͒ in the Schrödinger picture described by the density ma-

046129-8 FLUCTUATION THEOREMS FOR QUANTUM MASTER EQUATIONS PHYSICAL REVIEW E 73, 046129 ͑2006͒ trix ␳ˆ͑t͒. The evolution equation of ␳ˆ͑t͒ is not necessarily ˙ ͑ ͒ ͗ ͉ ˆ ͑ ͉͒ Ј͗͘ Ј͉ ͓͘ ͑ ͒ ͑ Ј͔͒ A t = ͚ mt H t mt mt m˙ t Pt m − Pt m unitary. mmЈ The energy of the system is given by = ͚ ͓͗m˙ ͉Hˆ ͑t͉͒m ͘ + ͗m ͉Hˆ ͑t͉͒m˙ ͔͘P ͑m͒. ͑A11͒ ͗ ˆ ͘ϵ ˆ ͑ ͒␳͑ ͒ ͗ ͉ ˆ ͑ ͉͒ Ј͗͘ Ј͉␳͑ ͉͒ ͘ ͑ ͒ t t t t t H TrH t ˆ t = ͚ at H t at at ˆ t at , A1 m Ј aa If we consider the time-dependent basis diagonalizing the ͕͉ ͖͘ ͑ ͕͉͖͒͘ ͕͉ ͖͘ where at is an arbitrary time-dependent basis set. The instantaneous Hamiltonian adiabatic basis at = it , energy change of the system can be written as ͗ ͉ ˆ ͑ ͉͒ Ј͘ ⑀ ͑ ͒␦ where it H t it = i t iiЈ, we have t d͗Hˆ ͑␶͒͘ ˙ d ⌬E͑t͒ϵ͵ d␶ = W͑t͒ + Q͑t͒ = W˜ ͑t͒ + Q˜ ͑t͒, ˜ Ј͑ ͒ ͚ ⑀ ͑ ͒ ͓͗ ͉␳ˆ͑ ͉͒ ͔͘ ˙ ͑ ͒ ˙ Ј͑ ͒ ͑ ͒ ␶ Q t = i t it t it = Q t − A t , A12 0 d i dt ͑A2͒ ˜˙ Ј͑ ͒ ⑀ ͑ ͒͗ ͉␳͑ ͉͒ ͘ ˙ ͑ ͒ ˙ Ј͑ ͒ ͑ ͒ where the heat and work are given by W t = ͚ ˙i t it ˆ t it = W t + A t , A13 i ˙ ˆ ˙ Q͑t͒ϵTrH͑t͒␳ˆ͑t͒, ͑A3͒ where

