Heat and work in Markovian quantum master equations: concepts, fluctuation theorems, and computations
Markovian quantum master equations (MQMEs) were established nearly half a century ago. They have often been used in the study of irreversible thermodynamics. However, the previous results were mainly concerned about ensemble averages; the stochastic thermodynamics of these systems went unnoticed for a very long time. This situation remained unchanged until a variety of fluctuation theorems in classical and quantum regimes were found in the past two decades. In this paper, we systematically summarize the two strategies of investigating the stochastic heat and work in the MQMEs. One strategy is to treat the system and its surrounding heat bath as a closed quantum system, to suppose that the evolution of the composite system is unitary under a time-dependent total Hamiltonian and to define the heat and work as the changes in energy by applying two energy measurements scheme to the composite system. The other strategy is to unravel these MQMEs into random quantum jump trajectories (QJTs) and to define the stochastic heat and work along the individual trajectories. Although these two strategies are based on these two very distinct measurement schemes, they lead to the same statistic features of these thermodynamic quantities. Many important concepts, mathematical techniques, and fluctuation theorems at different descriptive levels are given in as detailed a manner as possible. We also use concrete models to illustrate these results.
In the past two decades, there has been growing interest Mukamel (2003); De Roeck and Maes (2004); De Roeck (2007); Esposito et al. (2009); Horowitz and Parrondo (2013); Esposito and Mukamel (2006); Crooks (2008a); Talkner et al. (2009); Subaşı and Hu (2012); Horowitz (2012); Liu (2012); Chetrite and Mallick (2012); Jaksic et al. (2013); Hekking and Pekola (2013); Horowitz and Parrondo (2013); Leggio et al. (2013); Liu (2014a, b); Silaev et al. (2014); Suomela et al. (2014, 2015, 2016a, 2016b); Manzano et al. (2015); Gasparinetti et al. (2014); Cuetara et al. (2015); Alonso et al. (2016); Gong et al. (2016); Strasberg et al. (2017); Pigeon et al. (2015a); Pigeon and Xuereb (2016); Garrahan and Lesanovsky (2010); Hickey et al. (2012); Žnidarič (2014); Carrega et al. (2015, 2016); Manzano et al. (2015, 2016); Schmidt et al. (2015); Chiara et al. (2015); Wang et al. (2017); Blattmann and Mølmer (2017); Elouard et al. (2017); Perarnau-Llobet et al. (2017); Zhu et al. (2016); Wang et al. (2017); Smith et al. (2017) in the stochastic heat and work of nonequilibrium quantum processes. These studies include investigations of the definitions of these fundamental thermodynamic quantities, their statistical characteristics in general or specific physical models, computational methods, and experimental measurements and realizations. This research boom is driven mainly by two causes. One cause is the practical possibility of manipulating and controlling small quantum systems Gemmer et al. (2005); Boixo et al. (2014); Kosloff (2013); An et al. (2015); Rossnagel et al. (2016). Energy, one of the most important physical quantities, and its transfer are always of great concern, while randomness is an intrinsic characteristic of quantum systems. The other motivation is from theoretical interests. Over almost the same period, a breakthrough in non-equilibrium physics was the discovery of a variety of fluctuation theorems (FTs) Bochkov and Kuzovlev (1977); Evans et al. (1993); Gallavotti and Cohen (1995); Jarzynski (1997a); Kurchan (1998); Lebowitz and Spohn (1999); Maes (1999); Crooks (2000); Hatano and Sasa (2001); Seifert (2005); Kawai et al. (2007); Sagawa and Ueda (2010). These formulas concerning heat, work, and total entropy production were proved to be exact even in far-from-equilibrium regions, whereas in near-equilibrium regions, they are usually reduced to the famous fluctuation-dissipation theorems Degroot and Mazur (1964); Zubarev (1995); Kubo (1991). Because these remarkable results were usually obtained in classical physical systems, there has been intensive theoretical interest to extend them into the quantum regime Bochkov and Kuzovlev (1977); Kurchan (2000); Tasaki (2000); Yukawa (2000); Allahverdyan and Nieuwenhuizen (2005); Talkner et al. (2007); Andrieux and Gaspard (2008); Campisi et al. (2011a); Batalhão et al. (2014); An et al. (2015); Jarzynski et al. (2015).
