First and Second Law of Thermodynamics at Strong Coupling

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.

Thermodynamics and stochastic thermodynamics of strongly coupled systems

We further develop the strong-coupling theory of thermodynamics and stochastic thermodynamics for continuous systems, constructed in the previous work [Phys. Rev. Res. 4, 013015 (2022)2643-156410.1103/PhysRevResearch.4.013015]. A small system strongly interacting with a its environment, the dynamics of the system is assumed to be much slower than that of the bath. The system Hamiltonian is defined to be the Hamiltonian of mean force, whereas the system entropy is defined as the Gibbs-Shannon entropy. Equilibrium ensemble theories and thermodynamic theories are established for the system. Variations of three types of parameters are considered: (i) the system parameter λ which couples to the system and to the interaction, (ii) the bath parameter λ^{'} which couples to the bath only, and (iii) the temperature T=1/β. The work done to the system consists of three parts, proportional to dλ, dλ^{'}, and dβ respectively. The part proportional to dβ can be understood as the work done by the bath. As long as λ^{'} and β are not fixed, the work is not the change of total energy of the joint system. The differences between our strong-coupling equilibrium thermodynamics and the classical thermodynamics are discussed. The thermodynamic theory is promoted to the nonequilibrium level. Both the first and second laws of thermodynamics, as well as fluctuation theorems, are established for nonequilibrium processes. For processes with varying temperatures, fluctuation theorems cannot be expressed in terms of integrated work alone. Regardless of various subtleties, however, the stochastic thermodynamic theory is formulated in terms of system variables only, and dS-βd[over ¯]Q is the change of total entropy. Thermodynamic quantities of the system are related to those of the joint system, and the equivalence of theories at two levels of coarse-graining is explicitly demonstrated. Finally we show that there are infinite numbers of equivalent strong-coupling theories, each determined by its definition of system Hamiltonian. Our theory is distinguished by its maximal similarity with the weak-coupling theory.

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.

Quantum Otto cycle under strong coupling

Quantum heat engines are often discussed under the weak-coupling assumption that the interaction between the system and the reservoirs is negligible. Although this setup is easier to analyze, this assumption cannot be justified on the quantum scale. In this study, a quantum Otto cycle model that can be generally applied without the weak-coupling assumption is proposed. We replace the thermalization process in the weak-coupling model with a process comprising thermalization and decoupling. The efficiency of the proposed model is analytically calculated and indicates that, when the contribution of the interaction terms is neglected in the weak-interaction limit, it reduces to that of the earlier model. The sufficient condition for the efficiency of the proposed model not to surpass that of the weak-coupling model is that the decoupling processes of our model have a positive cost. Moreover, the relation between the interaction strength and the efficiency of the proposed model is numerically examined by using a simple two-level system. Furthermore, we show that our model's efficiency can surpass that of the weak-coupling model under particular cases. From analyzing the majorization relation, we also find a design method of the optimal interaction Hamiltonians, which are expected to provide the maximum efficiency of the proposed model. Under these interaction Hamiltonians, the numerical experiment shows that the proposed model achieves higher efficiency than that of its weak-coupling counterpart.

Assigning temperatures to eigenstates

In the study of thermalization in finite isolated quantum systems, an inescapable issue is the definition of temperature. We examine and compare different possible ways of assigning temperatures to energies or equivalently to eigenstates in such systems. A commonly used assignment of temperature in the context of thermalization is based on the canonical energy-temperature relationship, which depends only on energy eigenvalues and not on the structure of eigenstates. For eigenstates, we consider defining temperature by minimizing the distance between (full or reduced) eigenstate density matrices and canonical density matrices. We show that for full eigenstates, the minimizing temperature depends on the distance measure chosen and matches the canonical temperature for the trace distance; however, the two matrices are not close. With reduced density matrices, the minimizing temperature has fluctuations that scale with subsystem and system size but appears to be independent of distance measure. In particular limits, the two matrices become equivalent while the temperature tends to the canonical temperature.

Nonconjugate quantum subsystems

We introduce an alternative way to understand the decomposition of a quantum system into interacting parts and show that it is natural in several physical models. This enables us to define a reduced density operator for a working system interacting with a thermal bath that is consistent with the inclusion of the interaction Hamiltonian within the working system's energy. We subsequently provide a self-consistent formulation of quantum thermodynamics that incurs nontrivial physical corrections to thermodynamic relations and quantities previously defined within the literature.

Effective Gibbs State for Averaged Observables

We introduce the effective Gibbs state for the observables averaged with respect to fast free dynamics. We prove that the information loss due to the restriction of our measurement capabilities to such averaged observables is non-negative and discuss a thermodynamic role of it. We show that there are a lot of similarities between this effective Hamiltonian and the mean force Hamiltonian, which suggests a generalization of quantum thermodynamics including both cases. We also perturbatively calculate the effective Hamiltonian and correspondent corrections to the thermodynamic quantities and illustrate it with several examples.

