The thermodynamics of quantum systems coupled to periodically modulated heat baths and work reservoirs is developed. By identifying affinities and fluxes, the first and second law are formulated consistently. In the linear response regime, entropy production becomes a quadratic form in the affinities. Specializing to Lindblad-dynamics, we identify the corresponding kinetic coefficients in terms of correlation functions of the unperturbed dynamics. Reciprocity relations follow from symmetries with respect to time reversal. The kinetic coefficients can be split into a classical and a quantum contribution subject to a new constraint, which follows from a natural detailed balance condition. This constraint implies universal bounds on efficiency and power of quantum heat engines. In particular, we show that Carnot efficiency can not be reached whenever quantum coherence effects are present, i.e., when the Hamiltonian used for work extraction does not commute with the bare system Hamiltonian. For illustration, we specialize our universal results to a driven two-level system in contact with a heat bath of sinusoidally modulated temperature.
Periodically driven coherent conductors provide a universal platform for the development of quantum transport devices. Here, we lay down a comprehensive theory to describe the thermodynamics of these systems. We first focus on moderate thermo-electrical biases and low driving frequencies. For this linear response regime, we establish generalized Onsager-Casimir relations and an extended fluctuation-dissipation theorem. Furthermore, we derive a family of thermodynamic bounds proving that any local matter or heat current puts a non-trivial lower limit on the overall dissipation rate of a coheren…
We aim to give a pedagogic presentation of the open system dynamics of a periodically driven qubit in contact with a temperature bath. We are specifically interested in the thermodynamics of the qubit. It is well known that by combining the Markovian approximation with Floquet theory it is possible to derive a stochastic Schrödinger equation in $\mathbb{C}^2$ for the state of the qubit. We follow here a different approach. We use Floquet theory to embed the time-non autonomous qubit dynamics into time-autonomous yet infinite dimensional dynamics. We refer to the resulting infinite dimensional …
We present the stochastic thermodynamic analysis of a time-periodic single particle pump, including explicit results for flux, thermodynamic force, entropy production, work, heat and efficiency. These results are valid far from equilibrium. The deviations from the linear (Onsager) regime are discussed.
Theoretical treatments of periodically-driven quantum thermal machines (PD-QTMs) are largely focused on the limit-cycle stage of operation characterized by a periodic state of the system. Yet, this regime is not immediately accessible for experimental verification. Here, we present a general thermodynamic framework that can handle the performance of PD-QTMs both before and during the limit-cycle stage of operation. It is achieved by observing that periodicity may break down at the ensemble average level, even in the limit-cycle phase. With this observation, and using conventional thermodynamic…
Scattering theory is a standard tool for the description of transport phenomena in mesoscopic systems. Here, we provide a detailed derivation of this method for nano-scale conductors that are driven by oscillating electric or magnetic fields. Our approach is based on an extension of the conventional Lippmann-Schwinger formalism to systems with a periodically time dependent Hamiltonian. As a key result, we obtain a systematic perturbation scheme for the Floquet scattering amplitudes that describe the transition of a transport carrier through a periodically driven sample. Within a general multi-…
Recently, some general relations have been studied in nonequilibrium mesoscopic systems. In particular, the thermodynamic uncertainty relation (TUR) provides a universal internal relation among the cumulants of currents and the entropy production. In this paper, we give a simple derivation of TUR for thermally fluctuating particle current in a tilted periodic potential from a coarse grained point of view with the use of the transition probabilities. Also, we explore the condition for the bound of TUR to monotonically increase with the affinity.
We consider a harmonic oscillator under periodic driving and coupled to two harmonic-oscillator heat baths at different temperatures. We use the thermofield transformation with chain mapping for this setup, which allows us to study the unitary evolution of the system and the baths up to a time when the periodic steady state emerges in the system. We characterize this periodic steady state, and we show that, by tuning the system and the bath parameters, one can turn this system from an engine to an accelerator or even to a heater. The possibility to study the unitary evolution of the system and…
The problem of time-dependent particle transport in quantum conductors is nowadays a well established topic. In contrast, the way in which energy and heat flow in mesoscopic systems subjected to dynamical drivings is a relatively new subject that cross-fertilize both fundamental developments of quantum thermodynamics and practical applications in nanoelectronics and quantum information. In this short review, we discuss from a thermodynamical perspective recent investigations on nonstationary heat and work generated in quantum systems, emphasizing open questions and unsolved issues.