Among these theoretical efforts, quantum systems that can be described by Markovian quantum master equations (MQMEs) have attracted considerable attention Mukamel (2003); De Roeck and Maes (2004); De Roeck (2007); Esposito et al. (2009); Horowitz and Parrondo (2013); Žnidarič (2014); Esposito and Mukamel (2006); Crooks (2008a); Talkner et al. (2009); Horowitz (2012); Liu (2012); Chetrite and Mallick (2012); Jaksic et al. (2013); Hekking and Pekola (2013); Horowitz and Parrondo (2013); Leggio et al. (2013); Liu (2014a, b); Silaev et al. (2014); Suomela et al. (2014, 2015, 2016a, 2016b); Manzano et al. (2015); Gasparinetti et al. (2014); Cuetara et al. (2015); Alonso et al. (2016); Gong et al. (2016); Pigeon et al. (2015a); Pigeon and Xuereb (2016); Manzano et al. (2016); Wang et al. (2017); Elouard et al. (2017). We believe that this is not coincidental. On the one hand, as the closest extension of Hamiltonian systems, MQMEs Davies (1974); Lindblad (1976); Gorini et al. (1976) have solid mathematical and physical foundations Breuer and Petruccione (2002); Alicki and Lendi (2010); Rivas and Huelga (2012). On the other hand, in the statistical physics community, there has long been a tradition of studying irreversible thermodynamics using the MQMEs Spohn and Lebowitz (1978); Spohn (1978); Davies and Spohn (1978); Alicki (1979); Kohn (2001); Boukobza and Tannor (2006); Kosloff (2013); Alicki (2017). Mukamel Mukamel (2003) proved a quantum Jarzynski equality (JE) for specific MQMEs that can be transformed to the classic MEs Jarzynski (1997b); Crooks (2000); Seifert (2011) in the adiabatic basis. He also presented a formal analogy between the JE and the dephasing of quantum coherences in spectral line shapes. Using an analogous idea, for general Lindblad-type QMEs, Esposito and Mukamel Esposito and Mukamel (2006) obtained several FTs by recasting these equations into the classical birth-death MEs in a time-dependent basis that diagonalizes the reduced density matrix of the system. These authors then unraveled these MEs by classical stochastic jump processes and called them quantum trajectories. Although Esposito and Mukamel defined heat and work at these quantum trajectories as done in the classical cases Seifert (2005, 2011), the relations of these trajectories to measurement scheme remain unclear Teich and Mahler (1992); Wiseman and Milburn (1993). Additionally, to establish a quantum JE, De Roeck and Maes defined the quantum history of a subsystem (they called the subsystem and heat bath the total system) De Roeck and Maes (2004). A realization in their paper is a repeated discrete projective measurement on an open quantum system; between two successive measurements, the subsystem is weakly coupled to an infinite free reservoir and is subjected to an external time-dependent driving potential varying on the time scale of dissipation. After that, De Roeck and Maes Roeck and Maes (2006), De Roeck De Roeck (2007), and Derezinski et al Dereziński et al. (2008) systematically investigated the quantum extension of the Callavotti-Cohen FT for entropy production Gallavotti and Cohen (1995). Their results showed that this task could be realized either in microscopic Hamiltonian systems or in time-homogenous Lindblad QME with a stationary state. Importantly, the former was rigorously proved to converge to the latter in the weak-coupling limit. The important ingredients in defining the entropy production of the specific MQMEs are to unravel them into quantum jump trajectories (QJTs) Srinivas and Davies (1981); Plenio and Knight (1998); Carmichael (1993); Breuer and Petruccione (2002); Gardiner and Zoller (2004); Wiseman and Milburn (2010); Ueda (1990), to give probability measures on these trajectories and to interpret the fluctuating heat current as energy quanta that are transferred between the subsystem and reservoirs. Note that the QJTs here are not the same as those unravelled from the classical MEs as Espositor and Mukamel used Esposito and Mukamel (2006). Interestingly, Breuer Breuer (2003) independently defined the same heat current when he investigated the entropy production rate of the stochastic state vector dynamics. For Hamiltonian systems, heat was microscopically defined as energy eigenvalue differences between the reservoirs at the end and beginning of non-equilibrium processes. This is the famous two-energy-measurement (TEM) scheme defining stochastic thermodynamic quantities Kurchan (2000). Talkner et al. Talkner et al. (2009) proved that JE is always true for general open quantum systems if the system weakly interacts with its environment. In their paper, work was microscopically defined as energy eigenvalue differences between the composite system and its environment at the end and beginning of non-equilibrium processes. Their conclusion does not depend on whether the dynamics of the system is Markovian. Although MQMEs are typical open systems and fulfill the weakly coupling condition, since additional approximations are involved in these equations, whether the strong statement given by Talkner et al. remains true in these equations is not very clear. Crooks Crooks (2008b) discussed how to express the JE for open quantum systems whose dynamics can be described by general quantum dynamical semigroups Breuer and Petruccione (2002). To concisely represent the measurements of the heat flow from environment to the systems, a Hermitian map superoperator was used. In his another paper, Crooks Crooks (2008a) defined the time reversal of a quantum dynamical semigroup having a positive-definite invariant state. At arbitrary time the dynamical semigroup is a completely positive, trace-preserving (CPTP) quantum map, and the map admits a operator-sum representation in terms of a collection of Kraus operators. Because each Kraus operator represents an interaction with the environment that may be measured and recorded by an external observer, Crooks defined a quantum history by noting the observed sequence of Kraus operators. Under these two definitions, Crooks found that the probability of a quantum history and that of the time-reversed history are related by the heat exchanged with the environment. This very insightful paper was further generalized by Manzano et al. Manzano et al. (2015) recently, and a very general FT was obtained.