Numerical computation of the equilibrium-reduced density matrix for strongly coupled open quantum systems

We describe a numerical algorithm for approximating the equilibrium-reduced density matrix and the effective (mean force) Hamiltonian for a set of system spins coupled strongly to a set of bath spins when the total system (system + bath) is held in canonical thermal equilibrium by weak coupling with a "super-bath". Our approach is a generalization of now standard typicality algorithms for computing the quantum expectation value of observables of bare quantum systems via trace estimators and Krylov subspace methods. In particular, our algorithm makes use of the fact that the reduced system density, when the bath is measured in a given random state, tends to concentrate about the corresponding thermodynamic averaged reduced system density. Theoretical error analysis and numerical experiments are given to validate the accuracy of our algorithm. Further numerical experiments demonstrate the potential of our approach for applications including the study of quantum phase transitions and entanglement entropy for long range interaction systems.

Stochastic thermodynamics in a non-Markovian dynamical system

The developing field of stochastic thermodynamics extends concepts of macroscopic thermodynamics such as entropy production and work to the microscopic level of individual trajectories taken by a system through phase space. The scheme involves coupling the system to an environment-typically a source of Markovian noise that affects the dynamics of the system. Here we extend this framework to consider a non-Markovian environment, one whose dynamics have memory and which create additional correlations with the system variables, and illustrate this with a selection of simple examples. Such an environment produces a rich variety of behavior. In particular, for a case of thermal relaxation, the distributions of entropy produced under the non-Markovian dynamics differ from the equivalent case of Markovian dynamics only by a delay time. When a time-dependent external work protocol is turned on, the system's correlations with the environment can either assist or hinder its approach to equilibrium, and affect its production of entropy, depending on the coupling strength between the system and environment.

Information thermodynamics of encoding and encoders

Non-isolated systems have diverse coupling relations with the external environment. These relations generate complex thermodynamics and information transmission between the system and its environment. The framework depicted in the current research attempts to glance at the critical role of the internal orders inside the non-isolated system in shaping the information thermodynamics coupling. We characterize the coupling as a generalized encoding process, where the system acts as an information thermodynamics encoder to encode the external information based on thermodynamics. We formalize the encoding process in the context of the nonequilibrium second law of thermodynamics, revealing an intrinsic difference in information thermodynamics characteristics between information thermodynamics encoders with and without internal correlations. During the information encoding process of an external source Y, specific sub-systems in an encoder X with internal correlations can exceed the information thermodynamics bound on ( X , Y ) and encode more information than system X works as a whole. We computationally verify this theoretical finding in an Ising model with a random external field and a neural data set of the human brain during visual perception and recognition. Our analysis demonstrates that the stronger internal correlation inside these systems implies a higher possibility for specific sub-systems to encode more information than the global one. These findings may suggest a new perspective in studying information thermodynamics in diverse physical and biological systems.

Emergence and Breaking of Duality Symmetry in Generalized Fundamental Thermodynamic Relations

Thermodynamics as limiting behaviors of statistics is generalized to arbitrary systems with probability a priori where the thermodynamic infinite-size limit is replaced by a multiple-measurement limit. A duality symmetry between Massieu's and Gibbs's entropy arises in the limit of infinitely repeated observations, yielding the Gibbs equation and Hill-Gibbs-Duhem equation (HGDE) as a dual pair. If a system has a thermodynamic limit satisfying Callen's postulate, entropy being an Eulerian function, the symmetry is lost: the HGDE reduces to the Gibbs-Duhem equation. This theory provides a de-mechanized foundation for classical and nanothermodynamics and offers a framework for distilling emergence from large data, free from underlying details.

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.

Projection of strong coupling interaction with thermal bath in a polymer

We investigate modifications of a stochastic polymer picture through a shift in the boundary between the system and an external environment. A conventional bead-and-spring model serving as the coarse-graining model is given by the Langevin equation for all the monomers subject to white noise. However, stochastic motion for only a tagged monomer is observed to occur in the presence of colored noise. The qualitative change in the observations arises from the boundary shift decided by the observer. The Langevin dynamics analyses interpret the colored noise as the emergence of the polymeric elastic force, resulting in additional heat in the tagged monomer observation. Being distinguished from coarse-graining based on scale separation, the projection of comparable internal degrees of freedom is also discussed in light of the fluctuation theorem and the stochastic polymer thermodynamics.

Measurement-based quantum heat engine in a multilevel system

We compare quantum Otto engines based on two different cycle models: a two-bath model, with a standard heat source and sink, and a measurement-based protocol, where the role of heat source is played by a quantum measurement. We furthermore study these cycles using two different "working substances": a single qutrit (spin-1 particle) or a pair of qubits (spin-1/2 particles) interacting via the XXZ Heisenberg interaction. Although both cycle models have the same efficiency when applied on a single-qubit working substance, we find that both can reach higher efficiencies using these more complex working substances by exploiting the existence of "idle" levels, i.e., levels that do not shift while the spins are subjected to a variable magnetic field. Furthermore, with an appropriate choice of measurement, the measurement-based protocol becomes more efficient than the two-bath model.