We determine the quasistationary distribution of Floquet-state occupation probabilities for a parametrically driven harmonic oscillator coupled to a thermal bath. Since the system exhibits detailed balance, and the canonical representatives of its quasienergies are equidistant, these probabilities are given by a geometrical Boltzmann distribution, but its quasitemperature differs from the actual temperature of the bath, being affected by the functional form of the latter's spectral density. We provide two examples of quasithermal engineering, i.e., of deliberate manipulation of the quasistatio…
We study the thermodynamical properties of a one-dimensional gas with one-dimensional gravitational interactions, and placed in a uniform mass background. Periodic boundary conditions are implemented as a modification of the potential consisting of a sum over mirror images (Ewald sum), regularized with an exponential cut-off. The system has a phase transition at a critical temperature. Above the critical temperature the gas density is uniform, while below the critical point the system becomes inhomogeneous. Numerical simulations of the model confirms the existence of the phase transition, and …
We study a uniform matrix product state as a variational state for classical and quantum spin chains in the thermodynamic limit. Under a careful treatment of the translational symmetry, eigen values of the transfer matrix defined in the calculation of expectation values can reflect the periodicity of the ground state and indicate optimum periodicity of the matrix product state. We discuss the relation between the periodicity and accuracy of magnetization curves. This approach is free from the error due to finite system size, which works well especially for the magnetic plateau problem.
We introduce a general framework for analyzing the thermodynamics of small systems that are driven by both a periodic temperature variation and some external parameter modulating their energy. This set-up covers, in particular, periodic micro and nano-heat engines. In a first step, we show how to express total entropy production by properly identified time-independent affinities and currents without making a linear response assumption. In linear response, kinetic coefficients akin to Onsager coefficients can be identified. Specializing to a Fokker-Planck type dynamics, we show that these coeff…
Periodic driving is used to operate machines that go from standard macroscopic engines to small non-equilibrium micro-sized systems. Two classes of such systems are small heat engines driven by periodic temperature variations and molecular pumps driven by external stimuli. Well known results that are valid for nonequilibrium steady states of systems driven by fixed thermodynamic forces, instead of an external periodic driving, have been generalized to periodically driven heat engines only recently. These results include a general expression for entropy production in terms of currents and affin…
We consider two $s=1/2$ spins with Heisenberg coupling and a monochromatic, circularly polarized magnetic field acting only onto one of the two spins. This system turns out to be analytically solvable. Also the statistical distribution of the work performed by the driving forces during one period can be obtained in closed form and the Jarzynski equation can be checked. The mean value of this work, viewed as a function of the physical parameters, exhibits features that can be related to some kind of Rabi oscillation. Moreover, when coupled to a heat bath the two spin system will approach a non-…
We develop a generic model for a cyclic quantum heat engine that makes it possible to coherently amplify a periodically modulated input signal without the need to couple the working medium to multiple reservoirs at the same time. Instead, we suggest an operation principle that is based on the spontaneous creation of population inversion in incomplete relaxation processes induced by periodic temperature variations. Focusing on Lindblad dynamics and systems with equally spaced energy levels, e.g. qubits or quantum harmonic oscillators, we derive a general working criterion for such cyclic quantu…
A short introduction on quantum thermodynamics is given and three new topics are discussed: 1) Maximal work extraction from a finite quantum system. The thermodynamic prediction fails and a new, general result is derived, the ``ergotropy''. 2) In work extraction from two-temperature setups, the presence of correlations can push the effective efficiency beyond the Carnot bound. 3) In the presence of level crossing, non-slow changes may be more optimal than slow ones.
