Microcanonical quantum fluctuation theorems

Modified Jarzynski equality in a microcanonical ensemble

We show that the conventional Jarzynski equality does not hold for a system prepared in a microcanonical ensemble. We derive a modified equality that connects microcanonical work fluctuations to entropy production, in an analogous way to the Jarzynski equality, but with reference to an inverse temperature that depends on the path of the work protocol. For close to isothermal processes the modified equality can improve on the bound 〈W〉≥ΔF. Our result is a special case of a general expression for the microcanonical moment-generating function for any extensive quantity, which enables calculation of the breakdown of ensemble equivalence for thermodynamic fluctuations. We demonstrate our microcanonical Jarzynski equality in a system of driven two-level spins.

Sum rule for fluctuations of work

We study the fluctuations of work caused by applying cyclic perturbations and obtain an exact sum rule satisfied by the moments of work for a broad class of quantum stationary ensembles. In the case of the canonical ensemble, the sum rule reproduces the Jarzynski equality. The sum rule can also be simplified into a linear relationship between the work average and the second moment of work, which we numerically confirm via an exact diagonalization of a spin model system.

Entropy and temperature in finite isolated quantum systems

We investigate how the temperature calculated from the microcanonical entropy compares with the canonical temperature for finite isolated quantum systems. We concentrate on systems with sizes that make them accessible to numerical exact diagonalization. We thus characterize the deviations from ensemble equivalence at finite sizes. We describe multiple ways to compute the microcanonical entropy and present numerical results for the entropy and temperature computed in these various ways. We show that using an energy window whose width has a particular energy dependence results in a temperature with minimal deviations from the canonical temperature.

Integral fluctuation theorem and generalized Clausius inequality for microcanonical and pure states

Fluctuation theorems are cornerstones of modern statistical mechanics and their standard derivations routinely rely on the crucial assumption of a canonical equilibrium state. Yet rigorous derivations of certain fluctuation theorems for microcanonical states and pure energy eigenstates in isolated quantum systems are still lacking and constitute a major challenge to theory. In this work we tackle this challenge and present such a derivation of an integral fluctuation theorem (IFT) by invoking two central and physically natural conditions, i.e., the so-called "stiffness" and "smoothness" of transition probabilities. Our analytical arguments are additionally substantiated by numerical simulations for archetypal many-body quantum systems, including integrable as well as nonintegrable models of interacting spins and hard-core bosons on a lattice. These simulations strongly suggest that "stiffness" and "smoothness" are indeed of vital importance for the validity of the IFT for microcanonical and pure states. Our work contrasts with recent approaches to the IFT based on Lieb-Robinson speeds and the eigenstate thermalization hypothesis.

Fluctuation theorems from Bayesian retrodiction

Quantitative studies of irreversibility in statistical mechanics often involve the consideration of a reverse process, whose definition has been the object of many discussions, in particular for quantum mechanical systems. Here we show that the reverse channel very naturally arises from Bayesian retrodiction, in both classical and quantum theories. Previous paradigmatic results, such as Jarzynski's equality, Crooks' fluctuation theorem, and Tasaki's two-measurement fluctuation theorem for closed driven quantum systems, are all shown to be consistent with retrodictive arguments. Also, various corrections that were introduced to deal with nonequilibrium steady states or open quantum systems are justified on general grounds as remnants of Bayesian retrodiction. More generally, with the reverse process constructed on consistent logical inference, fluctuation relations acquire a much broader form and scope.

Anomalous Thermalization in Quantum Collective Models

We show that apparently thermalized states still store relevant amounts of information about their past, information that can be tracked by experiments involving nonequilibrium processes. We provide a condition for the microcanonical quantum Crook's theorem, and we test it by means of numerical experiments. In the Lipkin-Meshkov-Glick model, two different procedures leading to the same equilibrium states give rise to different statistics of work in nonequilibrium processes. In the Dicke model, two different trajectories for the same nonequilibrium protocol produce different statistics of work. Microcanonical averages provide the correct results for the expectation values of physical observables in all the cases; the microcanonical quantum Crook's theorem fails in some of them. We conclude that testing quantum fluctuation theorems is mandatory to verify if a system is properly thermalized.