Dependence of integrated, instantaneous, and fluctuating entropy production on the initial state in quantum and classical processes

We consider the additional entropy production (EP) incurred by a fixed quantum or classical process on some initial state ρ, above the minimum EP incurred by the same process on any initial state. We show that this additional EP, which we term the "mismatch cost of ρ," has a universal information-theoretic form: it is given by the contraction of the relative entropy between ρ and the least-dissipative initial state φ over time. We derive versions of this result for integrated EP incurred over the course of a process, for trajectory-level fluctuating EP, and for instantaneous EP rate. We also show that mismatch cost for fluctuating EP obeys an integral fluctuation theorem. Our results demonstrate a fundamental relationship between thermodynamic irreversibility (generation of EP) and logical irreversibility (inability to know the initial state corresponding to a given final state). We use this relationship to derive quantitative bounds on the thermodynamics of quantum error correction and to propose a thermodynamically operationalized measure of the logical irreversibility of a quantum channel. Our results hold for both finite- and infinite-dimensional systems, and generalize beyond EP to many other thermodynamic costs, including nonadiabatic EP, free-energy loss, and entropy gain.

Common Environmental Effects on Quantum Thermal Transistor

Quantum thermal transistor is a microscopic thermodynamical device that can modulate and amplify heat current through two terminals by the weak heat current at the third terminal. Here we study the common environmental effects on a quantum thermal transistor made up of three strong-coupling qubits. It is shown that the functions of the thermal transistor can be maintained and the amplification rate can be modestly enhanced by the skillfully designed common environments. In particular, the presence of a dark state in the case of the completely correlated transitions can provide an additional external channel to control the heat currents without any disturbance of the amplification rate. These results show that common environmental effects can offer new insights into improving the performance of quantum thermal devices.

Weak and Ultrastrong Coupling Limits of the Quantum Mean Force Gibbs State

The Gibbs state is widely taken to be the equilibrium state of a system in contact with an environment at temperature T. However, non-negligible interactions between system and environment can give rise to an altered state. Here, we derive general expressions for this mean force Gibbs state, valid for any system that interacts with a bosonic reservoir. First, we derive the state in the weak coupling limit and find that, in general, it maintains coherences with respect to the bare system Hamiltonian. Second, we develop a new expansion method suited to investigate the ultrastrong coupling regime. This allows us to derive the explicit form for the mean force Gibbs state, and we find that it becomes diagonal in the basis set by the system-reservoir interaction instead of the system Hamiltonian. Several examples are discussed including a single qubit, a three-level V-system, and two coupled qubits all interacting with bosonic reservoirs. The results shed light on the presence of coherences in the strong coupling regime, and provide key tools for nanoscale thermodynamics investigations.

Relating heat and entanglement in strong-coupling thermodynamics

Explaining the influence of strong coupling in the dynamics of open quantum systems is one of the most challenging issues in the rapidly growing field of quantum thermodynamics. By using a particular definition of heat, we develop an approach to study the thermodynamics in the strong-coupling regime, which takes into account quantum resources such as coherence and entanglement. We apply the method to calculate the time-dependent thermodynamic properties of a system and an environment interacting via the generalized amplitude-damping channel. The results indicate that the transient imbalance between heat dissipated and heat absorbed that occurs in the process is responsible for the generation of system-environment entanglement.

Numerically "exact" simulations of entropy production in the fully quantum regime: Boltzmann entropy vs von Neumann entropy

We present a scheme to evaluate thermodynamic variables for a system coupled to a heat bath under a time-dependent external force using the quasi-static Helmholtz energy from the numerically "exact" hierarchical equations of motion (HEOM). We computed the entropy produced by a spin system strongly coupled to a non-Markovian heat bath for various temperatures. We showed that when changes to the external perturbation occurred sufficiently slowly, the system always reached thermal equilibrium. Thus, we calculated the Boltzmann entropy and the von Neumann entropy for an isothermal process, as well as various thermodynamic variables, such as changes in internal energies, heat, and work, for a system in quasi-static equilibrium based on the HEOM. We found that although the characteristic features of the system entropies in the Boltzmann and von Neumann cases as a function of the system-bath coupling strength are similar, those for the total entropy production are completely different. The total entropy production in the Boltzmann case is always positive, whereas that in the von Neumann case becomes negative if we chose a thermal equilibrium state of the total system (an unfactorized thermal equilibrium state) as the initial state. This is because the total entropy production in the von Neumann case does not properly take into account the contribution of the entropy from the system-bath interaction. Thus, the Boltzmann entropy must be used to investigate entropy production in the fully quantum regime. Finally, we examined the applicability of the Jarzynski equality.

Statistical Mechanics at Strong Coupling: A Bridge between Landsberg's Energy Levels and Hill's Nanothermodynamics

We review and show the connection between three different theories proposed for the thermodynamic treatment of systems not obeying the additivity ansatz of classical thermodynamics. In the 1950s, Landsberg proposed that when a system comes into contact with a heat bath, its energy levels are redistributed. Based on this idea, he produced an extended thermostatistical framework that accounts for unknown interactions with the environment. A decade later, Hill devised his celebrated nanothermodynamics, where he introduced the concept of subdivision potential, a new thermodynamic variable that accounts for the vanishing additivity of increasingly smaller systems. More recently, a thermostatistical framework at strong coupling has been formulated to account for the presence of the environment through a Hamiltonian of mean force. We show that this modified Hamiltonian yields a temperature-dependent energy landscape as earlier suggested by Landsberg, and it provides a thermostatistical foundation for the subdivision potential, which is the cornerstone of Hill's nanothermodynamics.