The reader might note that the above studies were mainly concerned with the validity of various quantum FTs, and their considerations were usually abstract. This situation remained unchanged until Esposito et al. Esposito et al. (2009) derived a generalized quantum master equation (GQME) that can concretely calculate the characteristic function (CF, or the generating function in their paper) of a heat or matter current in time-homogenous MQMEs. The heat or matter current therein was also defined by the TEM scheme on the reservoirs. Importantly, the statistics of these stochastic quantities were implied in the GQME and its solution. This key equation quickly became popular in the literature. For instance, Silaev et al. Silaev et al. (2014) used the GQME to investigate the work and heat statistics in weakly driven open quantum systems Rivas and Huelga (2012). Gasparinetti et al Gasparinetti et al. (2014) and Cuetara et al. Cuetara et al. (2015) further extended the idea of Esposito et al. to periodically driven MQMEs Blümel et al. (1991); Kohler et al. (1997); Breuer and Petruccione (1997); Szczygielski et al. (2013). A very analogous GQME was derived as well. Garrahan and his coauthors Garrahan and Lesanovsky (2010); Garrahan et al. (2011); Hickey et al. (2012, 2013); Lesanovsky et al. (2013) applied the large-deviation method Touchette (2008) to study the “thermodynamics” of the QJTs of Lindblad QMEs, e.g., the dynamical crossover and dynamical phase transitions. Their discussions were also based on the GQME. In one of their papers, Garrahan and Lesanovsky Garrahan and Lesanovsky (2010) noted that the GQME has been in the literature of quantum optics for a long time Gardiner and Zoller (2004). However, in contrast to what Esposito et. al Esposito et al. (2009) did, the earlier version was based on the concept of QJTs and was obtained by finding a set of evolution equations for the reduced density matrix with given jump events Mollow (1975); Zoller et al. (1987). Hence, their observation could be thought of as a concrete demonstration of the statement given by Derezinski et al. Dereziński et al. (2008). Horowitz Horowitz (2012) formulated heat and work for the QJTs of a continuously monitored forced harmonic oscillator coupled to a thermal reservoir. A detailed FT for QJTs was derived. In contrast to previous results using QJTs Roeck and Maes (2006); Garrahan and Lesanovsky (2010), in his model, the Hamiltonian is time dependent. In the same spirit, Hekking and Pelako Hekking and Pekola (2013) investigated the statistics of work for the QJTs in a weakly driven two-level system coupled to a heat reservoir. They calculated the work distributions of the simple model and proved a quantum JE. Horowitz and Parrondom Horowitz and Parrondo (2013) and Horowitz and Sagawa Horowitz and Sagawa (2014) studied the quantum adiabatic and non-adiabatic entropy productions for the general Lindblad master equations. A quantum dual process analogous to the classical dual Markov process Esposito and Van den Broeck (2010) was defined by these authors. They found that these two entropy productions could be defined as ratios of the probabilities of QJTs in the original and dual processes. Chetrite and Mallick Chetrite and Mallick (2012) obtained a quantum Jarzynski-Hatano-Sasa equality for a general Lindblad equation having an instantaneous stationary state solution. To prove this result, they introduced a modified Lindblad equation and utilized a quantum Feynman-Kac (FK) formula. Their results were abstract and lacked probability interpretations. We Liu (2014a, b) noted that, in weakly driven and adiabatically driven MQMEs, the quantum FK formula was a book-keeping expression of the moment-generating function for work; it has a probability explanation if applying the measure of the QJTs. Additionally, in these two papers, in addition to proving a Bochkov-Kuzovlev equality (BKE) and JE, we presented two CF methods for calculating the exclusive and inclusive work distributions Jarzynski (2007); Campisi et al. (2011b). These equations are very analogous to those utilized in classical stochastic systems Imparato and Peliti (2005, 2007). We Liu (2016) studied whether the CF of the exclusive work is equivalent to the CF based on the TEM work defined on the composite system and heat reservoir Talkner et al. (2007); Campisi et al. (2011b). This affirmative result further stimulated us Liu and Xi (2016) to develop the CF methods of heat and work based on the QJT concept for several time-dependent MQMEs. Their connections with the CFs of heat and work based on the TEM scheme were established as well. Very recently, Suomela et al. Suomela et al. (2016a, b) developed a modified QJT model for systems interacting with a finite-size heat bath. This work was mainly motivated by a calorimetric measurement of work in driven open quantum system Pekola et al. (2013). Elouard et al. Elouard et al. (2017) considered the possibility of defining quantum heat that is purely measurement induced. In addition to QJTs, their formalism also includes the quantum sate diffusion (QSD) unravelling of MQMEs. Finally, Gong et al. Gong et al. (2016) considered these feedback effects on quantum fluctuation effects using the QJT concept.