Stiffness of probability distributions of work and Jarzynski relation for non-Gibbsian initial states

We consider closed quantum systems (into which baths may be integrated) that are driven, i.e., subject to time-dependent Hamiltonians. Our point of departure is the assumption that if systems start in non-Gibbsian states at some initial energies, the resulting probability distributions of work may be largely independent of the specific initial energies. It is demonstrated that this assumption has some far-reaching consequences, e.g., it implies the validity of the Jarzynski relation for a large class of non-Gibbsian initial states. By performing numerical analysis on integrable and nonintegrable spin systems, we find the above assumption fulfilled for all examples considered. Through an analysis based on Fermi's golden rule, we partially relate these findings to the applicability of the eigenstate thermalization ansatz to the respective driving operators.

Superconducting Qubits as Mechanical Quantum Engines

We propose the equivalence of superconducting qubits with a pistonlike mechanical quantum engine. The work reports a study on the nature of the nonequilibrium work exchanged with the quantum-nonadiabatic working medium, which is modeled as a multilevel coupled quantum well system subject to an external control parameter. The quantum dynamics is solved for arbitrary control protocols. It is shown that the work output has two components: one that depends instantaneously on the level populations and another that is due to the quantum coherences built in the system. The nonadiabatic coherent dynamics of the quantum engine gives rise to a resistance (friction) force that decreases the work output. We consider the functional equivalence of such a device and a rf-SQUID flux qubit.

Merits and qualms of work fluctuations in classical fluctuation theorems

Work is one of the most basic notions in statistical mechanics, with work fluctuation theorems being one central topic in nanoscale thermodynamics. With Hamiltonian chaos commonly thought to provide a foundation for classical statistical mechanics, here we present general salient results regarding how (classical) Hamiltonian chaos generically impacts on nonequilibrium work fluctuations. For isolated chaotic systems prepared with a microcanonical distribution, work fluctuations are minimized and vanish altogether in adiabatic work protocols. For isolated chaotic systems prepared at an initial canonical distribution at inverse temperature β, work fluctuations depicted by the variance of e^{-βW} are also minimized by adiabatic work protocols. This general result indicates that, if the variance of e^{-βW} diverges for an adiabatic work protocol, it diverges for all nonadiabatic work protocols sharing the same initial and final Hamiltonians. Such divergence is hence not an isolated event and thus greatly impacts on the efficiency of using Jarzynski's equality to simulate free-energy differences. Theoretical results are illustrated in a Sinai model. Our general insights shall boost studies in nanoscale thermodynamics and are of fundamental importance in designing useful work protocols.

Eigenstate thermalization hypothesis and quantum Jarzynski relation for pure initial states

Since the first suggestion of the Jarzynski equality many derivations of this equality have been presented in both the classical and the quantum context. While the approaches and settings differ greatly from one another, they all appear to rely on the condition that the initial state is a thermal Gibbs state. Here, we present an investigation of work distributions in driven isolated quantum systems, starting from pure states that are close to energy eigenstates of the initial Hamiltonian. We find that, for the nonintegrable quantum ladder studied, the Jarzynski equality is fulfilled to a good accuracy.

Occurrence of discontinuities in the performance of finite-time quantum Otto cycles

We study a quantum Otto cycle in which the strokes are performed in finite time. The cycle involves energy measurements at the end of each stroke to allow for the respective determination of work. We then optimize for the work and efficiency of the cycle by varying the time spent in the different strokes and find that the optimal value of the ratio of time spent on each stroke goes through sudden changes as the parameters of this cycle vary continuously. The position of these discontinuities depends on the optimized quantity under consideration such as the net work output or the efficiency.

