We present a fluctuation theorem for Floquet quantum master equations. This is a detailed version of the famous Gallavotti-Cohen theorem. In contrast to the latter theorem, which involves the probability distribution of the total heat current, the former involves the joint probability distribution of positive and negative heat currents and can be used to derive the latter. A quantum two-level system driven by a periodic external field is used to verify this result.
By directly using the probability formulas of quantum trajectories, we construct an auxiliary open quantum system for a periodically driven open quantum system whose dynamics is governed by the Floquet quantum master equation. This auxiliary system can generate a quantum trajectory ensemble that is consistent with the canonical quantum trajectory ensemble. We find that, at a long time limit, though the Lindblad operators are modified, the coherent dynamics of the auxiliary system is the same as that of the original system. A periodically driven two-level quantum system is used to illustrate th…
The stroboscopic evolution of a time-periodically driven isolated quantum system can always be described by an effective time-independent Hamiltonian. Whether this concept can be generalized to open Floquet systems, described by a Markovian master equation with time-periodic Lindbladian generator, remains an open question. By using a two level system as a model, we explicitly show the existence of two well-defined parameter regions. In one region the stroboscopic evolution can be described by a Markovian master equation with a time-independent Floquet Lindbladian. In the other it cannot; but h…
Based on the notion of quantum trajectory, we present a stochastic theoretical framework for Floquet quantum heat engines. As an application, the large deviation functions of two types of stochastic efficiencies for a two-level Floquet quantum heat engine are investigated. We find that the statistics of one efficiency agree well with the predictions of the universal theory of efficiency fluctuations developed by Verley et al. [Phys. Rev. E {\bf 90}, 052145 (2014)], whereas the statistics of the other efficiency do not. The reason for this discrepancy is attributed to the lack of fluctuation th…
As a paradigmatic model of open quantum system, the spin-boson model is widely used in theoretical and experimental investigations. Beyond the weak coupling limit, the spin dynamics can be described by a time-nonlocal generalized master equation with a memory kernel accounting for the dissipative effects induced by the bosonic environment. When the spin is additionally modulated by an external time-periodic electromagnetic field, the interplay between dissipation and modulations yields a spectrum of nontrivial asymptotic states, especially in the regime of nonlinear response. Here we implement…
We study probability distribution of a steady state of a periodically driven system coupled to a thermal bath by using a quantum master equation in the weak coupling limit. It is proved that, even when the external field is strong, the probability distribution is independent of the detailed nature of the thermal bath under the following conditions: (i) the Hamiltonian of the relevant system is bounded and the period of the driving field is short, (ii) the Hamiltonians for the driving field at different times commute, and (iii) the Hamiltonians of the driving field and of the interaction betwee…
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 (QME). 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 f…
A nonequilibrium master equation analysis for N interacting bosons, with Bogoliubov quasiparticles as the reservoir is presented. The analysis is based on a simplified Hamiltonian. The steady state solution yields the equilibrium density matrix. The results are in good agreement with and extend our previous rigorous canonical ensemble equilibrium statistical treatment leading to a quantum theory of the atom laser.