After the above long sketch of the literature, we clearly see that two main strategies are emerging in the studies of the quantum stochastic thermodynamics (QST) of MQMEs. One strategy is based on the TEM scheme defining thermodynamic quantities on the composite system and environments Kurchan (2000); Roeck and Maes (2006); Esposito et al. (2009); Campisi et al. (2011a); the basic tool of computations and statistical analysis is the GQME Esposito et al. (2009); Silaev et al. (2014); Gasparinetti et al. (2014); Cuetara et al. (2015). The other strategy is to utilize the fact that the MQMEs can be unraveled into QJTs Srinivas and Davies (1981); Ueda (1990); Carmichael (1993); Plenio and Knight (1998); Gardiner and Zoller (2004); Breuer and Petruccione (2002); Wiseman and Milburn (2010); along each QJT, heat, work, and entropy production can be well defined Breuer (2003); Roeck and Maes (2006); Horowitz (2012); Hekking and Pekola (2013); Liu (2014a, b); the computational and statistical analysis method could be straightforward Monte-Carlo simulation Horowitz (2012); Hekking and Pekola (2013) or the CFs based on the QJT concept Liu (2014a, b); Liu and Xi (2016). In this paper, we attempt to give a self-contained review of these two strategies. The concrete contents are as follows. (1) We systematically survey a variety of MQMEs that were often used in the study of quantum thermodynamics. Although the MQMEs have been criticized because many conditions or approximations are involved in their derivations, these equations indeed have elegant mathematical structures and are also easily accessible. (2) We provide a careful overview of the CFs for heat defined by the TEM scheme in these equations, and we check whether there is a unified GQME. In particular, we apply the same effort to the case of work. To our knowledge, few studies have examined this issue in a systematic manner. (3) We interpret QJTs in terms of the TEM scheme on the heat bath atoms. Many papers have applied this energy interpretation Roeck and Maes (2006); Horowitz and Parrondo (2013); Hekking and Pekola (2013); Liu (2014a), but few of them clearly explained their reasoning. Additionally, the concept of QJTs remains unfamiliar to many researchers who are concerned about quantum FTs, although there have been such concepts since as early as the 70s of the last century Davies (1969); now, they are broadly applied in quantum optics Gardiner and Zoller (2004); Wiseman and Milburn (2010). To attract additional attention, a detailed explanation seems to be essential. (4) We systematically establish the CFs of heat and work defined at QJTs and utilize the result to prove several finite-time FTs. In contrast to previous work, where almost all formulas were restricted to the MQMEs whose QJT has a wave-vector description Roeck and Maes (2006); Horowitz and Parrondo (2013); Hekking and Pekola (2013); Liu (2014a), our current results are sufficiently general to account for the cases in which the concept of QJTs is only available in the density matrix spaces of the quantum systems.
Let us present the scope of this paper’s discussion. First, we only consider one thermal heat bath interacting with a quantum open system. The particles of the system are not allowed to be exchanged with the environment. Hence, we do not discuss the quantum thermodynamics of quantum heat engines or quantum refrigerators here Kosloff (2013); Vinjanampathy and Anders (2016). Second, we are only concerned with the finite-time statistics for heat and work rather than their long-time behaviors. For the latter, the interested reader is referred to the excellent paper by Espositor et al Esposito et al. (2009) or the other relevant references Žnidarič (2014); Gasparinetti et al. (2014); Cuetara et al. (2015); Pigeon et al. (2015a, b). Finally, we focus on the concrete MQMEs rather than provide abstract discussions using the general CPTP quantum maps Crooks (2008a); Manzano et al. (2015).