Non-Clausius heat transfer: The method of the nonstationary Langevin equation

Compared to other formulations of the second law of thermodynamics, the Clausius statement that heat does not spontaneously flow from cold to hot concerns a system in nonequilibrium states, and in that respect is more ambitious but also more ambiguous. We discuss two scenarios when the Clausius statement in its plain form does not hold. First, for ergodic systems, the energy transfer may be consistent with the statement on a coarse-grained timescale, but be anomalously directed during time intervals shorter than the thermalization time. In particular, when an initially colder system is brought in contact to a hotter bath, the internal energy of the former increases with time in a long run but not monotonically. Second, the heat transfer may not respect the Clausius statement on any timescale in nonergodic systems due to the formation of localized vibrational modes. We illustrate the two scenarios with a familiar model of an isotope atom attached to a semi-infinite harmonic atomic chain. Technically, the discussion is based on a Langevin equation for the isotope, using the initial condition when the isotope and chain are initially prepared in uncorrelated canonical states under the constraint that the boundary atom between the isotope and chain is initially fixed and later released. In such setting, the noise in the Langevin equation is nonstationary, and the fluctuation-dissipation relation has a nonstandard form.

Faster Uphill Relaxation in Thermodynamically Equidistant Temperature Quenches

We uncover an unforeseen asymmetry in relaxation: for a pair of thermodynamically equidistant temperature quenches, one from a lower and the other from a higher temperature, the relaxation at the ambient temperature is faster in the case of the former. We demonstrate this finding on hand of two exactly solvable many-body systems relevant in the context of single-molecule and tracer-particle dynamics. We prove that near stable minima and for all quadratic energy landscapes it is a general phenomenon that also exists in a class of non-Markovian observables probed in single-molecule and particle-tracking experiments. The asymmetry is a general feature of reversible overdamped diffusive systems with smooth single-well potentials and occurs in multiwell landscapes when quenches disturb predominantly intrawell equilibria. Our findings may be relevant for the optimization of stochastic heat engines.

Strong Coupling and Nonextensive Thermodynamics

We propose a Hamiltonian-based approach to the nonextensive thermodynamics of small systems, where small is a relative term comparing the size of the system to the size of the effective interaction region around it. We show that the effective Hamiltonian approach gives easy accessibility to the thermodynamic properties of systems strongly coupled to their surroundings. The theory does not rely on the classical concept of dividing surface to characterize the system’s interaction with the environment. Instead, it defines an effective interaction region over which a system exchanges extensive quantities with its surroundings, easily producing laws recently shown to be valid at the nanoscale.

Quantum thermodynamics of single particle systems

Thermodynamics is built with the concept of equilibrium states. However, it is less clear how equilibrium thermodynamics emerges through the dynamics that follows the principle of quantum mechanics. In this paper, we develop a theory of quantum thermodynamics that is applicable for arbitrary small systems, even for single particle systems coupled with a reservoir. We generalize the concept of temperature beyond equilibrium that depends on the detailed dynamics of quantum states. We apply the theory to a cavity system and a two-level system interacting with a reservoir, respectively. The results unravels (1) the emergence of thermodynamics naturally from the exact quantum dynamics in the weak system-reservoir coupling regime without introducing the hypothesis of equilibrium between the system and the reservoir from the beginning; (2) the emergence of thermodynamics in the intermediate system-reservoir coupling regime where the Born-Markovian approximation is broken down; (3) the breakdown of thermodynamics due to the long-time non-Markovian memory effect arisen from the occurrence of localized bound states; (4) the existence of dynamical quantum phase transition characterized by inflationary dynamics associated with negative dynamical temperature. The corresponding dynamical criticality provides a border separating classical and quantum worlds. The inflationary dynamics may also relate to the origin of big bang and universe inflation. And the third law of thermodynamics, allocated in the deep quantum realm, is naturally proved.

Nonequilibrium Energy Transduction in Stochastic Strongly Coupled Rotary Motors

Living systems at the molecular scale are composed of many constituents with strong and heterogeneous interactions, operating far from equilibrium, and subject to strong fluctuations. These conditions pose significant challenges to efficient, precise, and rapid free energy transduction, yet nature has evolved numerous molecular machines that do just this. Using a simple model of the ingenious rotary machine F_oF₁-ATP synthase, we investigate the interplay between nonequilibrium driving forces, thermal fluctuations, and interactions between strongly coupled subsystems. This model reveals design principles for effective free energy transduction. Most notably, while tight coupling is intuitively appealing, we find that output power is maximized at intermediate-strength coupling, which permits lubrication by stochastic fluctuations with only minimal slippage.

Measurability of nonequilibrium thermodynamics in terms of the Hamiltonian of mean force

The nonequilibrium thermodynamics of an open (classical or quantum) system in strong contact with a single heat bath can be conveniently described in terms of the Hamiltonian of mean force. However, the conventional formulation is limited by the necessity to measure differences in equilibrium properties of the system-bath composite. We make use of the freedom involved in defining thermodynamic quantities, which leaves the thermodynamics unchanged, to show that the Hamiltonian of mean force can be inferred from measurements on the system alone, up to that irrelevant freedom. In doing so, we refute a key criticism expressed in the works by P. Talkner and P. Hänggi [Phys. Rev. E 94, 022143 (2016)10.1103/PhysRevE.94.022143 and arXiv:1911.11660]. We also discuss the remaining part of the criticism.

