A test of “fluctuation theorem” in non-Markovian open quantum systems
We study fluctuation theorems for open quantum systems with a non-Markovian heat bath using the approach of quantum master equations and examine the physical quantities that appear in those fluctuation theorems. The approach of Markovian quantum master equations to the fluctuation theorems was developed by Esposito and Mukamel [Phys. Rev. E 73, 046129 (2006)]. We show that their discussion can be formally generalized to the case of a non-Markovian heat bath when the local system is linearly connected to a Gaussian heat bath with the spectrum distribution of the Drude form. We found by numerically simulating the spin-boson model in non-Markovian regime that the “detailed balance” condition is well satisfied except in a strongly non-equilibrium transient situation, and hence our generalization of the definition of the “entropy production” is almost always legitimate. Therefore, our generalization of the fluctuation theorem seems meaningful in wide regions.
PACS Number: 05.70.Ln, 05.40.-a, 05.30.-d
Keywords: fluctuation theorem, quantum master equation, non-Markovian heat bath
The research of non-equilibrium statistical physics has been energetically developed for the last decade in the context of fluctuation theorems and the Jarzynski equality Kurchan (); Tasaki (); Talkner et al. (2007); Talkner and Hänggi (2007); Andrieux and Gaspard (2008); Monnai (2005); Monnai and Tasaki (2003); Yukawa (2000). In particular, the case of open quantum systems, where a reservoir with an infinite number of degrees of freedom is attached to a local quantum system with a finite number of degrees of freedom, is of great interest presently Monnai (2010); Talkner et al. (2009); Campisi et al. (2009); Talkner et al. (2007); Esposito et al. (2009); Crooks (2008); Esposito and Mukamel (2006); Deffner and Lutz (); Campisi et al. (2010). The most common approach to discussing the fluctuation theorems for an open quantum system is by the twice energy measurement on the local system together with the reservoir. There, one often assumes that the coupling between the local system and the reservoir is weak. This approach, since the measurements are done on both the local system and the reservoir, can be viewed as manipulating the total isolated system.
In contrast, we here consider whether there exist any fluctuation theorems for an open quantum system that are expressed solely in terms of quantities of its local system. Fluctuation theorems of such kind were indeed proposed by Esposito and Mukamel Esposito and Mukamel (2006) under the context of Markovian quantum master equations. Their approach is a quantum analog to the fluctuation theorems of classical stochastic processes discussed by Crooks Crooks (1998, 1999) and Seifert Seifert (2005). Note, however, that the entropy production that appears in their formalism is not a thermodynamic entropy production; it is an information entropy production. We will use the term entropy production in this sense in the following.
In the present paper, we will first show that we can formally generalize the discussion by Esposito and Mukamel to the case of the dynamics in a non-Markovian heat bath. To this end, we employ a method called hierarchy equations of motion, which describes the reduced dynamics in a non-Markovian heat bath. The generalization to the non-Markovian dynamics in classical systems was already done in Refs. Zamponi et al. (2005); Ohkuma and Ohta (2007); Speck and Seifert (2007).
When we discuss the fluctuation theorems, defining the work, the heat, and the entropy production is also an essential problem Monnai and Tasaki (2003); Allahverdyan and Nieuwenhuizen (2005); Esposito et al. (2010); J. Gemmer, M. Michel, & G. Mahler (2010); Deffner and Lutz (). As was done by Esposito and Mukamel Esposito and Mukamel (2006), we can proceed with formal definitions based on quantum analogs to classical stochastic processes as working assumptions. However, it is necessary to examine whether the definitions are indeed appropriate quantum-mechanically. We will examine the validity of the “entropy production,” the “entropy flow,” and the “heat” that appear in the fluctuation theorems under our generalization by analyzing the spin-boson model, numerically in the non-Markovian regions and analytically in the limit where the Born-Markov approximation and the rotating-wave approximation are applicable. We will find that the “entropy production” indeed vanishes in equilibrium, and thus we can regard that our fluctuation theorem is legitimate even in non-Markovian regions except for strongly non-equilibrium transient states.
As another point, if the “entropy flow” is related to the “heat” by the “microscopic reversibility,” then we can rewrite our fluctuation theorem into the Crooks-type when the initial and the final states are the canonical states. We will find, however, that the “microscopic reversibility” is broken in wide parameter regions although it does hold in the limit where the Born-Markov approximation and the rotating-wave approximation are applicable in the driving protocols that we will investigate in the present paper. Furthermore, we will find that the “detailed balance” is almost satisfied in some parameter regions.