The paper is organized as follows. In Sec. II, we sketch the general mathematical structure of MQMEs. Then, four types of equations that have often been used in the literature are derived. We show that all types can be unified into a general formula. In Sec. III, we introduce the definition of work based on the TEM scheme in closed quantum systems. The time evolution equation of the work characteristic operator (WCO) is established. In Sec. IV, we investigate the derivations of the time evolution equations of the heat characteristic operator (HCO) and WCO for individual MQMEs. We find that they have unified forms as well. In addition, we compare the means of the stochastic heat and work to the ensemble heat and work that are typically proposed and used in ensemble quantum thermodynamics. In Sec. V, we utilize the evolution equations of the HCO and WCO to prove several integral and detailed FTs. In Sec. VI, we introduce the concept of QJTs using a repeated interaction model in detail. We explicitly apply the TEM scheme to heat bath atoms. Based on this concept, we define the stochastic heat and work at the QJTs in Sec. VII, and their corresponding COs and CFs are investigated in Sec. VIII. In Sec. IX, the FTs at the QJT level are proved. Section X concerns the total entropy production of open quantum systems. In Sec. XI, we use a periodically driven two-level system to numerically illustrate several results obtained in this paper. Section XII is the conclusion of this paper.
Ii Markovian quantum master equations
ii.1 General mathematical structure
An open quantum system is a quantum system that interacts with another quantum system. The latter is usually called the environment. We denote the system and the environment by capital letters A and B, respectively. Because the environment B in which we are interested has an infinite number of degrees of freedom and is assumed to be in a thermal equilibrium state initially, we also call it the heat bath. The system A evolves under its own dynamics, the interaction with the heat bath B, and some external classical forces or fields; therefore, its dynamics is usually very complex. A conventional method of treating such a complicated situation is to regard the open quantum system A and heat bath B as a global system. Because the composite system is closed, we can describe its dynamics by a unitary evolution with a total Hamiltonian
where and are the Hamiltonians of system A and heat bath B, respectively. The interaction Hamiltonian between these two subsystems is
where and are the Hermitian operators of these two respective subsystems. This interaction is supposed to be very weak to ensure the validity of MQMEs. In addition, throughout this paper we also suppose that the composite system has an uncorrelated initial density matrix,
Here, and are the initial density matrix of system A and heat bath B, respectively. Because heat bath B is in a thermal equilibrium state at time 0, has a canonical form: is the inverse temperature, , and is the partition function. Note that is the trace taken over the heat bath. In the remainder of this paper, we always use the subscript to denote the degree of freedoms over which the trace is being taken. At a later time , the density matrix of the composite system, , satisfies the Liouville-von Neumann equation
Planck’s constant has been set equal to 1 here. If one could solve the equation, the dynamics of the open quantum system A is then obtained by taking the trace over the degree of freedom of the heat bath, i.e.,
Because this is a reduction procedure of the global density matrix, is also called the reduced density matrix of the reduced system A.
The solution of Eq. (4) can always be written formally as
by introducing the time evolution operator of the composite system,
where the symbol denotes the chronological time-ordering operator. Considering that the open system A and heat bath B are initially uncorrelated, as in Eq. (3), the reduced density matrix of system A at time can also be expressed using a dynamical map :
is called the Kraus operator. To obtain Eq. (8), we have used a spectral decomposition of the density matrix of the heat bath:
where and are the eigenvalue and eigenvector of the Hamiltonian , respectively, and
The dynamical map has three important properties: convex linearity, trace preservation, and complete positivity Breuer and Petruccione (2002); Alicki et al. (2006); Rivas and Huelga (2012). The dynamic map at arbitrary time forms a one-parameter family . Although in principle this family describes the entire evolution of the open quantum system A, due to the complexity of the composite unitary evolution for a general , it could be very involved. The simplest one-parameter family should be a Markovian family, namely,
Such a family is also called quantum dynamical semigroups Breuer and Petruccione (2002); Alicki et al. (2006); Rivas and Huelga (2012). Importantly, one can alternatively express the dynamical semigroup using a linear differential equation or a MQME:
where the superoperator is called the semigroup generator. Obviously, Eq. (12) is the quantum analogue to the Chapman-Kolmogorov equation in classical Markovian processes Gardiner (1983a). In particular, for a finite -dimensional semigroup, the most general form of the generator was shown to be Rivas and Huelga (2012)
where is the effective Hamiltonian, which is not necessarily identical to the system’s free Hamiltonian, ; is a complete orthonormal basis with respect to the Hilbert-Schmidt inner product; in the space of operators for the open quantum system A; and the matrix is Hermitian and positive semidefinite. We call the commutator in the above equation the coherent dynamics and call the summation term the dissipator. Because is positive, one may always introduce a new set of operators to replace the set of operators and rewrite Eq. (14) in the “diagonal” form:
where the non-negative coefficients are the eigenvalues of . Historically, Gorini, Kossakowski, and Sudarshan Gorini et al. (1976) were credited with establishing Eq. (14) for a time-homogenous quantum dynamic map, that is, depends only on the difference . In this situation, the matrix is time independent. Independently, also for the homogenous case, Lindblad Lindblad (1976) obtained Eq. (15) for the infinite-dimensional semigroups with bounded generators Alicki et al. (2006). In the literature, Eqs. (14) and (15) are called the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation.