Strong Coupling Thermodynamics of Open Quantum Systems

A general thermodynamic framework is presented for open quantum systems in fixed contact with a thermal reservoir. The first and second law are obtained for arbitrary system-reservoir coupling strengths, and including both factorized and correlated initial conditions. The thermodynamic properties are adapted to the generally strong coupling regime, approaching the ones defined for equilibrium, and their standard weak-coupling counterparts as appropriate limits. Moreover, they can be inferred from measurements involving only system observables. Finally, a thermodynamic signature of non-Markovianity is formulated in the form of a negative entropy production rate.

Effective thermodynamics of two interacting underdamped Brownian particles

Starting from the stochastic thermodynamics description of two coupled underdamped Brownian particles, we showcase and compare three different coarse-graining schemes leading to an effective thermodynamic description for the first of the two particles: marginalization over one particle, bipartite structure with information flows, and the Hamiltonian of mean force formalism. In the limit of time-scale separation where the second particle with a fast relaxation time scale locally equilibrates with respect to the coordinates of the first slowly relaxing particle, the effective thermodynamics resulting from the first and third approach are shown to capture the full thermodynamics and to coincide with each other. In the bipartite approach, the slow part does not, in general, allow for an exact thermodynamic description as the entropic exchange between the particles is ignored. Physically, the second particle effectively becomes part of the heat reservoir. In the limit where the second particle becomes heavy and thus deterministic, the effective thermodynamics of the first two coarse-graining methods coincide with the full one. The Hamiltonian of mean force formalism, however, is shown to be incompatible with that limit. Physically, the second particle becomes a work source. These theoretical results are illustrated using an exactly solvable harmonic model.

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.

Negative Thermophoretic Force in the Strong Coupling Regime

Negative thermophoresis (a particle moving up the temperature gradient) is a somewhat counterintuitive phenomenon which has thus far eluded a simple thermostatistical description. The purpose of this Letter is to show that a thermodynamic framework based on the formulation of a Hamiltonian of mean force has the descriptive ability to capture this interesting and elusive phenomenon in an unusually elegant and straightforward fashion. We propose a mechanism that describes the advent of a thermophoretic force acting from cold to hot on systems that are strongly coupled to a nonisothermal heat bath. When a system is strongly coupled to the heat bath, the system's eigenenergies E_{j} become effectively temperature dependent. This adjustment of the energy levels allows the system to take heat from the environment, +d⟨E_{j}⟩, and return it as work, -d⟨TdE_{j}/dT⟩. This effect can make the temperature dependence of the effective energy profile nonmonotonic. As a result, particles may experience a force in either direction depending on the temperature.

Repeated Interactions and Quantum Stochastic Thermodynamics at Strong Coupling

The thermodynamic framework of repeated interactions is generalized to an arbitrary open quantum system in contact with a heat bath. Based on these findings, the theory is then extended to arbitrary measurements performed on the system. This constitutes a direct experimentally testable framework in strong coupling quantum thermodynamics. By construction, it provides many quantum stochastic processes and quantum causal models with a consistent thermodynamic interpretation. The setting can be further used, for instance, to rigorously investigate the interplay between non-Markovianity and nonequilibrium thermodynamics.

Theory of Nonequilibrium Free Energy Transduction by Molecular Machines

Biomolecular machines are protein complexes that convert between different forms of free energy. They are utilized in nature to accomplish many cellular tasks. As isothermal nonequilibrium stochastic objects at low Reynolds number, they face a distinct set of challenges compared with more familiar human-engineered macroscopic machines. Here we review central questions in their performance as free energy transducers, outline theoretical and modeling approaches to understand these questions, identify both physical limits on their operational characteristics and design principles for improving performance, and discuss emerging areas of research.

Heat capacities of thermally manipulated mechanical oscillator at strong coupling

Coherent quantum oscillators are basic physical systems both in quantum statistical physics and quantum thermodynamics. Their realizations in lab often involve solid-state devices sensitive to changes in ambient temperature. We represent states of the solid-state optomechanical oscillator with temperature-dependent frequency by equivalent states of the mechanical oscillator with temperature-dependent energy levels. We interpret the temperature dependence as a consequence of strong coupling between the oscillator and the heat bath. We explore parameter regimes corresponding to anomalous behavior of mechanical and thermodynamic characteristics as a consequence of the strong coupling: (i) The localization and the purification induced by heating, and (ii) the negativity of two generalized heat capacities. The capacities can be used to witness non-linearity in the temperature dependency of the energy levels. Our phenomenological experimentally-oriented approach can stimulate development of new optomechanical and thermomechanical experiments exploring basic concepts of strong coupling thermodynamics.

Quantum Thermodynamics in the Refined Weak Coupling Limit

We present a thermodynamic framework for the refined weak coupling limit. In this limit, the interaction between system and environment is weak, but not negligible. As a result, the system dynamics becomes non-Markovian breaking divisibility conditions. Nevertheless, we propose a derivation of the first and second law just in terms of the reduced system dynamics. To this end, we extend the refined weak coupling limit for allowing slowly-varying external drivings and reconsider the definition of internal energy due to the non-negligible interaction.