The paper is organized as follows. In Sec. II, we will derive the fluctuation theorems for a quantum master equation in a non-Markovian heat bath. In this Section, we will assume that the heat bath is composed of the ensemble of harmonic oscillators that are linearly connected to the local system, but the Hamiltonian of the local system can be arbitrary. In Sec. III, we will show our numerical results for the spin-boson model; we will demonstrate the time evolution of the deviation from the “detailed balance.” In equilibrium states, the “detailed balance” condition means that the “entropy production” is zero. We will also analyze the case where the local system is weakly driven by a Zeeman magnetic field periodically. The “entropy flow” and the “entropy production” are well-defined in this case, although they might be ill-defined when we drive the local system strongly or in different schedules. We will also examine whether any other useful relation holds; we will find that, although the “microscopic reversibility” which we will discuss in detail below, does not hold in general, there exist a case where the equation of the “detailed balance” instead is almost satisfied.
Ii Fluctuation theorems for a quantum master equation in a non-Markovian heat bath
We consider a local system which is linearly connected to a Gaussian heat bath. The Hamiltonian of the total system reads
The Hamiltonian of the local system is arbitrary. The heat-bath Hamiltonian consists of the harmonic oscillators of an infinite number of modes , where and are the creation and annihilation operators of each mode. We omitted the ground-state energy of the harmonic oscillators. The interaction between the system and the heat bath is linear in the position of the harmonic oscillator of each mode with the coupling strength . The operator is an arbitrary one of the local system which depends on time in general. We approximate that the spectral distribution of the bath mode is of the Drude form:
which is the Ohmic distribution with the Lorentzian cutoff and a coefficient . The cutoff gives the decay rate of the canonical time correlation function of the heat bath, whereas is the coefficient related to the system-bath coupling strength . For the region where the Born-Markov approximation is appropriate, i.e. ,
we refer to it as a Markovian heat bath; otherwise we refer to it as a non-Markovian heat bath.
Esposito and Mukamel Esposito and Mukamel (2006) started their discussion with a Markovian quantum master equation of the form
where is the reduced density matrix of the local system and they used the following notation:
In (6), is a dynamical semi-group acting on the reduced Liouville space of the local system. (We use the single-hatted letter as an operator acting on the Hilbert space and the double-hatted letter as an operator acting on the Liouville space.) The evolution of the reduced density matrix does not directly lead to the fluctuation theorem. In order to investigate the fluctuation theorem analogously to classical stochastic processes, they translated the Markovian quantum master equation in a form of a classical master equation using a time-dependent basis. They then constructed a “quantum trajectory” for the dynamics and discussed the forward and backward probabilities of the trajectory. The time-dependent basis is a basis which diagonalizes the reduced density matrix at each time, and hence the reduced density matrix is represented as
where we suppressed the subscript for and on the right-hand side. Then we can regard the basis as a set of states with probability at time . By connecting these states, Esposito and Mukamel were able to construct a quantum trajectory (Fig. 1 in Ref. Esposito and Mukamel (2006)) of the local system. Under this basis, they Esposito and Mukamel (2006) had the classical master equation representation of the quantum master equation (6):
We will show that the discussion above by Esposito and Mukamel Esposito and Mukamel (2006) can be formally generalized to the dynamics in the non-Markovian heat bath. In their discussion, it was essential that the equation of motion is expressed in a time-local form. In order to keep the time-local form for the non-Markovian heat bath, we employ a set of master equations called hierarchy equations of motion Tanimura and Kubo (1989); Ishizaki and Tanimura (2005); Tanimura (2006); Tanimura and Wolynes (1991). In the formalism of the hierarchy equations of motion, a state of the system is expressed with a set of infinite matrices in an extended space instead of a reduced density matrix. Thereby we can obtain a time-local equation of motion by taking account of the time correlation of the heat bath in the form of correlations among the matrices.
ii.1 Hierarchy equations of motion
In the formalism of the hierarchy equations of motion Tanimura and Kubo (1989); Ishizaki and Tanimura (2005); Tanimura (2006); Tanimura and Wolynes (1991), a state of the local system is expressed as a set of an infinite number of matrices as follows:
where is the reduced density matrix in the usual sense, while are the set of auxiliary matrices which possess the information of non-Markovian effects. These matrices follow from the equations of motion (76)–(79) in Appendix A. The auxiliary matrices are introduced for computational purposes only and possess no physical meaning themselves. The superscript indicates the th correction with respect to the decay rate of the heat bath and the subscript indicates the corrections with respect to the temperature, i.e. the Matsubara frequencies . We define its inner product as
where is the generator that represents all the operations on the right-hand sides of Eqs. (76)–(79). Note that Eq. (14) is derived under the assumption that the local system is linearly coupled to the Gaussian heat bath (2) with the spectrum distribution of the Drude form (4). This setting is often assumed when we derive a Markovian quantum master equation microscopically.
