Path Integral Approach to Quantum Thermodynamics

Structure of quantum mean force Gibbs states for coupled harmonic systems

An open quantum system interacting with a heat bath at a given temperature is expected to reach the mean force Gibbs (MFG) state as a steady state. The MFG state is given by tracing out the bath degrees of freedom from the equilibrium Gibbs state of the total system plus bath. When the interaction between the system and the bath is not negligible, it is different from the usual system Gibbs state obtained from the system Hamiltonian only. Using the path integral method, we present the exact MFG state for a coupled system of quantum harmonic oscillators in contact with multiple thermal baths at the same temperature. We develop a nonperturbative method to calculate the covariances with respect to the MFG state. By comparing them with those obtained from the system Gibbs state, we find that the effect of coupling to the bath decays exponentially as a function of the distance from the system-bath boundary. This is similar to the skin effect found recently for a quantum spin chain interacting with an environment. Using the exact results, we also investigate the ultrastrong coupling limit where the coupling between the system and the bath gets arbitrarily large and makes a connection with the recent result found for a general quantum system.

Dynamical quantum phase transitions and quantum thermodynamics: An approach through dynamical transformations

Quantum mechanics provides various pictures for understanding physical phenomena, yet the full potential of these pictures' equivalence remains underutilized. This work introduces a novel picture, opposite to the interaction picture, achieved through a transformation of the time evolution operator. This new approach permits the manipulation of the relative weight of Hamiltonian components. Utilizing picture equivalence, we investigate dynamical quantum phase transitions and quantum thermodynamics. Our findings reveal a significant relationship between dynamical topological quantum phase transitions and accidental ones through time-reversal operations, offering new insights into their fundamental connections. Moreover, our picture-based transformation addresses the challenge of defining thermodynamic quantities in systems with strong system-reservoir coupling, maintaining the robustness of weak coupling analyses.

Work distribution for unzipping processes

A simple zipper model is introduced, representing in a simplified way, e.g., the folded DNA double helix or hairpin structures in RNA. The double stranded hairpin is connected to a heat bath at temperature T and subject to an external force f, which couples to the free length L of the unzipped sequence. The leftmost zipped position can be seen as the position of a random walker in a special external field. Increasing the force leads to a zipping-unzipping first-order phase transition at a critical force f_{c}(T) in the thermodynamic limit of a very large chain. We compute analytically, as a function of temperature T and force f, the full distribution P(L) of free lengths in the thermodynamic limit and show that it is qualitatively very different for ff_{c}. Next we consider quasistatic work processes where the force is incremented according to a linear protocol. Having obtained P(L) already allows us to derive an analytical expression for the work distribution P(W) in the zipped phase f Collapse

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Affiliation(s)

Peter Werner Institut für Physik, Universität Oldenburg, 26111 Oldenburg, Germany
Alexander K Hartmann Institut für Physik, Universität Oldenburg, 26111 Oldenburg, Germany
Satya N Majumdar Laboratoire de Physique Théorique et Modèles Statistique, Université Paris-Saclay, 91405 Orsay CEDEX, France

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Statistics of quantum heat in the Caldeira-Leggett model

Nonequilibrium fluctuation relation lies at the heart of the quantum thermodynamics. Many previous studies have demonstrated that the heat exchange between a quantum system and a thermal bath initially prepared in their own Gibbs states at different temperatures obeys the famous Jarzynski-Wójcik fluctuation theorem. However, this conclusion is obtained under the assumption of Born-Markovian approximation. In this paper, going beyond the Born-Markovian limitation, we investigate the statistics of quantum heat in an exactly non-Markovian relaxation process described by the well-known Caldeira-Leggett model. It is revealed that the Jarzynski-Wójcik fluctuation theorem breaks down in the strongly non-Markovian regime. Moreover, we find the steady-state quantum heat within the non-Markovian framework can be widely tunable by using the quantum reservoir-engineering technique. These results are sharply contrary to the common Born-Markovian predictions. Our results presented in this paper may update the understanding of the quantum thermodynamics in strongly coupled and low-temperature systems. Moreover, the controllable heat may have some potential applications in improving the performance of a quantum heat engine.

Quantum Work Statistics at Strong Reservoir Coupling

Determining the statistics of work done on a quantum system while strongly coupled to a reservoir is a formidable task, requiring the calculation of the full eigenspectrum of the combined system and reservoir. Here, we show that this issue can be circumvented by using a polaron transformation that maps the system into a new frame where weak-coupling theory can be applied. Crucially, this polaron approach reproduces the Jarzynski fluctuation theorem, thus ensuring consistency with the laws of stochastic thermodynamics. We apply our formalism to a system driven across the Landau-Zener transition, where we identify clear signatures in the work distribution arising from a non-negligible coupling to the environment. Our results provide a new method for studying the stochastic thermodynamics of driven quantum systems beyond Markovian, weak-coupling regimes.