Contributions to single-shot energy exchanges in open quantum systems

The exchange of energy between a classical open system and its environment can be analyzed for a single run of an experiment using the phase-space trajectory of the system. By contrast, in the quantum regime such energy exchange processes must be defined for an ensemble of runs of the same experiment based on the reduced system density matrix. Single-shot approaches based on stochastic wave functions have been proposed for quantum systems that are continuously monitored or weakly coupled to a heat bath. However, for systems strongly coupled to the environment and not continuously monitored, a single-shot analysis has not been attempted because no system wave function exists for such systems within the standard formulation of quantum theory. Using the notion of the conditional wave function of a quantum system, we derive here an exact formula for the rate of total energy change in an open quantum system, valid for arbitrary coupling between the system and the environment. In particular, this allows us to identify three distinct contributions to the total energy flow: an external contribution coming from the explicit time dependence of the Hamiltonian, an interaction contribution associated with the interaction part of the Hamiltonian, and an entanglement contribution, directly related to the presence of entanglement between the system and its environment. Given the close connection between weak values and the conditional wave function, the approach presented here provides a new avenue for experimental studies of energy fluctuations in open quantum systems.

Thermal power of heat flow through a qubit

In this paper we consider the thermal power of a heat flow through a qubit between two baths. The baths are modeled as a set of harmonic oscillators initially at equilibrium, at two temperatures. Heat is defined as the change of energy of the cold bath, and thermal power is defined as expected heat per unit time, in the long-time limit. The qubit and the baths interact as in the spin-boson model, i.e., through qubit operator σ_{z}. We compute thermal power in an approximation analogous to a "noninteracting blip" (NIBA) and express it in the polaron picture as products of correlation functions of the two baths, and a time derivative of a correlation function of the cold bath. In the limit of weak interaction we recover known results in terms of a sum of correlation functions of the two baths, a correlation functions of the cold bath only, and the energy split.

Role of coherence in the nonequilibrium thermodynamics of quantum systems

Exploiting the relative entropy of coherence, we isolate the coherent contribution in the energetics of a driven nonequilibrium quantum system. We prove that a division of the irreversible work can be made into a coherent and incoherent part. This provides an operational criterion for quantifying the coherent contribution in a generic nonequilibrium transformation on a closed quantum system. We then study such a contribution in two physical models of a driven qubit and kicked rotor. In addition, we also show that coherence generation is connected to the nonadiabaticity of a processes, for which it gives the dominant contribution for slow-enough transformations. The amount of generated coherence in the energy eigenbasis is equivalent to the change in diagonal entropy, and here we show that it fulfills a fluctuation theorem.

Non-Markovianity and negative entropy production rates

Entropy production plays a fundamental role in nonequilibrium thermodynamics to quantify the irreversibility of open systems. Its positivity can be ensured for a wide class of setups, but the entropy production rate can become negative sometimes. This is often taken as an indicator of non-Markovianity. We make this link precise by showing under which conditions a negative entropy production rate implies non-Markovianity and when it does not. For a system coupled to a single heat bath, this can be established within a unified language for two setups: (i) the dynamics resulting from a coarse-grained description of a Markovian master equation and (ii) the classical Hamiltonian dynamics of a system coupled to a bath. The quantum version of the latter result is shown not to hold despite the fact that the integrated thermodynamic description is formally equivalent to the classical case. The instantaneous fixed point of a non-Markovian dynamics plays an important role in our study. Our key contribution is to provide a consistent theoretical framework to study the finite-time thermodynamics of a large class of dynamics with a precise link to its non-Markovianity.

Path integral approach to heat in quantum thermodynamics

We study the heat statistics of a quantum Brownian motion described by the Caldeira-Leggett model. By using the path integral approach, we introduce a concept of the quantum heat functional along every pair of Feynman paths. This approach has the advantage of improving our understanding about heat in quantum systems. First, we demonstrate the microscopic reversibility of the system by connecting the heat functional to the forward and time-reversed probabilities. Second, we analytically prove the quantum-classical correspondence of the heat functional and their statistics, which allows us to obtain better intuitions about the difference between classical and quantum heat.

Steady-State Coherences by Composite System-Bath Interactions

We identify sufficient conditions on the structure of the interaction Hamiltonian between a two-level quantum system and a thermal bath that, without any external drive or coherent measurement, guarantee the generation of steady-state coherences (SSC). The SSC obtained this way, remarkably, turn out to be independent of the initial state of the system, which could therefore be taken as initially incoherent. We characterize in detail this phenomenon, first analytically in the weak coupling regime for two paradigmatic models, and then numerically in more complex systems without any assumption on the coupling strength. In all of these cases, we find that SSC become increasingly significant as the bath is cooled down. These results can be directly verified in many experimental platforms.

Path Integral Approach to Quantum Thermodynamics

Work belongs to the most basic notions in thermodynamics but it is not well understood in quantum systems, especially in open quantum systems. By introducing a novel concept of the work functional along an individual Feynman path, we invent a new approach to study thermodynamics in the quantum regime. Using the work functional, we derive a path integral expression for the work statistics. By performing the ℏ expansion, we analytically prove the quantum-classical correspondence of the work statistics. In addition, we obtain the quantum correction to the classical fluctuating work. We can also apply this approach to an open quantum system in the strong coupling regime described by the quantum Brownian motion model. This approach provides an effective way to calculate the work in open quantum systems by utilizing various path integral techniques. As an example, we calculate the work statistics for a dragged harmonic oscillator in both isolated and open quantum systems.