ii.2 Microscopic models
Eq. (14), and its equivalent Eq. (15), is only formally exact; its derivation does not depend on the concrete physical models. From the viewpoint of physics, it shall be more desirable if we could derive it from first principles, e.g., the microscopic expressions for and . Indeed, under the weak-coupling assumption, that is, the interaction term in Eq. (1) is so weak that the influence of the open quantum system A on the heat bath B is small, and under several other key approximations, including the Born-Makovian approximation and the rotating wave approximation, one can indeed obtain a variety of MQMEs having the form of Eq. (14). In this paper, we will discuss four types of equations. The first type is the time-homegeneous MQMEs Davies (1974); Lindblad (1975); Gorini et al. (1976); Rivas and Huelga (2012), which we occasionally call the standard master equations in this paper. These equations have mainly been applied to issues related to how a system relaxes into a thermal equilibrium state Breuer and Petruccione (2002); Alicki and Lendi (2010) or arrives at non-equilibrium steady state De Roeck and Maes (2004); Dereziński et al. (2008); Esposito et al. (2009); Garrahan and Lesanovsky (2010); Žnidarič (2014); Schaller (2014). The second type has time-dependent coherent dynamics, but the dissipator is static Geva and Kosloff (1994); Hekking and Pekola (2013); Liu (2014a); Silaev et al. (2014). These equations have often been used in quantum optics Breuer and Petruccione (2002); Carmichael (1993); Wiseman and Milburn (2010), e.g., a two-level atom interacting with a radiation field while being driven by a classical time-varying electric field Mollow (1975). Their physical validity is ensured if the external driving field is so weak that its effect on the environment is negligible. The equations of the last two types fully depend on time, including the adiabatically (slowly) driven Davies and Spohn (1978); Alicki (1979); Albash et al. (2012); Horowitz (2012); Liu (2014b); Suomela et al. (2015) and periodically driven MQME Blümel et al. (1991); Kohler et al. (1997); Breuer and Petruccione (1997); Szczygielski et al. (2013); Gasparinetti et al. (2014); Cuetara et al. (2015). Although the applicable regions of these four types of master equations are very distinct Alicki et al. (2006), all of them are very analogue to Eq. (14). In the remainder of this section, we will survey their derivations.
We first write Eq. (4) in the interaction picture:
These notations with tildes indicate that they are interaction picture operators with respect to the free Hamiltonians, namely, an arbitrary operator ,
where the time evolution operator for the free Hamiltonians is
The explicit expression of the interaction picture operator for is
It is not difficult to prove that
The second equation has a formal solution:
Here, we have considered the vanishing initial condition, , and have defined the propagator
Considering that the Liouville-von Neumann equation (4) is unitary, we may connect at two times and () by
where is the time evolution operator of the interaction picture operator ,
The reader is reminded that this result is a time-convolutionless form with respect to : the density matrix on the right-hand side (RHS) depends on the time instead of the earlier time Breuer and Petruccione (2002).