ii.2 Birth-death master equation for the hierarchy equations of motion
Analogously to Ref. Esposito and Mukamel (2006), let be the states that satisfy
where denotes an infinite set of zero matrices in the space of the auxiliary matrices. Our purpose is to extract out the equation of motion for the probability . Taking the inner product with in (14), we have
Next, we decompose into the following form:
The left-hand side of (19) can be also recast as
Then we have
we can rewrite (21) as
Conservation of the probability during the evolution leads to
Then we have
We thereby obtain an equation of motion for the hierarchy equations of motion in the form of a classical birth-death master equation:
This is a generalization of Eq. (10) to the case of a non-Markovian heat bath. It is essential to consider an open system, since we can easily show that the “transition rate” becomes zero for an isolated system; see Appendix B. Indeed, the transition of the local system occurs thanks to the coupling to the heat bath.
Note, however, that the positivity of the “transition rate” in (22) is not guaranteed. For the birth-death master equation in classical stochastic processes, we construct the equation of motion based on physically given transition rates; the transition rate there is positive by definition. In contrast, Eq. (27) is the equation that we derived from the Liouville-von Neumann equation of the total system; Eq. (27) does not necessarily have the meaning of the birth-death master equation in the sense of stochastic processes.
Algebraically, Eq. (27) is correct in any parameter regions, except for the restriction that the local system is linearly coupled to the Gaussian heat bath with the spectrum distribution of the Drude form. Our calculation starts from the exact Liouville-von Neumann dynamics of the total system and uses no approximations before arriving at (27). Therefore, the simulations that we will show in Sec. III are numerically exact.
ii.3 Forward and Backward Trajectories
Equation (27) naturally implies a quantum trajectory analogous to the trajectory in classical stochastic processes. We can regard the basis as a set of states with probability and the local system hops from a state to another with the “transition rate” at time . We can construct a forward quantum trajectory by labeling the times when the transitions occur and connecting these states (Fig. 1):
where represents the state after the th transition at time . Hence, for when and for . We set and .
Similarly, we can construct a backward trajectory corresponding to the forward one. The label of time for the backward trajectory is related to the forward one as and the value (duration) of the time is related as . The label of the state is related as , so that the backward trajectory corresponding to is (Fig. 1)
where we set and .
The probabilities for the forward and backward quantum trajectories are obtained just by replacing with in Eqs. (62) and (66) of Ref. Esposito and Mukamel (2006):
Hereafter, we set the probability of the final state of the forward trajectory equal to the probability of the initial state of the backward trajectory, i.e. .
In the argument in Ref. Esposito and Mukamel (2006), the dynamics that gives the backward process was a hypothetical one; the real time-reversed dynamics does not satisfy the relation because the quantum master equation with the Markov approximation breaks the time reversal symmetry. In contrast, the backward process that we consider in (31) is truly the process of the time-reversed dynamics because the hierarchy equations of motion formally solve the total system and do not break the time reversal symmetry. Then we exactly have
Note that this condition dictates that the auxiliary matrices also evolve backward because the dynamics of the heat bath must be also reversed. Therefore, precisely speaking, the backward process which satisfies (32) also becomes a hypothetical one if we truncate the hierarchy, although it would be effectively a real time-reversed process if the conditions (72) and (80) in Appendix A are satisfied.
ii.4 Fluctuation theorems
By repeating the same argument as in Ref. Esposito and Mukamel (2006), we can show the fluctuation theorem for the dynamics in a non-Markovian heat bath.
We define the entropy change , the “entropy flow” , and the “entropy production” along a single trajectory as
This type of definitions were first introduced by Schnakenberg Schnakenberg (1976) and used by Seifert Seifert (2005) in order to discuss the fluctuation theorems in classical stochastic processes. Esposito and Mukamel Esposito and Mukamel (2006) extended these definitions for the classical dynamics to the ones for the Markovian quantum dynamics. Our definitions (33)–(35) are their generalization.
We also define the logarithm of the ratio of the probabilities for a forward trajectory and a backward trajectory as
This relation immediately leads to the integrated fluctuation theorem
where the average is the one over all forward trajectories.