We study a dynamical phase transition in optical bistable systems subject to a time-periodic driving field. The phase transition occurs in the structure of limit cycle as a function of the frequency of the driving field. In the thermodynamic limit, a single limit cycle is divided into two separated limit cycles at the transition point. In finite-size systems, however, there is always a single limit cycle due to the quantum tunneling effect. We use a Floquet dissipative map, which is a time-evolution operator over one period in a dynamics given by a quantum master equation, and discuss the deca…
In Floquet engineering, periodic driving is used to realize novel phases of matter which are inaccessible in thermal equilibrium. For this purpose, the Floquet theory provides us a recipe of obtaining a static effective Hamiltonian. Although many existing works have treated closed systems, it is important to consider the effect of dissipation, which is ubiquitous in nature. Understanding interplay of periodic driving and dissipation is not only a fundamental problem of nonequilibrium statistical physics, but also receiving growing interest because of the fact that experimental advances have al…
The very notion of a current fluctuation is problematic in the quantum context. We study that problem in the context of nonequilibrium statistical mechanics, both in a microscopic setup and in a Markovian model. Our answer is based on a rigorous result that relates the weak coupling limit of fluctuations of reservoir observables under a global unitary evolution with the statistics of the so-called quantum trajectories. These quantum trajectories are frequently considered in the context of quantum optics, but they remain useful for more general nonequilibrium systems. In contrast with the appro…
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 current understanding on the stochastic heat and work in MQMEs using two distinct strategie…
We develop a comprehensive Renormalization Group (RG) approach to criticality in open Floquet systems, where dissipation enables the system to reach a well-defined Floquet steady state of finite entropy, and all observables are synchronized with the drive. We provide a detailed description of how to combine Keldysh and Floquet formalisms to account for the critical fluctuations in the weakly and rapidly driven regime. A key insight is that a reduction to the time-averaged, static sector, is not possible close to the critical point. This guides the design of a perturbative dynamic RG approach, …
The fluctuation-dissipation relation is calculated for a class of stochastic models obeying a master equation. The transition rates are assumed to obey detailed balance also in the presence of a field. It is shown that in general the linear response cannot be expressed via time-derivatives of the correlation function alone, but an additional function $ξ(t,t_w)$, which has been rarely discussed before is required. This function depends on the two times also relevant for the response and the correlation and vanishes under equilibrium conditions. It can be expressed in terms of the propagators an…
Physical and chemical stochastic processes described by the master equation are investigated. In this paper, we examine the entropy production both for the master equation and for the corresponding Fokker-Planck equation. For the master equation, the exact expression of the entropy production was recently derived by Gaspard using the Kolmogorov-Sinai entropy ({\em J.Stat.Phys.}, \textbf{117} (2004), 599; [Errata; \textbf{126} (2006), 1109 ]). Although Gaspard's expression is derived from a stochastic consideration, it should be noted that Gaspard's expression conincides with the thermodynamica…
An open quantum system interacting with its environment can be modeled under suitable assumptions as a Markov process, described by a Lindblad master equation. In this work, we derive a general set of fluctuation relations for systems governed by a Lindblad equation. These identities provide quantum versions of Jarzynski-Hatano-Sasa and Crooks relations. In the linear response regime, these fluctuation relations yield a fluctuation-dissipation theorem (FDT) valid for a stationary state arbitrarily far from equilibrium. For a closed system, this FDT reduces to the celebrated Callen-Welton-Kubo …
Fluctuations and optimal control in a Floquet Quantum Thermal Transistor
Indian Institute of Technology Kanpur（坎普尔印度理工学院）；Universidad de Santiago de Chile（智利圣地亚哥大学）；Millennium Institute for Research in Optics（光学研究千年研究所）；IISER Berhampur（贝拉姆浦尔印度科学教育与研究所）；University of Pennsylvania（宾夕法尼亚大学）
The operation of autonomous finite-time quantum heat engines rely on the existence of a stable limit cycle in which the dynamics becomes periodic. The two main questions that naturally arise are therefore whether such a limit cycle will eventually be reached and, once it has, what is the state of the system within the limit cycle. In this paper we show that the application of Floquet's theory to Lindblad dynamics offers clear answers to both questions. By moving to a generalized rotating frame, we show that it is possible to identify a single object, the Floquet Liouvillian, which encompasses …
Inspired on the continued-fraction technique to solve the classical Fokker--Planck equation, we develop continued-fraction methods to solve quantum master equations in phase space (Wigner representation of the density matrix). The approach allows to study several classes of nonlinear quantum systems subjected to environmental effects (fluctuations and dissipation), with the only limitations that the starting master equations may have. We illustrate the method with the canonical problem of quantum Brownian motion in periodic potentials.
We study a two-state statistical process with a non-Poisson distribution of sojourn times. In accordance with earlier work, we find that this process is characterized by aging and we study three different ways to define the correlation function of arbitrary age of the corresponding dichotomous fluctuation based respectively on the Generalized Master Equation formalism, on a Liouville-like approach and on a trajectory perspective.