Temperature fluctuations in mesoscopic systems

Temperature is a fundamental concept in thermodynamics. In macroscopic thermodynamics, systems possess their own intrinsic temperature which equals the reservoir temperature when they equilibrate. In stochastic thermodynamics for simple systems at the microscopic level, thermodynamic quantities other than temperature (a deterministic parameter of the reservoir) are stochastic. To bridge the disparity in the perspectives about temperature between the micro- and macroregimes, we assign a generic mesoscopic N-body system an intrinsic fluctuating temperature T in this work. We simplify the complicated dynamics of numerous particles to one stochastic differential equation with respect to T, where the noise term accounts for finite-size effects arising from random energy transfer between the system and the reservoir. Our analysis reveals that these fluctuations make the extensive quantities (in the thermodynamic limit) deviate from being extensive. Moreover, we derive finite-size corrections, characterized by heat capacity of the system, to the Jarzynski equality. A possible violation of the principle of maximum work that scales with N^{-1} is also discussed. Additionally, we examine the impact of temperature fluctuations in a finite-size Carnot engine. We show that irreversible entropy production resulting from the temperature fluctuations of the working substance diminishes the average efficiency of the cycle as η_{C}-〈η〉∼N^{-1}, highlighting the unattainability of the Carnot efficiency η_{C} for mesoscopic heat engines even under the quasistatic limit. Our general framework paves the way for further exploration of nonequilibrium thermodynamics and the corresponding finite-size effects in a mesoscopic regime.

Heat distribution in quantum Brownian motion

We investigate the heat statistics in a relaxation process of quantum Brownian motion described by the Caldeira-Leggett model. By employing the normal mode transformation and the phase-space formulation approach, we can analyze the quantum heat distribution within an exactly dynamical framework beyond the traditional paradigm of Born-Markovian and weak-coupling approximations. It is revealed that the exchange fluctuation theorem for quantum heat generally breaks down in the strongly non-Markovian regime. Our results may improve the understanding about the nonequilibrium thermodynamics of open quantum systems when the usual Markovian treatment is no longer appropriate.

Heat statistics in the relaxation process of the Edwards-Wilkinson elastic manifold

The stochastic thermodynamics of systems with a few degrees of freedom has been studied extensively so far. We would like to extend the study to systems with more degrees of freedom and even further-continuous fields with infinite degrees of freedom. The simplest case for a continuous stochastic field is the Edwards-Wilkinson elastic manifold. It is an exactly solvable model of which the heat statistics in the relaxation process can be calculated analytically. The cumulants require a cutoff spacing to avoid ultraviolet divergence. The scaling behavior of the heat cumulants with time and the system size as well as the large deviation rate function of the heat statistics in the large size limit is obtained.

Hierarchical structure of fluctuation theorems for a driven system in contact with multiple heat reservoirs

For driven open systems in contact with multiple heat reservoirs, we find the marginal distributions of work or heat do not satisfy any fluctuation theorem, but only the joint distribution of work and heat satisfies a family of fluctuation theorems. A hierarchical structure of these fluctuation theorems is discovered from microreversibility of the dynamics by adopting a step-by-step coarse-graining procedure in both classical and quantum regimes. Thus, we put all fluctuation theorems concerning work and heat into a unified framework. We also propose a general method to calculate the joint statistics of work and heat in the situation of multiple heat reservoirs via the Feynman-Kac equation. For a classical Brownian particle in contact with multiple heat reservoirs, we verify the validity of the fluctuation theorems for the joint distribution of work and heat.

Work Measurement in OPEN Quantum System

Work is an important quantity in thermodynamics. In a closed quanutm system, the two-point energy measurements can be applied to measure the work but cannot be utilized in an open quantum system. With the two-point energy measurements, it has been shown that the work fluctuation satisfies the Jarzynski equality. We propose a scheme to measure the work in an open quantum system through the technique of reservoir engineering. Based on this scheme, we show that the work fluctuation in open quantum system may violate the Jarzynski equality. We apply our scheme to a two-level atom coupled to an engineered reservoir and numerically justify the general results, especially demonstrating that the second law of thermodynamics can be violated.