The detailed fluctuation theorem also holds. The probability that is equal to a value of the “entropy production” is
then (37) leads to our fluctuation theorem
where we used . Although Eq. (40) is formally the same as the fluctuation theorem for the case in a Markovian heat bath, the “transition rate” contains the effect of non-Markovian properties in the auxiliary matrices .
Note that the derivation of the fluctuation theorems here are totally formal; we need to examine whether the definition of the “entropy production” (35) is physically legitimate.
ii.5 Points to be checked
It is crucial for a fluctuation theorem whether the quantities and that we defined above are indeed appropriate as the entropy production and the entropy flow, respectively. We here list two points to be checked in Sec. III.
First, as a property of the entropy production, we expect it to be zero in equilibrium. Therefore, defined in (35) should vanish there, i.e.
which is nothing but a detailed balance condition. If this condition is satisfied and the “transition rate” is positive, then we can regard the formal equality (37) as the fluctuation theorem. Otherwise, (37) is an equality without any meanings of entropies. As we mentioned at the end of Sec. II.2, the “transition rate” can be negative; we will numerically demonstrate in Sec. III.1 that it indeed becomes negative during a strongly non-equilibrium transient state. In such a case, the definition (35) of the entropy production as well as (34) of the entropy flow are inappropriate because the inside of the logarithm of (35) and (34), or the left-hand side of (41), becomes negative. Remarkably, we will find in Sec. III.2 that except for the transient case, the “transition rate” is almost always positive and the “detailed balance” is almost satisfied. Furthermore, we will show in Appendix C that the “transition rate” is always positive and the equilibrium state satisfies the “detailed balance” in the limit where the Born-Markov approximation (5) and the rotating-wave approximation (85) are applicable.
Next, following Ref. Esposito and Mukamel (2006), we define the “heat” that flows out of the local system along a trajectory by
As a property of the entropy flow, we expect that it satisfies the following equality:
After defining to be , we can recast our expectation (43) to
The equality (44) is what we call the “microscopic reversibility.” Note that this relation does not affect the validity of (37). Although the probability distributions of the initial and the final states are arbitrary in general, when they are both the canonical state, i.e. , and , we can rewrite (37) into the form of the Crooks-type fluctuation theorem:
where and we defined the “work”
where . Note that we can only check the consistency of the “entropy flow” and the “heat” ; while the left-hand side of (44) is not an established quantity as the ratio of the transition rates as we mentioned at the end of Sec. II.2, the “heat” on the right-hand side is also a hypothetical quantity. In Sec. III.2 and in Appendix C, we will show that the equality (44) holds when the Born-Markov approximation (5) and the rotating-wave approximation (85) are appropriate, although it does not hold in general.
Iii Numerical examination of the “detailed balance”
and the “microscopic reversibility”
In order to examine the above points numerically, let us consider the spin-boson model:
The operators and are the Pauli matrices and and are the corresponding coefficients. The local system Hamiltonian is the two-level system with its energy difference . We added a counter-term to the local system Hamiltonian in order to maintain the translation invariance of the system with respect to the heat bath Weiss (2008); Breuer and Petruccione (2002). We set below.
In numerical simulation of the hierarchy equations of motion, the set of parameters in (72) and (80) (see Appendix A) determines the accuracy of the calculation. As we take the values of larger, the physical quantities converge to a certain value, which is numerically exact. The present paper is based on the program nonMarkovian09, which is distributed on the web site of Yoshitaka Tanimura tan ().
We will mainly examine Eq. (41) in Sec. III.1 and examine Eq. (44) in Sec. III.2, while we will also investigate some other properties of the dynamics. We again stress that the calculation is numerically exact in the sense that we did not use any approximations in deriving Eq. (14). In the region where the Born-Markov approximation and the rotating wave approximation are applicable, we can analytically investigate the behavior with the quantum optical master equation (Appendix C), and therefore we will compare our results to those cases.
iii.1 Thermalization after the energy measurement and the “detailed balance”
Let us consider the following protocol: we measure the energy of the isolated two-level system at and find it in the ground state. Then we connect the two-level system to the heat bath which is in the thermal equilibrium state at the inverse temperature and let the total system evolve without any other perturbation. As long as the coupling between the local two-level system and the bath contains a non-vanishing term of , the local two-level system exchanges the energy with the heat bath and relaxes to a stationary state. Since we are not perturbing the total system with any external fields, we take this stationary state as the equilibrium state. Note that the equilibrium state of the two-level system differs from the canonical state in general; the total system should be in the canonical state in thermal equilibrium, but the reduced system is not.
The relation that we examine here is the “detailed balance” (41), or equivalently
The set of states is the basis that diagonalizes the reduced density matrix at time . We label the elements of the basis