˙ Ј͑ ͒ ͚ ⑀ ͑ ͓͒͗˙ ͉␳ˆ͑ ͉͒ ͘ ͗ ͉␳ˆ͑ ͉͒˙ ͔͘ ˙ ˆ˙ A t =− i t it t it + it t it W͑t͒ϵTrH͑t͒␳ˆ͑t͒, ͑A4͒ i in a time-independent basis, and by ͓⑀ ͑ ͒ ⑀ ͑ ͔͒͗ ͉˙Ј͗͘ Ј͉␳͑ ͉͒ ͘ ͑ ͒ = ͚ i t − iЈ t it it it ˆ t it . A14 Ј ˙ d ii Q˜ ͑t͒ϵ͚ ͗a ͉Hˆ ͑t͉͒aЈ͘ ͓͗aЈ͉␳ˆ͑t͉͒a ͔͘, ͑A5͒ t t dt t t It is interesting to notice the similarity between the two bases aaЈ ͕͉ ͖͘ ͕͉ ͖͘ mt and it . In both cases, the heat results from changes in the population of the states ͑and therefore from transitions ˙ d ͓͒ ͑ ͒ ͑ ͔͒ W˜ ͑t͒ϵ͚ ͓͗a ͉Hˆ ͑t͉͒aЈ͔͗͘aЈ͉␳ˆ͑t͉͒a ͘, ͑A6͒ between states see Eqs. A9 and A12 and the work from dt t t t t ͓ aaЈ the evolution of the Hamiltonian along the states see Eqs. ͑A10͒ and ͑A13͔͒. Using Eqs. ͑A11͒ with ͑A14͒ one gets in a time-dependent basis. ˙ ˙ How does Q͑t͓͒W͑t͔͒ relate to Q˜ ͑t͓͒W˜ ͑t͔͒? We ﬁnd A˙ ͑t͒ − A˙ Ј͑t͒ = W˜ ͑t͒ − W˜ Ј͑t͒ ˙ ˙ ˙ Q˜ ͑t͒ = Q˙ ͑t͒ − A˙ ͑t͒, = Q˜ Ј͑t͒ − Q˜ ͑t͒ d ͑ ͒⑀ ͑ ͒ ͉͑͗ ͉ ͉͘2͒ ͑ ͒ ˙ = ͚ Pt m i t it mt . A15 W˜ ͑t͒ = W˙ ͑t͒ + A˙ ͑t͒, ͑A7͒ i,m dt where Let us assume now that the density matrix of the system obeys the von Neumann equation A˙ ͑t͒ =−͚ ͗a ͉Hˆ ͑t͉͒aЈ͓͗͘a˙Ј͉␳ˆ͑t͉͒a ͘ + ͗aЈ͉␳ˆ͑t͉͒a˙ ͔͘ t t t t t t ˆ aaЈ ␳ˆ˙͑t͒ =−i͓Hˆ ͑t͒,␳ˆ͑t͔͒ = Lˆ ͑t͒␳ˆ͑t͒, ͑A16͒ ͓͗ ͉ ˆ ͑ ͉͒ Ј͘ ͗ ͉ ˆ ͑ ͉͒ Ј͔͗͘ Ј͉␳͑ ͉͒ ͘ ͑ ͒ = ͚ a˙ t H t at + at H t a˙ t at ˆ t at . A8 whose solution reads aaЈ ˆ ˆ ˆ † ͗ ͉ Ј͘ ͗ ͉ Ј͘ ␳ˆ͑t͒ = U͑t͒␳ˆ͑0͒ = U͑t͒␳ˆ͑0͒U ͑t͒, ͑A17͒ We have used the fact that at a˙ t =− a˙ t at which come d ͑͗ ͉ Ј͒͘ from dt at at =0. where If we consider the time-dependent basis set which diago- ͕͉ ͖͘ ˆ t ˆ nalizes the instantaneous density density matrix at Uˆ ͑t͒ = exp ͭ͵ d␶Lˆ ͑␶͒ͮ, ͕͉ ͖͘ ͗ ͉␳͑ ͉͒ Ј͘ ͑ ͒␦ + = mt , where mt ˆ t mt = Pt m mmЈ, we have 0 ˙ Q˜ ͑t͒ = ͚ ͗m ͉Hˆ ͑t͉͒m ͘P˙ ͑m͒ = Q˙ ͑t͒ − A˙ ͑t͒, ͑A9͒ t t t t Uˆ ͑t͒ = exp ͭ− i͵ d␶Hˆ ͑␶͒ͮ. ͑A18͒ m + 0 d ˆ ˜˙ ͑ ͒ ͓͗ ͉ ˆ ͑ ͉͒ ͔͘ ͑ ͒ ˙ ͑ ͒ ˙ ͑ ͒ ͓ ͔ ˆ ͑ ͓͒Uˆ ͑ ͔͒ W t = ͚ mt H t mt Pt m = W t + A t , The evolution operator superoperator UT t T t is uni- dt ͕͉ ͖͘ m tary. In this case, the expressions in the basis mt simplify ͑A10͒ to where Q˜ ͑t͒ = Q͑t͒ =0, ͑A19͒

046129-9 M. ESPOSITO AND S. MUKAMEL PHYSICAL REVIEW E 73, 046129 ͑2006͒

APPENDIX B: PROBABILITY OF THE BACKWARD W˜ ͑t͒ = W͑t͒ = ⌬E͑t͒. ͑A20͒ TRAJECTORY This is due to the fact that The probability of a backward trajectory ñ͑˜␶͒ reads N ␶ ͑ ͒ ͗ ͉␳͑ ͉͒ ͘ ͗ ͉ ˆ †͑ ͒ ˆ ͑ ͒␳͑ ͒ ˆ †͑ ͒ ˆ ͑ ͉͒ ͘ ˜j P0 m = m0 ˆ 0 m0 = m0 U t U t ˆ 0 U t U t m0 ␮ ͓ñ͑␶͔͒ = ˜P ͑ñ ͒ͫ͟ expͩ− ͵ d˜␶Ј͚ W˜ ␶ ͑ñ ,m˜ ͒ͪ B ˜ 0 0 ˜Ј j−1 ͗ ͉␳͑ ͉͒ ͘ ͑ ͒ ͑ ͒ ␶ = mt ˆ t mt = Pt m . A21 j=1 ˜j−1 m˜ This means that no heat is produced by the driving force for t ϫW˜ ␶ ͑ñ ,ñ ͒ͬexpͩ− ͵ d˜␶Ј͚ W˜ ␶ ͑ñ ,m˜ ͒ͪ, ˜ ˜ a unitary evolution. This is reasonable since there is no en- j j−1 j Ј N ˜␶ m˜ vironment. The only way in which the energy of the system N may increase is via the work done on the system. Notice that ͑B1͒ ˜ Ј͑ ͒ ˜ Ј͑ ͒ ␶ ␶ ͑ ͒ ͑ ͒ ␶ in the adiabatic basis both W t and Q t are finite for a where ˜0 =0 and Ñ+1=t. Using Eqs. 64 and 65 and ˜j =t ␶ unitary evolution. − N−j+1, we can rewrite this probability as