To simplify the complex integral in Eq. (29), we must resort to several approximations. Let us redefine as , where is a small perturbation parameter. We expand the above equation up to the second order of and then arrive at
Here, we have performed a change of variables, , and used an approximation
Inserting the concrete formulas of the projector operator and into Eq. (30), we obtain
where and its initial condition is . To proceed further, we need to perform a spectrum decomposition on the interaction picture operators or . Unfortunately, this is not always well-defined for general . We have to restrict our discussion to the four types of MQMEs. After individually investigating them, we will see that these equations have a unified formula.
ii.2.1 Static Hamiltonian
This is the standard case and is also the simplest case in which the Hamiltonian of the quantum system is time independent Lindblad (1975); Gorini et al. (1976). The time evolution operator of system A is
where and are the discrete energy eigenvector and eigenvalue of , respectively. Then, the interaction picture operator has the following spectrum decomposition Breuer and Petruccione (2002):
The energy differences are called Bohr frequencies, which may be positive or negative but always appear in pairs, and is the Kronecker delta. These operators satisfy simple properties:
Therefore, and are also called the eigenoperators of . The reader is particularly reminded that different operators denoted by different indices may have the same frequency , as with the operators . This point is very relevant to the descriptions of QJT using either wave vectors or density matrices. Substituting the decomposition (34) into Eq. (32), we obtain
Here, we have used the time-homogeneous property of the two-time correlation function,
which is a result of the stationary heat bath .
There are two further approximations involved in the following derivations Breuer and Petruccione (2002). The first approximation is that the upper integral limit is replaced by . This approximation is justified if the evolution time scale of system A is far larger than the decay time of the heat bath. Specifically, significant non-zero contributions of these correlation functions in the integrals are around . Under this approximation, it shall be convenient to write these one-side Fourier integrals by the double-side Fourier integrals, e.g.,
and denotes the Cauchy principal value of the integral. The other integrals can be rewritten analogously. Because the heat bath is at thermal equilibrium, these correlation functions satisfy the so-called Kubo-Martin-Schwinger (KMS) condition Kubo (1957); Martin and Schwinger (1959). In particular, this condition on the Fourier transform is
We will show later that this relation plays a key role in proving various FTs. The second approximation is the rotation wave approximation, or the secular approximation. Specifically, the terms with different and on the RHS of Eq. (38) are neglected. This approximation would be valid if a typical value of is far larger than the reciprocal of time, . Given these two key approximations, doing a simple algebraic manipulation, we finally obtain
where the Lamb-shift term is
Here, we have reabsorbed the small parameter into the interaction Hamiltonian . Note that these two approximations can be rigorously justified by taking the weak coupling limit, , , and keeping constant Davies (1974); Gorini et al. (1976); Alicki et al. (2006); Rivas and Huelga (2012).
Eq. (44) can be transformed back into the Schrdinger picture. First, we know that
we immediately find that the second term on the RHS of Eq. (46) is exactly the same as the RHS of Eq. (44) except that all therein are replaced by . After the above discussion, we have an conclusion that, if the decomposition (34) is true, the following equations, (38)-(45), would be available. This observation will guide our discussion about the time-dependent MQMEs.
ii.2.2 Weakly driven Hamiltonian
The first time-dependent case is the weakly driven Hamiltonian,
where is the constant Hamiltonian of the bare system, represents the coupling term of the bare system and some time-varying external force or fields, and the parameter accounts for the coupling strength. We assume that is of the same order of magnitude as in the interaction Hamiltonian . This is the precise meaning of being weakly driven. Under this assumption, the time evolution operator of system A is approximated to be
When we substitute Eq. (49) into Eq. (32), the higher orders should be . Because we only retain the terms up to second order and neglect the other higher order terms, we re-obtain Eq. (44). However, we shall bear in mind that the decomposition, Eq. (34), is with respect to the eigenvectors and eigenvalues of the bare Hamiltonian, . Additionally, when we transform the time evolution equation into the Schrdinger picture, will be recovered because this Hamiltonian has a contribution to first order in .
ii.2.3 Periodically driven Hamiltonian
It will be more desirable if we have a decomposition of Eq. (34) even if the Hamiltonian of the quantum system A is strongly driven. This is indeed possible if is periodic with a frequency Blümel et al. (1991); Kohler et al. (1997); Breuer and Petruccione (1997); Grifoni and Hänggi (1998); Breuer et al. (2000); Kohn (2001); Szczygielski et al. (2013); Langemeyer and Holthaus (2014); Gasparinetti et al. (2014); Cuetara et al. (2015), namely,
where these time-independent are called quasi-energies, and the Floquet basis satisfies
The Floquet basis is periodic, that is,
It is not difficult to find that the solution
is still in the Floquet basis but with a shifted quasi-energy
where is an arbitrary integer. Therefore, for the periodic Hamiltonian, the Floquet bases connected by Eq. (54) are equivalent in physics. We may project them into one Brillouin zone of width . Using Eq. (51), we may see that still has a decomposition similar to Eq. (34) Breuer (2004):
or the Fourier coefficient with frequency . Notice that in contrast to the static Hamiltonian case, the Bohr frequency here depends on the external periodicity . Substituting the new decomposition into Eq. (32), applying the same approximations as those in the static Hamiltonian case, and going back to the Schrdinger picture, we finally obtain
where the Lamb shift is now
In this derivation, we have used the formula Breuer (2004)
Compared with the evolution equation for the static Hamiltonian case, Eq. (46), the time dependence in Eq. (59) is through the operators and in addition to the time-dependent coherent dynamics. Nevertheless, the coefficients remain static.