Aspects of quantum work

Various approaches of defining and determining work performed on a quantum system are compared. Any operational definition of work, however, must allow for two facts: first, that work characterizes a process rather than an instantaneous state of a system and, second, that quantum systems are sensitive to the interactions with a measurement apparatus. We compare different measurement scenarios on the basis of the resulting postmeasurement states and the according probabilities for finding a particular work value. In particular, we analyze a recently proposed work meter for the case of a Gaussian pointer state and compare it with the results obtained by two projective and, alternatively, two Gaussian measurements. In the limit of a strong effective measurement strength the work distribution of projective two energy measurements can be recovered. In the opposite limit the average of work becomes independent of any measurement. Yet the fluctuations about this value diverge. The performance of the work meter is illustrated by the example of a spin in a suddenly changing magnetic field.

Irreversible processes without energy dissipation in an isolated Lipkin-Meshkov-Glick model

For a certain class of isolated quantum systems, we report the existence of irreversible processes in which the energy is not dissipated. After a closed cycle in which the initial energy distribution is fully recovered, the expectation value of a symmetry-breaking observable changes from a value differing from zero in the initial state to zero in the final state. This entails the unavoidable loss of a certain amount of information and constitutes a source of irreversibility. We show that the von Neumann entropy of time-averaged equilibrium states increases in the same magnitude as a consequence of the process. We support this result by means of numerical calculations in an experimentally feasible system, the Lipkin-Meshkov-Glick model.

Fluctuations of work in nearly adiabatically driven open quantum systems

We extend the quantum jump method to nearly adiabatically driven open quantum systems in a way that allows for an accurate account of the external driving in the system-environment interaction. Using this framework, we construct the corresponding trajectory-dependent work performed on the system and derive the integral fluctuation theorem and the Jarzynski equality for nearly adiabatic driving. We show that such identities hold as long as the stochastic dynamics and work variable are consistently defined. We numerically study the emerging work statistics for a two-level quantum system and find that the conventional diabatic approximation is unable to capture some prominent features arising from driving, such as the continuity of the probability density of work. Our results reveal the necessity of using accurate expressions for the drive-dressed heat exchange in future experiments probing jump time distributions.

Thermodynamic laws in isolated systems

The recent experimental realization of exotic matter states in isolated quantum systems and the ensuing controversy about the existence of negative absolute temperatures demand a careful analysis of the conceptual foundations underlying microcanonical thermostatistics. Here we provide a detailed comparison of the most commonly considered microcanonical entropy definitions, focusing specifically on whether they satisfy or violate the zeroth, first, and second laws of thermodynamics. Our analysis shows that, for a broad class of systems that includes all standard classical Hamiltonian systems, only the Gibbs volume entropy fulfills all three laws simultaneously. To avoid ambiguities, the discussion is restricted to exact results and analytically tractable examples.

Quantum fluctuation theorems and generalized measurements during the force protocol

Generalized measurements of an observable performed on a quantum system during a force protocol are investigated and conditions that guarantee the validity of the Jarzynski equality and the Crooks relation are formulated. In agreement with previous studies by M. Campisi, P. Talkner, and P. Hänggi [Phys. Rev. Lett. 105, 140601 (2010); Phys. Rev. E 83, 041114 (2011)], we find that these fluctuation relations are satisfied for projective measurements; however, for generalized measurements special conditions on the operators determining the measurements need to be met. For the Jarzynski equality to hold, the measurement operators of the forward protocol must be normalized in a particular way. The Crooks relation additionally entails that the backward and forward measurement operators depend on each other. Yet, quite some freedom is left as to how the two sets of operators are interrelated. This ambiguity is removed if one considers selective measurements, which are specified by a joint probability density function of work and measurement results of the considered observable. We find that the respective forward and backward joint probabilities satisfy the Crooks relation only if the measurement operators of the forward and backward protocols are the time-reversed adjoints of each other. In this case, the work probability density function conditioned on the measurement result satisfies a modified Crooks relation. The modification appears as a protocol-dependent factor that can be expressed by the information gained by the measurements during the forward and backward protocols. Finally, detailed fluctuation theorems with an arbitrary number of intervening measurements are obtained.