Characteristic functions of quantum heat with baths at different temperatures

This paper is about quantum heat defined as the change in energy of a bath during a process. The presentation takes into account recent developments in classical strong-coupling thermodynamics and addresses a version of quantum heat that satisfies quantum-classical correspondence. The characteristic function and the full counting statistics of quantum heat are shown to be formally similar. The paper further shows that the method can be extended to more than one bath, e.g., two baths at different temperatures, which opens up the prospect of studying correlations and heat flow. The paper extends earlier results on the expected quantum heat in the setting of one bath [E. Aurell and R. Eichhorn, New J. Phys. 17, 065007 (2015)NJOPFM1367-263010.1088/1367-2630/17/6/065007; E. Aurell, Entropy 19, 595 (2017)ENTRFG1099-430010.3390/e19110595].

Energy-temperature uncertainty relation in quantum thermodynamics

It is known that temperature estimates of macroscopic systems in equilibrium are most precise when their energy fluctuations are large. However, for nanoscale systems deviations from standard thermodynamics arise due to their interactions with the environment. Here we include such interactions and, using quantum estimation theory, derive a generalised thermodynamic uncertainty relation valid for classical and quantum systems at all coupling strengths. We show that the non-commutativity between the system's state and its effective energy operator gives rise to quantum fluctuations that increase the temperature uncertainty. Surprisingly, these additional fluctuations are described by the average Wigner-Yanase-Dyson skew information. We demonstrate that the temperature's signal-to-noise ratio is constrained by the heat capacity plus a dissipative term arising from the non-negligible interactions. These findings shed light on the interplay between classical and non-classical fluctuations in quantum thermodynamics and will inform the design of optimal nanoscale thermometers.

Additivity of multiple heat reservoirs in the Langevin equation

The Langevin equation greatly simplifies the mathematical expression of the effects of thermal noise by using only two terms, a dissipation term, and a random-noise term. The Langevin description was originally applied to a system in contact with a single heat reservoir; however, many recent studies have also adopted a Langevin description for systems connected to multiple heat reservoirs. This is accomplished through the introduction of a simple summation for the dissipation and random-noise terms associated with each reservoir. However, the validity of this simple addition has been the focus of only limited discussion and has raised several criticisms. Moreover, this additive description has never been either experimentally or numerically verified, rendering its validity is still an open question. Here we perform molecular dynamics simulations for a Brownian system in simultaneous contact with multiple heat reservoirs to check the validity of this additive approach. Our simulation results confirm that the effect of multiple heat reservoirs is additive in general. A very small deviation in the total amount of dissipation and associated noise is found but seems not significant within statistical errors. We find that the steady-state properties satisfy the additivity perfectly and are not affected by this deviation.

Quantum Thermodynamics at Strong Coupling: Operator Thermodynamic Functions and Relations

Identifying or constructing a fine-grained microscopic theory that will emerge under specific conditions to a known macroscopic theory is always a formidable challenge. Thermodynamics is perhaps one of the most powerful theories and best understood examples of emergence in physical sciences, which can be used for understanding the characteristics and mechanisms of emergent processes, both in terms of emergent structures and the emergent laws governing the effective or collective variables. Viewing quantum mechanics as an emergent theory requires a better understanding of all this. In this work we aim at a very modest goal, not quantum mechanics as thermodynamics, not yet, but the thermodynamics of quantum systems, or quantum thermodynamics. We will show why even with this minimal demand, there are many new issues which need be addressed and new rules formulated. The thermodynamics of small quantum many-body systems strongly coupled to a heat bath at low temperatures with non-Markovian behavior contains elements, such as quantum coherence, correlations, entanglement and fluctuations, that are not well recognized in traditional thermodynamics, built on large systems vanishingly weakly coupled to a non-dynamical reservoir. For quantum thermodynamics at strong coupling, one needs to reexamine the meaning of the thermodynamic functions, the viability of the thermodynamic relations and the validity of the thermodynamic laws anew. After a brief motivation, this paper starts with a short overview of the quantum formulation based on Gelin & Thoss and Seifert. We then provide a quantum formulation of Jarzynski's two representations. We show how to construct the operator thermodynamic potentials, the expectation values of which provide the familiar thermodynamic variables. Constructing the operator thermodynamic functions and verifying or modifying their relations is a necessary first step in the establishment of a viable thermodynamics theory for quantum systems. We mention noteworthy subtleties for quantum thermodynamics at strong coupling, such as in issues related to energy and entropy, and possible ambiguities of their operator forms. We end by indicating some fruitful pathways for further developments.