We develop the Floquet-Magnus expansion for a classical equation of motion under a periodic drive that is applicable to both isolated and open systems. For classical systems, known approaches based on the Floquet theorem fail due to the nonlinearity and the stochasticity of their equations of motion (EOMs) in contrast to quantum ones. Here, employing their master equation, we successfully extend the Floquet methodology to classical EOMs to obtain their Floquet-Magnus expansions, thereby overcoming this difficulty. Our method has a wide range of application from classical to quantum as long as …
We present a general quantum fluctuation theorem for the entropy production of an open quantum system coupled to multiple environments, not necessarily at equilibrium. Such a general theorem, when restricted to the weak-coupling and Markovian regime, holds for both local and global master equations, corroborating the thermodynamic consistency of local quantum master equations. The theorem is genuinely quantum, as it can be expressed in terms of conservation of a Hermitian operator, describing the dynamics of the system state operator and of the entropy change in the baths. The integral fluctua…
The atom fluctuations statistics of an ideal, mesoscopic, Bose-Einstein condensate is investigated from several different perspectives. By generalizing the grand canonical analysis (applied to the canonical ensemble problem), we obtain a self-consistent equation for the mean condensate particle number that coincides with the microscopic result calculated from the laser master equation approach. For the case of a harmonic trap, we obtain an analytic expression for the condensate particle number that is very accurate at all temperatures, when compared with numerical canonical ensemble results. A…
We present a theoretical investigation of thermal fluctuation statistics in a molecular motor. Energy transfer in the motor is described using a multidimensional discrete master equation with nearest-neighbor hopping. In this theory, energy transfer leads to statistical correlations between thermal fluctuations in different degrees of freedom. For long times, the energy transfer is a multivariate diffusion process with a constant drift and diffusion. The fluctuations and drift align in the strong-coupling limit enabling a one-dimensional description along the coupled coordinate. We derive form…
Lower bounds on fluctuations of thermodynamic currents depend on the nature of time: discrete or continuous. To understand the physical reason, we compare current fluctuations in discrete-time Markov chains and continuous-time master equations. We prove that current fluctuations in the master equations are always more likely, due to random timings of transitions. This comparison leads to a mapping of the moments of a current between discrete and continuous time. We exploit this mapping to obtain new uncertainty bounds. Our results reduce the quests for uncertainty bounds in discrete and contin…
We present a characteristic function method to calculate the probability density functions of the inclusive work in the adiabatic two-level quantum Markovian master equations. These systems are steered by some slowly varying parameters and the dissipations may depend on time. Our theory is based on the interpretation of the quantum jump for the master equations. In addition to the calculation, we also find that the fluctuation properties of the work can be described by the symmetry of the characteristic functions, which is exactly the same as the case of the isolated systems. A periodically dr…
We develop a variational scheme in a field theoretic approach to a stochastic process. While various stochastic processes can be expressed using master equations, in general it is difficult to solve the master equations exactly, and it is also hard to solve the master equations numerically because of the curse of dimensionality. The field theoretic approach has been used in order to study such complicated master equations, and the variational scheme achieves tremendous reduction in the dimensionality of master equations. For the variational method, only the Poisson ansatz has been used, in whi…
We derive an extended fluctuation theorem for a geometric pumping in a spin-boson system under a periodic control of environmental temperatures by using a Markovian quantum master equation. We perform the Monte-Carlo simulation and obtain the current distribution, the average current and the fluctuation. Using the extended fluctuation theorem we try to explain the results of our simulation. The fluctuation theorem leads to the fluctuation dissipation relations but the absence of the conventional reciprocal relation.