Work fluctuations for Bose particles in grand canonical initial states

We consider bosons in a harmonic trap and investigate the fluctuations of the work performed by an adiabatic change of the trap curvature. Depending on the reservoir conditions such as temperature and chemical potential that provide the initial equilibrium state, the exponentiated work average (EWA) defined in the context of the Crooks relation and the Jarzynski equality may diverge if the trap becomes wider. We investigate how the probability distribution function (PDF) of the work signals this divergence. It is shown that at low temperatures the PDF is highly asymmetric with a steep fall-off at one side and an exponential tail at the other side. For high temperatures it is closer to a symmetric distribution approaching a Gaussian form. These properties of the work PDF are discussed in relation to the convergence of the EWA and to the existence of the hypothetical equilibrium state to which those thermodynamic potential changes refer that enter both the Crooks relation and the Jarzynski equality.

Quantum-trajectory approach to the stochastic thermodynamics of a forced harmonic oscillator

I formulate a quantum stochastic thermodynamics for the quantum trajectories of a continuously monitored forced harmonic oscillator coupled to a thermal reservoir. Consistent trajectory-dependent definitions are introduced for work, heat, and entropy, through engineering the thermal reservoir from a sequence of two-level systems. Within this formalism the connection between irreversibility and entropy production is analyzed and confirmed by proving a detailed fluctuation theorem for quantum trajectories. Finally, possible experimental verifications are discussed.

Nonequilibrium work statistics of an Aharonov-Bohm flux

We investigate the statistics of work performed on a noninteracting electron gas confined in a ring as a threaded magnetic field is turned on. For an electron gas initially prepared in a grand canonical state it is demonstrated that the Jarzynski equality continues to hold in this case, with the free energy replaced by the grand potential. The work distribution displays a marked dependence on the temperature. While in the classical (high-temperature) regime, the work probability density function follows a Gaussian distribution and the free energy difference entering the Jarzynski equality is null, the free energy difference is finite in the quantum regime, and the work probability distribution function becomes multimodal. We point out the dependence of the work statistics on the number of electrons composing the system.

Work statistics of charged noninteracting fermions in slowly changing magnetic fields

We consider N fermionic particles in a harmonic trap initially prepared in a thermal equilibrium state at temperature β^{-1} and examine the probability density function (pdf) of the work done by a magnetic field slowly varying in time. The behavior of the pdf crucially depends on the number of particles N but also on the temperature. At high temperatures (β≪1) the pdf is given by an asymmetric Laplace distribution for a single particle, and for many particles it approaches a Gaussian distribution with variance proportional to N/β(2). At low temperatures the pdf becomes strongly peaked at the center with a variance that still linearly increases with N but exponentially decreases with the temperature. We point out the consequences of these findings for the experimental confirmation of the Jarzynski equality such as the low probability issue at high temperatures and its solution at low temperatures, together with a discussion of the crossover behavior between the two temperature regimes.

Autonomous modular quantum systems: contextual Jarzynski relations

For autonomous quantum systems with modular structure we demonstrate that the Jarzynski relation can be reinterpreted to apply even locally: For this purpose certain contexts have to be introduced by selecting the system of interest versus its environment. The respective energy exchange is then divided into heat and work based on a generalized definition of these notions. In this way we are able to identify functional parts of the environment as either heat or work sources, respectively. We investigate different combinations of these functional parts with respect to contextual Jarzynski relations. Our analytical results are confirmed by numerical investigations on small multipartite systems.