Quantum-Classical Correspondence Principle for Heat Distribution in Quantum Brownian Motion

Quantum Brownian motion, described by the Caldeira-Leggett model, brings insights to the understanding of phenomena and essence of quantum thermodynamics, especially the quantum work and heat associated with their classical counterparts. By employing the phase-space formulation approach, we study the heat distribution of a relaxation process in the quantum Brownian motion model. The analytical result of the characteristic function of heat is obtained at any relaxation time with an arbitrary friction coefficient. By taking the classical limit, such a result approaches the heat distribution of the classical Brownian motion described by the Langevin equation, indicating the quantum-classical correspondence principle for heat distribution. We also demonstrate that the fluctuating heat at any relaxation time satisfies the exchange fluctuation theorem of heat and its long-time limit reflects the complete thermalization of the system. Our research study justifies the definition of the quantum fluctuating heat via two-point measurements.

Work statistics in non-Hermitian evolutions with Hermitian endpoints

Non-Hermitian systems with specific forms of Hamiltonians can exhibit novel phenomena. However, it is difficult to study their quantum thermodynamical properties. In particular, the calculation of work statistics can be challenging in non-Hermitian systems due to the change of state norm. To tackle this problem, we modify the two-point measurement method in Hermitian systems. The modified method can be applied to non-Hermitian systems which are Hermitian before and after the evolution. In Hermitian systems, our method is equivalent to the two-point measurement method. When the system is non-Hermitian, our results represent a projection of the statistics in a larger Hermitian system. As an example, we calculate the work statistics in a non-Hermitian Su-Schrieffer-Heeger model. Our results reveal several differences between the work statistics in non-Hermitian systems and the one in Hermitian systems.

Exact density matrix elements for a driven dissipative system described by a quadratic Hamiltonian

For a prototype quadratic Hamiltonian describing a driven, dissipative system, exact matrix elements of the reduced density matrix are obtained from a generating function in terms of the normal characteristic functions. The approach is based on the Heisenberg equations of motion and operator calculus. The special and limiting cases are discussed.

Nonequilibrium Green's Function's Approach to the Calculation of Work Statistics

The calculation of work distributions in a quantum many-body system is of significant importance and also of formidable difficulty in the field of nonequilibrium quantum statistical mechanics. To solve this problem, inspired by the Schwinger-Keldysh formalism, we propose the contour-integral formulation for work statistics. Based on this contour integral, we show how to do the perturbation expansion of the characteristic function of work (CFW) and obtain the approximate expression of the CFW to the second order of the work parameter for an arbitrary system under a perturbative protocol. We also demonstrate the validity of fluctuation theorems by utilizing the Kubo-Martin-Schwinger condition. Finally, we use noninteracting identical particles in a forced harmonic potential as an example to demonstrate the powerfulness of our approach.

Heat and Work Along Individual Trajectories of a Quantum Bit

We use a near quantum limited detector to experimentally track individual quantum state trajectories of a driven qubit formed by the hybridization of a waveguide cavity and a transmon circuit. For each measured quantum coherent trajectory, we separately identify energy changes of the qubit as heat and work, and verify the first law of thermodynamics for an open quantum system. We further establish the consistency of these results by comparison with the master equation approach and the two-projective-measurement scheme, both for open and closed dynamics, with the help of a quantum feedback loop that compensates for the exchanged heat and effectively isolates the qubit.

Path-integral approach to the calculation of the characteristic function of work

Work statistics characterizes important features of a nonequilibrium thermodynamic process, but the calculation of the work statistics in an arbitrary nonequilibrium process is usually a cumbersome task. In this work, we study the work statistics in quantum systems by employing Feynman's path-integral approach. We derive the analytical work distributions of two prototype quantum systems. The results are proved to be equivalent to the results obtained based on Schrödinger's formalism. We also calculate the work distributions in their classical counterparts by employing the path-integral approach. Our study demonstrates the effectiveness of the path-integral approach for the calculation of work statistics in both quantum and classical thermodynamics, and brings important insights to the understanding of the trajectory work in quantum systems.

Work distribution in thermal processes

We find the moment generating function (mgf) of the nonequilibrium work for open systems undergoing a thermal process, i.e., when the stochastic dynamics maps thermal states into time-dependent thermal states. The mgf is given in terms of a temperaturelike scalar satisfying a first-order ordinary differential equation. We apply the result to some paradigmatic situations: a levitated nanoparticle in a breathing optical trap, a Brownian particle in a box with a moving piston, and a two-state system driven by an external field, where the work mgfs are obtained for different timescales and compared with Monte Carlo simulations.