N ␶ t− N−j+1 t ␮ ͓˜ ͔ ˜ ͑ ͒ͫ͟ ͵ ˜␶Ј͚ ͑ ͒ ͑ ͒ͬ ͵ ˜␶Ј͚ ͑ ͒ ͑ ͒ n͑␶͒ = P n expͩ− d W ␶ n ,m ͪW␶ n ,n expͩ− d W ␶ n ,m ͪ. B2 B ˜ 0 N t−˜Ј N−j+1 N−j+1 N−j t−˜Ј 0 ␶ N−j+1 ␶ j=1 t− N−j+2 m t− 1 m Using the change of variables ␶=t−˜␶,weget

␶ N ␶ 1 N−j+2 ␮ ͓˜ ͔ ˜ ͑ ͒ ͵ ␶Ј͚ ͑ ͒ ͫ͟ ͵ ␶Ј͚ ͑ ͒ ͑ ͒ͬ ͑ ͒ n͑␶͒ = P n expͩ− d W␶ n ,m ͪ expͩ− d W␶ n ,m ͪW␶ n ,n . B3 B ˜ 0 N Ј 0 Ј N−j+1 N−j+1 N−j ␶ N−j+1 0 m j=1 N−j+1 m

With the help of j=N− j +2, Eq. ͑B3͒ finally becomes ␮ ͓͉ ͘ → ͉ Ј͔͘ ͗ ͉␴͑ ͒␳eq͉ ͘ old F m0b mtb = m0b ˆ 0 ˆ B m0b Eq. ͑66͒. ϫ͉͗ ͉ ˆ ͑ ͉͒ ͉͘2 ͑ ͒ mtbЈ U t m0b , C2 APPENDIX C: QUANTUM FLUCTUATION THEOREM Uˆ ͑t͒ FOR UNCORRELATED SUBSYSTEM AND BATH where is the unitary evolution operator of the total system. The probability of the backward process to go from ͉ ͘ ͉ ͘ We derive a general quantum fluctuation theorem ͑not re- mtbЈ at time t to m0b at time 0 by the time reversed stricted to steady states͒ for a driven quantum subsystem in evolution ͓27,30,40͔ is given by contact with its environment. The derivation is similar to that ͓ ͔ ␮ ͓͉ Ј͘ → ͉ ͔͘ ͗ Ј͉␴͑ ͒␳eq͉ Ј͘ of Monnai in 27 and is given for completeness. B mtb m0b = mtb ˆ t ˆ B mtb We assume weak coupling between the subsystem and ϫ͉͗ ͉ ˆ ͑ ͉͒ ͉͘2 ͑ ͒ environment and that the environment is infinitely large so mtbЈ U t m0b . C3 that at all times the density matrix of the total system ͑sub- system plus environment͒ can be written as We therefore have that