The context of the present paper is stochastic thermodynamics - an approach to nonequilibrium thermodynamics rooted within the broader framework of stochastic control. In contrast to the classical paradigm of Carnot engines, we herein propose to consider thermodynamic processes with periodic continuously varying temperature of a heat bath and study questions of maximal power and efficiency for two idealized cases, overdamped (first-order) and underdamped (second-order) stochastic models. We highlight properties of optimal periodic control, derive and numerically validate approximate formulae f…
The nature of the behaviour of an isolated many-body quantum system periodically driven in time has been an open question since the beginning of quantum mechanics. After an initial transient, such a system is known to synchronize with the driving; in contrast to the non-driven case, no fundamental principle has been proposed for constructing the resulting non-equilibrium state. Here, we analytically show that, for a class of integrable systems, the relevant ensemble is constructed by maximizing an appropriately defined entropy subject to constraints, which we explicitly identify. This result c…
Onsager's phenomenological equations successfully describe irreversible thermodynamic processes. They assume a symmetric coupling matrix between thermodynamic fluxes and forces. It is easily shown that the antisymmetric part of a coupling matrix does not contribute to dissipation. Therefore, entropy production is exclusively governed by the symmetric matrix even in the presence of antisymmetric terms. In this work we focus on the antisymmetric contributions which describe isentropic oscillations and well-defined equations of motion. The formalism contains variables that are equivalent to momen…
Quantum thermodynamics seeks to extend non-equilibrium stochastic thermodynamics to small quantum systems where non-classical features are essential to its description. Such a research area has recently provided meaningful theoretical and experimental advances by exploring the wealth and the power of quantum features along with informational aspects of a system's thermodynamics. The relevance of such investigations is related to the fact that quantum technological devices are currently at the forefront of science and engineering applications. This short review article provides an overview of s…
We introduce an alternative route for obtaining reliable cyclic engines, based on interacting Brownian particles under time-periodic drivings. General expressions for the thermodynamic fluxes, such as power and heat, are obtained using the framework of Stochastic Thermodynamics. Several protocols for optimizing the engine performance are considered, by looking at system parameters such as the output forces and their phase-difference. We study both work-to-work and heat-to-work engines. Our results suggest that carefully designed interactions between particles can lead to more efficient engines…
Investigation of states with a periodic time dependence of physical quantities attracts a considerable interest now. Although it has been proposed initially that such states (coined Quantum Time Crystals) might be macroscopic and thermodynamically stable, results of a more careful study of the problem seemed to indicate that quantum time crystals could be realized only in systems out of equilibrium. Here we show that, in contrast to the general belief, thermodynamically stable macroscopic quantum time crystals can exist. The order parameter of this new state of matter is periodic in both real …
This book provides an introduction to the emerging field of quantum thermodynamics, with particular focus on its relation to quantum information and its implications for quantum computers and next generation quantum technologies. The text, aimed at graduate level physics students with a working knowledge of quantum mechanics and statistical physics, provides a brief overview of the development of classical thermodynamics and its quantum formulation in Chapter 1. Chapter 2 then explores typical thermodynamic settings, such as cycles and work extraction protocols, when the working material is ge…
According to the second law of thermodynamics the total entropy of a system is increased during almost any dynamical process. The positivity of the specific heat implies that the entropy increase is associated with heating. This is generally true both at the single particle level, like in the Fermi acceleration mechanism of charged particles reflected by magnetic mirrors, and for complex systems in everyday devices. Notable exceptions are known in noninteracting systems of particles moving in periodic potentials. Here the phenomenon of dynamical localization can prevent heating beyond certain …
The thermodynamics of small quantum many-body systems strongly coupled to a heat bath at low temperatures with non-Markovian behavior are new challenges for quantum thermodynamics, as traditional thermodynamics is built on large systems vanishingly weakly coupled to a non-dynamical reservoir. Important also are the quantum attributes, as in quantum coherence, correlations, entanglement and fluctuations. All told, one needs to reexamine the meaning of the thermodynamic functions, the viability of the thermodynamic relations and the validity of the thermodynamic laws anew. In one popular approac…
A two-dimensional lattice gas model is proposed. The ground state of this model with a fixed density is neither periodic nor quasi-periodic. It also depends on system size in an irregular manner. On the other hand, it is ordered in the sense that the entropy density is zero in the thermodynamic limit. The existence of a thermodynamic transition associated with such irregularly ordered ground states is conjectured from a duality relation for a thermodynamic function. This conjecture is supported by a phenomenological argument and numerical experiments.