Symmetry and its breaking in a path-integral approach to quantum Brownian motion

We study the Caldeira-Leggett model where a quantum Brownian particle interacts with an environment or a bath consisting of a collection of harmonic oscillators in the path-integral formalism. Compared to the contours that the paths take in the conventional Schwinger-Keldysh formalism, the paths in our study are deformed in the complex time plane as suggested by the recent study by C. Aron, G. Biroli, and L. F. Cugliandolo [SciPost Phys. 4, 008 (2018)10.21468/SciPostPhys.4.1.008]. This is done to investigate the connection between the symmetry properties in the Schwinger-Keldysh action and the equilibrium or nonequilibrium nature of the dynamics in an open quantum system. We derive the influence functional explicitly in this setting, which captures the effect of the coupling to the bath. We show that in equilibrium the action and the influence functional are invariant under a set of transformations of path-integral variables. The fluctuation-dissipation relation is obtained as a consequence of this symmetry. When the system is driven by an external time-dependent protocol, the symmetry is broken. From the terms that break the symmetry, we derive a quantum Jarzynski-like equality for a quantum mechanical worklike quantity given as a function of fluctuating quantum trajectory. In the classical limit, the transformations becomes those used in the functional integral formalism of the classical stochastic thermodynamics to derive the classical fluctuation theorem.

Three heats in a strongly coupled system and bath

We investigate three kinds of heat produced in a system and a bath strongly coupled via an interaction Hamiltonian. By studying the energy flows between the system, the bath, and their interaction, we provide rigorous definitions of two types of heat, Q_{S} and Q_{B}, from the energy loss of the system and the energy gain of the bath, respectively. This is in contrast to the equivalence of Q_{S} and Q_{B}, which is commonly assumed to hold in the weak-coupling regime. The bath we consider is equipped with a thermostat which enables it to reach an equilibrium. We identify another kind of heat Q_{SB} from the energy dissipation of the bath into the superbath that provides the thermostat. We derive the fluctuation theorems (FTs) for the system variables and various heats, which are discussed in comparison with the FT for the total entropy production. We take an example of a sliding harmonic potential of a single Brownian particle in a fluid and calculate the three heats in a simplified model. These heats are found to equal, on average, in the steady state of energy, but show different fluctuations at all times.

Computing characteristic functions of quantum work in phase space

In phase space, we analytically obtain the characteristic functions (CFs) of a forced harmonic oscillator [Talkner et al., Phys. Rev. E 75, 050102(R) (2007)PLEEE81539-375510.1103/PhysRevE.75.050102], a time-dependent mass and frequency harmonic oscillator [Deffner and Lutz, Phys. Rev. E 77, 021128 (2008)PLEEE81539-375510.1103/PhysRevE.77.021128], and coupled harmonic oscillators under driving forces in a simple and unified way. For general quantum systems, a numerical method that approximates the CFs to ℏ^{2} order is proposed. We exemplify the method with a time-dependent frequency harmonic oscillator and a family of quantum systems with time-dependent even power-law potentials.

Quantum Martingale Theory and Entropy Production

We employ martingale theory to describe fluctuations of entropy production for open quantum systems in nonequilbrium steady states. Using the formalism of quantum jump trajectories, we identify a decomposition of entropy production into an exponential martingale and a purely quantum term, both obeying integral fluctuation theorems. An important consequence of this approach is the derivation of a set of genuine universal results for stopping-time and infimum statistics of stochastic entropy production. Finally, we complement the general formalism with numerical simulations of a qubit system.

Path-integral methodology and simulations of quantum thermal transport: Full counting statistics approach

We develop and test a computational framework to study heat exchange in interacting, nonequilibrium open quantum systems. Our iterative full counting statistics path integral (iFCSPI) approach extends a previously well-established influence functional path integral method, by going beyond reduced system dynamics to provide the cumulant generating function of heat exchange. The method is straightforward; we implement it for the nonequilibrium spin boson model to calculate transient and long-time observables, focusing on the steady-state heat current flowing through the system under a temperature difference. Results are compared to perturbative treatments and demonstrate good agreement in the appropriate limits. The challenge of converging nonequilibrium quantities, currents and high order cumulants, is discussed in detail. The iFCSPI, a numerically exact technique, naturally captures strong system-bath coupling and non-Markovian effects of the environment. As such, it is a promising tool for probing fundamental questions in quantum transport and quantum thermodynamics.

Quantum heat distribution in thermal relaxation processes

We analyze the heat exchange distribution of open quantum systems undergoing a thermal relaxation process with a time-dependent effective temperature. We show that such processes arise, for example, if the dynamics maximizes the entropy production. Using a two-point measurement scheme, we find an expression for the heat moment generating function that depends solely on the system's partition function and on the thermalization function (i.e., the law of cooling) describing the effective temperature. Applications include the relaxation of free bosonic and fermionic modes, for which closed-form expressions for the time-dependent heat distribution function are derived. Multiple free modes with arbitrary dispersion relations are also briefly discussed. In the semiclassical limit our formula agrees with previous results of the literature for the heat distribution of an optically trapped nanoscopic particle far from equilibrium.