ii.2.4 Adiabatically driven Hamiltonian
If the Hamiltonian varies in time very slowly, the adiabatical approximation can be applied to the operators Davies and Spohn (1978); Alicki (1979); Amin et al. (2008); Albash et al. (2012). In this case, the time evolution operator of system A is
where and are the instantaneous energy eigenvector and eigenvalues of , respectively, and the phase in the exponential function is
With the above approximation, we have the following decomposition:
Note that there still exist two identities analogous to Eq. (37):
where is brief notation of and the Bohr frequency is
Different from previous cases, these frequencies are generally time dependent. For notational simplicity, we suppose that both and are uniquely determined by the quantum numbers . The decomposition (64) remains inadequate; we still need another key approximation:
Under this approximation, we have the following decomposition:
The approximation (70) shall be justified if is almost unchanged in the time interval . Obviously, is preferred. Importantly, because of the very short decay time of these two-time correlation functions in Eq. (32), this condition is always fulfilled. We may relax the seemly strict condition of . To ensure the validity of the adiabatical approximation, Eq. (62), the duration of the non-equilibrium process shall be sufficiently large Messiah (1962). In this situation, the change in the Hamiltonian would be very slow. Hence, in the interval with nonzero , the Hamiltonian could still be reasonably thought to be a constant operator. In particular, we expect that in the quantum adiabatic limit, or , may be any finite value.
where the Lamb-shift term is
where the operator is in fact the defined in Eq. (65), when we return to the Schrdinger picture, the time evolution equation for the reduced density matrix of the quantum open system A is
Distinct from the three previous cases, the coefficients in the current case generally vary with time through the time-dependent Bohr frequencies, . Finally, we want to emphasize that Eq. (76) can be proved to be rigorous if one considers the weak coupling limit and quantum adiabatic limit simultaneously Davies and Spohn (1978); Thunström et al. (2005); Oreshkov and Calsamiglia (2010).
After a survey of the above derivations of these four types of MQMEs, we find that all of them can be unified into a general formula:
and the positive and negative Bohr frequencies appear in pairs. Obviously, this formula follows the structure of Eq. (14). We can write its solution formally as
Iii Work in closed quantum systems
Let us start with the stochastic work of closed quantum systems. This concept is relatively simple and is also the foundation of studying the stochastic heat and work in open quantum systems. For convenience, throughout this paper, we suppose that the time dependence of the Hamiltonian of the quantum system A is always realized by a protocol, , namely,
as is the case for the time dependence of the instantaneous energy eigenvectors and eigenvalues of the Hamiltonian,
Because there is only one quantum system, we do not explicitly write the subscript A here. Consider that the non-equilibrium process of a closed quantum system is performed within the time interval . According to the TME scheme Kurchan (2000); Tasaki (2000), the stochastic work done by some external agents on the quantum system is defined as the difference in the instantaneous energy eigenvalues between the end and beginning of the process:
Following the terms named by Jarzynski Jarzynski (2007), we call Eq. (82) the inclusive work. The reason for this will be given shortly. By repeating this process many times, we can construct the probability distribution of the work as
where is the time evolution operator of the system and is the probability of finding the system at the eigenvector at time . Because of the involvement of the Dirac function, Eq. (81) is not the most convenient form if we want to analyze the statistical properties of the work distribution. An alternative method is to resort to its Fourier transform or CF Gardiner (1983a); Risken (1984); Kurchan (1998); Lebowitz and Spohn (1999); Imparato and Peliti (2007); Talkner et al. (2007); Esposito et al. (2009); Campisi et al. (2011a):
At first glance, both Eqs. (83) and (84) depend on the time evolution operator of the quantum system, ; no apparent advantages are shown in the latter expression. However, the CF can be rewritten as taking the trace over some operators, that is,