eq ␮ ͓͉ ͘ → ͉ Ј͔͘ ͑ ͒ ␳ˆ͑t͒ = ␴ˆ ͑t͒␳ˆ , ͑C1͒ F m0b mtb P0 m0 ␤ B = e− QbbЈ, ͑C4͒ ␮ ͓͉ ͘ → ͉ ͔͘ ͑ ͒ B mtbЈ m0b Pt mt ␤ ˆ where ␳ˆ eq=e− HB /Z is the time-independent equilibrium re- B B ͑ ͒ ͗ ͉␴͑ ͉͒ ͘ ͑ ͒ ͗ ͉␴͑ ͉͒ ͘ duced density matrix of the environment and ␴ˆ ͑t͒ the time- where P0 m0 = m0 ˆ 0 m0 , Pt mt = mt ˆ t mt , and QbbЈ dependent reduced density matrix of the subsystem. Assum- =Eb −EbЈ. ing the form ͑C1͒ is not very different from assuming that The entropy of a state mt is defined as the subsystem density matrix obeys a QME since the QME ͑ ͒ ͑ ͒ ͑ ͒ derivation implicitly assumes an invariant environment den- s t =−lnPt mt . C5 sity matrix ͑e.g., the Born approximation ͓34͔͒. ͕͉ ͖͘ ͕͉ ͖͘ ͕͉ ͖͘ ␴͑ ͒ Let us define the basis mtb , where mt diagonalize This definition makes sense because mt diagonalizes ˆ t , the subsystem density matrix at time t and where ͕͉b͖͘ diag- so that by averaging over the different states we recover the onalize the time-independent environment Hamiltonian. The von Neumann entropy. The entropy difference between the ͉ ͘ ͉ ͘ probability to go from m0b at time 0 to mtbЈ at time t is initial and final states of the subsystem starting at time 0 in ͉ ͘ ͉ ͘ given by m0 and ending at time t in mt is given by

046129-10 FLUCTUATION THEOREMS FOR QUANTUM MASTER EQUATIONS PHYSICAL REVIEW E 73, 046129 ͑2006͒

P ͑m ͒ states, we get the general fluctuation theorem ⌬s͑s ,s ;t͒ =ln 0 0 . ͑C6͒ 0 t ͑ ͒ Pt mt ͑⌬ ͑ ͒͒ ␮ ͓͉ ͘ → ͉ ͔͘ p Si t = ͚ F m0b mtbЈ The entropy production of this same process is given by m0,mt,b,bЈ ϫ␦ ⌬ ͑ ͒ ⌬ ͑ Ј ͒ P ͑m ͒ Q „ Si t − si m0,mt,b,b ;t … ⌬ ͑ Ј ͒ 0 0 bbЈ ͑ ͒ si m0,mt,b,b ;t =ln − , C7 ⌬ ͑ ͒ ͑ ͒ ␮ ͓͉ ͘ → ͉ ͔͘ si m0,mt,b,bЈ;t Pt mt T = ͚ B m0b mtbЈ e Ј because one assumes that the entropy flow difference is m0,mt,b,b ϫ␦ ⌬ ͑ ͒ ⌬ ͑ Ј ͒ given by „ Si t − si m0,mt,b,b ;t … ⌬ ͑ ͒ Q = p − ⌬S ͑t͒ e Si t . ͑C10͒ ⌬ ͑ ͒ bbЈ ͑ ͒ „ i … se b,bЈ;t = . C8 T This result agrees with Eq. ͑74͒ and is not restricted to steady states. This approach is based on the time reversal invariance Using Eqs. ͑C6͒–͑C8͒, Eq. ͑C4͒ becomes of the evolution of the total system and does not provide a ␮ ͓͉m b͘ → ͉m bЈ͔͘ trajectory picture. We note that in the total system space, the F 0 t ⌬ ͑ ͒ ln = si m ,mt,b,bЈ;t heat ͑or the entropy flow͒ going from the subsystem to the ␮ ͓͉m bЈ͘ → ͉m b͔͘ 0 B t 0 environment only depends on the end points and not on the ⌬ ͑ ͒ ⌬ ͑ ͒ = s s0,st;t − se b,bЈ;t . path itself. If one derive a Jarzynski relation from this result ͓ ͔ ͑C9͒ as in 27 , the work is found to be path independent. When considering the reduced dynamics of the system alone, as This result is the analog of our fundamental relation ͑70͒.By done in this paper, these quantities become path dependent averaging the probabilities over all possible initial and final and a trajectory picture is provided.