Unified picture of strong-coupling stochastic thermodynamics and time reversals

Strong-coupling statistical thermodynamics is formulated as the Hamiltonian dynamics of an observed system interacting with another unobserved system (a bath). It is shown that the entropy production functional of stochastic thermodynamics, defined as the log ratio of forward and backward system path probabilities, is in a one-to-one relation with the log ratios of the joint initial conditions of the system and the bath. A version of strong-coupling statistical thermodynamics where the system-bath interaction vanishes at the beginning and at the end of a process is, as is also weak-coupling stochastic thermodynamics, related to the bath initially in equilibrium by itself. The heat is then the change of bath energy over the process, and it is discussed when this heat is a functional of the system history alone. The version of strong-coupling statistical thermodynamics introduced by Seifert and Jarzynski is related to the bath initially in conditional equilibrium with respect to the system. This leads to heat as another functional of the system history which needs to be determined by thermodynamic integration. The log ratio of forward and backward system path probabilities in a stochastic process is finally related to log ratios of the initial conditions of a combined system and bath. It is shown that the entropy production formulas of stochastic processes under a general class of time reversals are given by the differences of bath energies in a larger underlying Hamiltonian system. The paper highlights the centrality of time reversal in stochastic thermodynamics, also in the case of strong coupling.

Strong Coupling Corrections in Quantum Thermodynamics

Quantum systems strongly coupled to many-body systems equilibrate to the reduced state of a global thermal state, deviating from the local thermal state of the system as it occurs in the weak-coupling limit. Taking this insight as a starting point, we study the thermodynamics of systems strongly coupled to thermal baths. First, we provide strong-coupling corrections to the second law applicable to general systems in three of its different readings: As a statement of maximal extractable work, on heat dissipation, and bound to the Carnot efficiency. These corrections become relevant for small quantum systems and vanish in first order in the interaction strength. We then move to the question of power of heat engines, obtaining a bound on the power enhancement due to strong coupling. Our results are exemplified on the paradigmatic non-Markovian quantum Brownian motion.

Achieving the classical Carnot efficiency in a strongly coupled quantum heat engine

Generally, the efficiency of a heat engine strongly coupled with a heat bath is less than the classical Carnot efficiency. Through a model-independent method, we show that the classical Carnot efficiency is achieved in a strongly coupled quantum heat engine. First, we present the first law of quantum thermodynamics in strong coupling. Then, we show how to achieve the Carnot cycle and the classical Carnot efficiency at strong coupling. We find that this classical Carnot efficiency stems from the fact that the heat released in a nonequilibrium process is balanced by the absorbed heat. We also analyze the restrictions in the achievement of the Carnot cycle. The first restriction is that there must be two corresponding intervals of the controllable parameter in which the corresponding entropies of the work substance at the hot and cold temperatures are equal, and the second is that the entropy of the initial and final states in a nonequilibrium process must be equal. Through these restrictions, we obtain the positive work conditions, including the usual one in which the hot temperature should be higher than the cold, and a new one in which there must be an entropy interval at the hot temperature overlapping that at the cold. We demonstrate our result through a paradigmatic model-a two-level system in which a work substance strongly interacts with a heat bath. In this model, we find that the efficiency may abruptly decrease to zero due to the first restriction, and that the second restriction results in the control scheme becoming complex.

Quantum thermodynamics from the nonequilibrium dynamics of open systems: Energy, heat capacity, and the third law

In a series of papers, we intend to take the perspective of open quantum systems and examine from their nonequilibrium dynamics the conditions when the physical quantities, their relations, and the laws of thermodynamics become well defined and viable for quantum many-body systems. We first describe how an open-system nonequilibrium dynamics (ONEq) approach is different from the closed combined system +  environment in a global thermal state (CGTs) setup. Only after the open system equilibrates will it be amenable to conventional thermodynamics descriptions, thus quantum thermodynamics (QTD) comes at the end rather than assumed in the beginning. The linkage between the two comes from the reduced density matrix of ONEq in that stage having the same form as that of the system in the CGTs. We see the open-system approach having the advantage of dealing with nonequilibrium processes as many experiments in the near future will call for. Because it spells out the conditions of QTD's existence, it can also aid us in addressing the basic issues in quantum thermodynamics from first principles in a systematic way. We then study one broad class of open quantum systems where the full nonequilibrium dynamics can be solved exactly, that of the quantum Brownian motion of N strongly coupled harmonic oscillators, interacting strongly with a scalar-field environment. In this paper, we focus on the internal energy, heat capacity, and the third law. We show for this class of physical models, amongst other findings, the extensive property of the internal energy, the positivity of the heat capacity, and the validity of the third law from the perspective of the behavior of the heat capacity toward zero temperature. These conclusions obtained from exact solutions and quantitative analysis clearly disprove claims of negative specific heat in such systems and dispel allegations that in such systems the validity of the third law of thermodynamics relies on quantum entanglement. They are conceptually and factually unrelated issues. Entropy and entanglement will be the main theme of our second paper on this subject matter.

Stochastic thermodynamics of interacting degrees of freedom: Fluctuation theorems for detached path probabilities

Systems with interacting degrees of freedom play a prominent role in stochastic thermodynamics. Our aim is to use the concept of detached path probabilities and detached entropy production for bipartite Markov processes and elaborate on a series of special cases including measurement-feedback systems, sensors, and hidden Markov models. For these special cases we show that fluctuation theorems involving the detached entropy production recover known results which have been obtained separately before. Additionally, we show that the fluctuation relation for the detached entropy production can be used in model selection for data stemming from a hidden Markov model. We discuss the relation to previous approaches including those which use information flow or learning rate to quantify the influence of one subsystem on the other. In conclusion, we present a complete framework with which to find fluctuation relations for coupled systems.