A general formulation is presented to derive the equation of motion and to demonstrate thermodynamic consistency for several classes of phase field models at once. It applies to models with a conserved phase field, describing either uniform or periodic stable states, and containing slow as well as fast thermodynamic variables. The approach is based on an entropy functional formalism previously developed in the context of phase field models for uniform states [P. Galenko and D. Jou, Phys. Rev. E {\bf 71}, 046125 (2005)] and thus allows to extend several properties of the latter to phase field m…
Quantum decoherence is seen as an undesired source of irreversibility that destroys quantum resources. Quantum coherences seem to be a property that vanishes at thermodynamic equilibrium. Away from equilibrium, quantum coherences challenge the classical notions of a thermodynamic bath in a Carnot engines, affect the efficiency of quantum transport, lead to violations of Fourier's law, and can be used to dynamically control the temperature of a state. However, the role of quantum coherence in thermodynamics is not fully understood. Here we show that the relative entropy of a state with quantum …
This article sets up a new formalism to investigate stochastic thermodynamics of out-of-equilibrium quantum systems, where stochasticity primarily comes from quantum measurement. In the absence of any bath, we define a purely quantum component to heat exchange, that corresponds to energy fluctuations induced by measurement back-action. By providing a central position to quantum measurement, our formalism bridges the gap between quantum optics and quantum thermodynamics: Thermal signatures of measurement induced irreversibility are identified in a simple implementation of Jarzynski equality, wh…
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 nonequil…
The thermodynamic uncertainty relation is a universal trade-off relation connecting the precision of a current with the average dissipation at large times. For continuous time Markov chains (also called Markov jump processes) this relation is valid in the time-homogeneous case, while it fails in the time-periodic case. The latter is relevant for the study of several small thermodynamic systems. We consider here a time-periodic Markov chain with continuous time and a broad class of functionals of stochastic trajectories, which are general linear combinations of the empirical flow and the empiri…
Exact expressions of correlation functions between two spins in the boundary row of the two-dimensional rectangular Ising model with periodic-free boundary conditions and finite size
Using Brownian motion in periodic potentials $V(x)$ tilted by a force $f$, we provide physical insight into the thermodynamic uncertainty relation, a recently conjectured principle for statistical errors and irreversible heat dissipation in nonequilibrium steady states. According to the relation, nonequilibrium output generated from dissipative processes necessarily incurs an energetic cost or heat dissipation $q$, and in order to limit the output fluctuation within a relative uncertainty $ε$, at least $2k_BT/ε^2$ of heat must be dissipated. Our model shows that this bound is attained not only…
We give nonequilibrium Green's function (NEGF) perspective on thermodynamics formulations for open quantum systems strongly coupled to baths. Scattering approach implying thermodynamic consideration of a super-system (system plus baths) weakly coupled to external super-baths is compared with consideration of thermodynamics of a system strongly coupled to its baths. We analyze both approaches from the NEGF perspective and argue that the latter yields a possibility of thermodynamic formulation consistent with dynamical (quantum transport) description.
Accurately describing work extraction from a quantum system is a central objective for the extension of thermodynamics to individual quantum systems. The concepts of work and heat are surprisingly subtle when generalizations are made to arbitrary quantum states. We formulate an operational thermodynamics suitable for application to an open quantum system undergoing quantum evolution under a general quantum process by which we mean a completely-positive and trace-preserving map. We derive an operational first law of thermodynamics for such processes and show consistency with the second law. We …
Shanon's fundamental coding theorems relate classical information theory to thermodynamics. More recent theoretical work has been successful in relating quantum information theory to thermodynamics. For example, Schumacher proved a quantum version of Shannon's 1948 classical noiseless coding theorem. In this note, we extend the connection between quantum information theory and thermodynamics to include quantum error correction. There is a standard mechanism for describing errors that may occur during the transmission, storage, and manipulation of quantum information. One can formulate a criter…
Quantum work is usually determined from two projective measurements of the energy at the beginning and at the end of a thermodynamic process. However, this paradigm cannot be considered thermodynamically consistent as it does not account for the thermodynamic cost of these measurements. To remedy this conceptual inconsistency we introduce a novel paradigm that relies only on the expected change of the average energy given the initial energy eigenbasis. In particular, we completely omit quantum measurements in the definition of quantum work, and hence quantum work is identified as a thermodynam…
This article sets up a formalism to describe stochastic thermodynamics for driven out-of-equilibrium open quantum systems. A stochastic Schrödinger equation allows to construct quantum trajectories describing the dynamics of the system state vector in presence of an eventually monitored environment. Thermodynamic quantities are defined at the single quantum trajectory level, independently of any energy measurement, at any time of the protocol. We thereby identify coherent contributions, without classical counterparts, leading to quantum fluctuations of thermodynamic quantities. This formalism …
Classical 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 to study the exact nonequilibrium thermodynamics of quantum systems that is applicable to arbitrary small systems, even for single particle systems, in contact with a reservoir. We generalize the concept of temperature into nonequilibrium regime that depends on the detailed dynamics of quantum states. When we apply the theory to the cavity system and the…