͓1͔ C. Jarzynski, Phys. Rev. Lett. 78, 2690 ͑1997͒; Phys. Rev. E ͓23͔ W. De Roeck and C. Maes, Phys. Rev. E 69, 026115 ͑2004͒. 56, 5018 ͑1997͒. ͓24͔ J. Kurchan, e-print cond-mat/0007360. ͓2͔ G. E. Crooks, J. Stat. Phys. 90, 1481 ͑1998͒. ͓25͔ T. Monnai and S. Tasaki, e-print cond-mat/0308337. ͓ ͔ 3 C. Jarzynski, J. Stat. Mech.: Theory Exp. 2004, P09005. ͓26͔ W. De Roeck and C. Maes, e-print cond-mat/0406004. ͓ ͔ 4 E. G. D. Cohen and D. Mauzerall, J. Stat. Mech.: Theory Exp. ͓27͔ T. Monnai, Phys. Rev. E 72, 027102 ͑2005͒. 2004, P07006. ͓28͔ C. Jarzynski and D. K. Wojcik, Phys. Rev. Lett. 92, 230602 ͓5͔ C. Maes, Séminaire Poincaré 2,29͑2003͒. ͑2004͒. ͓6͔ D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Phys. Rev. ͓ ͔ Lett. 71, 2401 ͑1993͒. 29 A. E. Allahverdyan and Th. M. Nieuwenhuizen, Phys. Rev. E ͑ ͒ ͓7͔ G. Gallavotti and E. G. D. Cohen, Phys. Rev. Lett. 74, 2694 71, 066102 2005 . ͑1995͒; J. Stat. Phys. 80, 931 ͑1995͒. ͓30͔ I. Callens, W. De Roeck, T. Jacobs, C. Maes, and K. Netocny, ͓8͔ D. J. Evans and D. J. Searles, Phys. Rev. E 50, 1645 ͑1994͒; Physica D 187, 383 ͑2004͒. 52, 5839 ͑1995͒; 53, 5808 ͑1996͒. ͓31͔ F. Haake, Statistical Treatment of Open Systems, Springer ͓9͔ G. N. Bochkov and Yu. E. Kuzovlev, Zh. Eksp. Teor. Fiz. 72, Tracts in Modern Physics No. 66 ͑Springer, New York, 1973͒. 238 ͑1977͓͒Sov. Phys. JETP 45, 125 ͑1977͔͒; 76, 1071 ͓32͔ H. Spohn, Rev. Mod. Phys. 53, 569 ͑1980͒. ͑1979͓͒49, 543 ͑1979͔͒; Physica A 106, 443 ͑1981͒; 106, 480 ͓33͔ C. W. Gardiner and P. Zoller, Quantum Noise ͑Springer, Ber- ͑1981͒. lin, 2000͒. ͓10͔ R. L. Stratonovich, Nonlinear Nonequilibrium Thermodynam- ͓34͔ H.-P. Breuer and F. Petruccione, The Theory of Open Quantum ics II ͑Springer, Berlin, 1994͒. Systems ͑Oxford University Press, Oxford, 2002͒. ͓11͔ C. Jarsynski, J. Stat. Phys. 98,77͑2000͒. ͓35͔ The trace invariance used in Eq. ͑7͒ could be a delicate matter ͓12͔ J. Kurchan, J. Phys. A 31, 3719 ͑1998͒. for systems with a continuous spectrum where the trace may ͓13͔ J. L. Lebowitz and H. Spohn, J. Stat. Phys. 95, 333 ͑1999͒. diverge. This can be always formally addressed by using a ͓14͔ D. J. Searles and D. J. Evans, Phys. Rev. E 60, 159 ͑1999͒. dense quasicontinuum. Physically, Eq. ͑7͒ implies energy con- ͓15͔ P. Gaspard, J. Chem. Phys. 120,19͑2004͒; D. Andrieux and P. servation. Gaspard, ibid. 121, 6167 ͑2004͒. ͓36͔ S. Mukamel, Principles of Nonlinear Optical Spectroscopy ͓16͔ P. Gaspard, J. Stat. Phys. 117, 599 ͑2004͒; e-print cond-mat/ ͑Oxford University Press, New York, 1995͒. 0603382. ͓37͔ D. Kondepudi and I. Prigogine, Modern Thermodynamics ͓17͔ G. E. Crooks, Phys. Rev. E 60, 2721 ͑1999͒. ͑Wiley, Chichester, 1998͒. ͓18͔ G. E. Crooks, Phys. Rev. E 61, 2361 ͑2000͒. ͓38͔ S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynam- ͓19͔ U. Seifert, Phys. Rev. Lett. 95, 040602 ͑2005͒. ics ͑Dover, New York, 1984͒. ͓20͔ S. Yukawa, J. Phys. Soc. Jpn. 69, 2367 ͑2000͒. ͓39͔ N. G. van Kampen, Stochastic Processes in Physics and ͓21͔ H. Tasaki, e-print cond-mat/0009244. Chemistry, 2nd ed. ͑North-Holland, Amsterdam, 1997͒. ͓22͔ S. Mukamel, Phys. Rev. Lett. 90, 170604 ͑2003͒. ͓40͔ T. Yacobs and C. Maes, Phys. Mag. 27,119͑2005͒.

046129-11