Motivated by recent progresses in nonequilibrium Fluctuation Relations, we present a generalized time reversal for stochastic master equation systems with discrete states that is defined as a splitting of the rate matrix into irreversible and reversible parts. An immediate advantage of this definition is that a variety of fluctuation relations can be attributed to different matrix splitting. Additionally, we also find that, the accustomed total entropy production formula and conditions of the detailed balance must be modified appropriately to account for the presence of the reversible part, wh…
We investigate the sensitivity of the time evolution of a kinetic Ising model with Glauber dynamics against the initial conditions. To do so we apply the "damage spreading" method, i.e., we study the simultaneous evolution of two identical systems subjected to the same thermal noise. We derive a master equation for the joint probability distribution of the two systems. We then solve this master equation within an effective-field approximation which goes beyond the usual mean-field approximation by retaining the fluctuations though in a quite simplistic manner. The resulting effective-field the…
We calculate a current and its fluctuation in a two-state stochastic system under a periodic perturbation. The system could be interpreted as a channel on a cell surface or a single Michaelis-Menten catalyzing enzyme. It has been shown that the periodic perturbation induces so-called pump current, and the pump current and its fluctuation are calculated with the aid of the geometrical phase interpretation. We give a simple calculation recipe for the statistics of the current, especially in a non-adiabatic case. The calculation scheme is based on the non-adiabatic geometrical phase interpretatio…
We show a microscopic derivation of a quantum master equation with counting terms which describes the electron statistics. A localized spin behaves as a probe whose precession angle monitors the net electron current by the magnetic-moment interaction. The probe Hamiltonian is proportional to the current, and is determined self-consistently for a model of a quantum dot. Then it turns out that the quantum master equation for the spin-precession contains the counting terms. As an application, we show the fluctuation theorem for the electron current.
We propose a piecewise deterministic Markovian jump process in Hilbert space such that the covariance matrix of this stochastic process solves the thermodynamic quantum master equation. The proposed stochastic process is particularly simple because the normalization of the vectors in Hilbert space is preserved only on average. As a consequence of the nonlinearity of the thermodynamic master equation, the construction of stochastic trajectories involves the density matrix as a running ensemble average. We identify a principle of detailed balance and a fluctuation-dissipation relation for our Ma…
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 {\bf73}, 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 num…
Reaction networks in the bulk and on surfaces are widespread in physical, chemical and biological systems. In macroscopic systems, which include large populations of reactive species, stochastic fluctuations are negligible and the reaction rates can be evaluated using rate equations. However, many physical systems are partitioned into microscopic domains, where the number of molecules in each domain is small and fluctuations are strong. Under these conditions, the simulation of reaction networks requires stochastic methods such as direct integration of the master equation. However, direct inte…
We derive a general set of fluctuation relations for a nonequilibrium open quantum system described by a Lindblad master equation. In the special case of conservative Hamiltonian dynamics, these identities allow us to retrieve quantum versions of Jarzynski and Crooks relations. In the linear response regime, these fluctuation relations yield a fluctuation-dissipation theorem (FDT) valid for a stationary state arbitrarily far from equilibrium. For a closed system, this FDT reduces to the celebrated Callen-Welton-Kubo formula.
We present a statistical physics framework for description of nonlinear non-equilibrium stochastic processes, modeled via chemical master equation, in the weak-noise limit. Using the Poisson representation approach and applying the large-deviation principle we first solve the master equation. Then we use the notion of the non-equilibrium free energy to derive an integral fluctuation relation for nonlinear non-equilibrium systems under feedback control. We point out that the free energy as well as some functionals can serve as non-equilibrium Lyapunov function which has an important property to…
We derive the Markov process equivalent to She-Leveque scaling in homogeneous and isotropic turbulence. The Markov process is a jump process for velocity increments $u(r)$ in scale $r$ in which the jumps occur randomly but with deterministic width in $u$. From its master equation we establish a prescription to simulate the She-Leveque process and compare it with Kolmogorov scaling. To put the She-Leveque process into the context of other established turbulence models on the Markov level, we derive a diffusion process for $u(r)$ from two properties of the Navier-Stokes equation. This diffusion …
With a modulated oscillator, we study several effects of quantum fluctuations far from thermal equilibrium. One of them is quantum heating, where quantum fluctuations lead to a finite-width distribution of a resonantly modulated oscillator over its quasienergy (Floquet) states. We also analyze large rare fluctuations responsible for the tail of the quasienergy distribution and switching between the states of forced vibrations. We find an observable characteristic of these fluctuations, the most probable paths followed by the quasienergy in rare events, and in particular in switching. We also e…