The Carnot statement of the second law of thermodynamics poses an upper limit on the efficiency of all heat engines. Recently, it has been studied whether generic quantum features such as coherence and quantum entanglement could allow for quantum devices with efficiencies larger than the Carnot efficiency. The present study shows that this is not permitted by the laws of thermodynamics. In particular, we will show that rather the definition of heat has to be modified to account for the thermodynamic cost for maintaining coherence and entanglement. Our theoretical findings are numerically illus…
In the present paper, we study the power output and efficiency of overdamped stochastic thermodynamic engines that are in contact with a heat bath having a temperature that varies periodically with time. This is in contrast to most of the existing literature that considers the Carnot paradigm of alternating contact with heat baths having different fixed temperatures, hot and cold. Specifically, we consider a periodic and bounded but otherwise arbitrary temperature profile and derive explicit bounds on the power and efficiency achievable by a suitable controlling potential that couples the ther…
We revisit the equilibrium one-dimensional $ϕ^4$ model from the dynamical systems point of view. We find an infinite number of periodic orbits which are computationally stable. At the same time some of the orbits are found to exhibit positive Lyapunov exponents! The periodic orbits confine every particle in a periodic chain to trace out either the same or a mirror-image trajectory in its two-dimensional phase space. These "computationally stable" sets of pairs of single-particle orbits are either symmetric or antisymmetric to the very last computational bit. In such a periodic chain the odd-nu…
Using stochastic thermodynamics, the properties of interacting linear chains subject to periodic drivings are investigated. The systems are described by Fokker-Planck-Kramers equation and exact (explicit) solutions are obtained for periodic drivings as functions of the modulation frequency and strength constants. The limit of long chains is analyzed by means of a protocol in which the intermediate temperatures are self-consistently chosen and the entropy production is decomposed as a sum of two individual contributions, one coming from real baths (placed at extremities of lattice) and other fr…
We analyze an engine whose working fluid consists of a single quantum particle, paralleling Szilard's construction of a classical single-particle engine. Following his resolution of Maxwell's Second Law paradox using the latter, which turned on physically instantiating the demon (control subsystem), the quantum engine's design mirrors the classically-chaotic Szilard Map that operates a thermodynamic cycle of measurement, thermal-energy extraction, and memory reset. Focusing on the thermodynamic costs to observe and control the particle and comparing these in the quantum and classical limits, w…
We establish quantum thermodynamics for open quantum systems weakly coupled to their reservoirs when the system exhibits degeneracies. The first and second law of thermodynamics are derived, as well as a finite-time fluctuation theorem for mechanical work and energy and matter currents. Using a double quantum dot junction model, local eigenbasis coherences are shown to play a crucial role on thermodynamics and on the electron counting statistics.
We study thermodynamic processes in contact with a heat bath that may have an arbitrary time-varying periodic temperature profile. Within the framework of stochastic thermodynamics, and for models of thermo-dynamic engines in the idealized case of underdamped particles in the low-friction regime, we derive explicit bounds as well as optimal control protocols that draw maximum power and achieve maximum efficiency at any specified level of power.
The quantum Nash equilibrium in the thermodynamic limit is studied for games like quantum Prisoner's dilemma and the quantum game of chicken. A phase transition is seen in both games as a function of the entanglement in the game. We observe that for maximal entanglement irrespective of the classical payoffs, a majority of players choose Quantum strategy over Defect in the thermodynamic limit.
A strengthened Lindblad inequality has been proved. We have applied this result for proving a generalized $H$-theorem in non equilibrium thermodynamics. Information processing also can be considered as some thermodynamic process. From this point of view we have proved a strengthened data processing inequality in quantum information theory.
The second law of ordinary thermodynamics and the second law of steady state thermodynamics, as proposed by Oono and Paniconi, are investigated from the microscopic point of view for the open quantum system. Based on the H-theorem of Lindblad, we show that both second laws are consistent with the quantum dynamics generated by the completely positive map. In addition, microscopic expressions of entropy production and ``housekeeping heat'' are obtained for some classes of dynamical quantum systems.
Thermodynamics of nanoscale devices is an active area of research. Despite their noisy surrounding they often produce mechanical work (e.g. micro-heat engines), display rectified Brownian motion (e.g. molecular motors). This invokes research in terms of experimentally quantifiable thermodynamic efficiencies. Here, a Brownian particle is driven by a harmonic confinement with time-periodic contraction and expansion. The system produces work by being alternately (time-periodically) connected to baths with different dissipations. We analyze the system theoretically using stochastic thermodynamics.…
Several models of quantum open systems are known at present to violate, according to principles of the standard quantum theory of open systems, the second law of thermodynamics. Here, a new and rather trivial model of another type is suggested describing mechanism that violates, according to the same principles, the zeroth and the second laws of thermodynamics simultaneously. Up to a technically minor modification, the model resembles some models already known, solved by standard means, and properly understood. Universal validity of two basic principles of thermodynamics in strictly quantum si…