Quantum heat-fluctuation theorem of a reduced system: an exactly solvable case

For a system in which free quantum Brownian particles interact with a non-Markovian heat bath with product initial states, the quantum heat-fluctuation theorems and the Jarzynski relation of the reduced system are derived using the path integral technique. They are not a direct generalization of the classical case. The validity of the second law of quantum thermodynamics is shown. The forms of these theorems and relations have to do with the specified nonequilibrium initial states, and are dependent on the structure of the bath (ergodic or nonergodic) in the quantum case as well as the classical case under some initial states.

Quantum Bochkov-Kuzovlev work fluctuation theorems

The quantum version of the Bochkov-Kuzovlev identity is derived on the basis of the appropriate definition of work as the difference of the measured internal energies of a quantum system at the beginning and the end of an external action on the system given by a prescribed protocol. According to the spirit of the original Bochkov-Kuzovlev approach, we adopt the 'exclusive' viewpoint, meaning that the coupling to the external work source is not counted as part of the internal energy. The corresponding canonical and microcanonical quantum fluctuation theorems are derived as well, and are compared with the respective theorems obtained within the 'inclusive' approach. The relations between the quantum inclusive work w, the exclusive work w(0) and the dissipated work w(dis), are discussed and clarified. We show by an explicit example that w(0) and w(dis) are distinct stochastic quantities obeying different statistics.

Quantum quenches in the Dicke model: statistics of the work done and of other observables

We study the statistics of the work done in a zero temperature quench of the coupling constant in the Dicke model describing the interaction between an ensemble of two level systems and a single bosonic mode. When either the final or the initial coupling constants approach the critical coupling lambdac that separates the normal and superradiant phases of the system, the probability distribution of the work done displays singular behavior. The average work tends to diverge as the initial coupling parameter is brought closer to the critical value lambdac. In contrast, for quenches ending close to criticality, the distribution of work has finite moments but displays a sequence of edge singularities. This contrasting behavior is related to the difference between the processes of compression and expansion of a particle subject to a sudden change in its confining potential. We confirm this by studying in detail the time-dependent statistics of other observables, such as the quadratures of the photons and the total occupation of the bosonic modes.

Finite bath fluctuation theorem

We demonstrate that a finite bath fluctuation theorem of the Crooks type holds for systems that have been thermalized via weakly coupling them to a bath with energy independent finite specific heat. We show that this theorem reduces to the known canonical and microcanonical fluctuation theorems in the two respective limiting cases of infinite and vanishing specific heat of the bath. The result is elucidated by applying it to a two-dimensional hard disk colliding elastically with few other hard disks in a rectangular box with perfectly reflecting walls.

Increase of Boltzmann entropy in a quantum forced harmonic oscillator

Recently, a quantum-mechanical proof of the increase of Boltzmann entropy in quantum systems that are coupled to an external classical source of work has been given. Here we illustrate this result by applying it to a forced quantum harmonic oscillator. We show plots of the actual temporal evolution of work and entropy for various forcing protocols. We note that entropy and work can be partially or even fully returned to the source, although both work and entropy balances are non-negative at all times in accordance with the minimal work principle and the Clausius principle, respectively. A necessary condition for the increase of entropy is that the initial distribution is decreasing (e.g., canonical). We show evidence that for a nondecreasing distribution (e.g., microcanonical), the quantum expectation of entropy may decrease slightly. Interestingly, the classical expectation of entropy cannot decrease, irrespective of the initial distribution, in the forced harmonic oscillator.

Quantum free-energy differences from nonequilibrium path integrals. II. Convergence properties for the harmonic oscillator

Nonequilibrium path-integral methods for computing quantum free-energy differences are applied to a quantum particle trapped in a harmonic well of uniformly changing strength with the purpose of establishing the convergence properties of the work distribution and free energy as the number of degrees of freedom M in the regularized path integrals goes to infinity. The work distribution is found to converge when M tends to infinity regardless of the switching speed, leading to finite results for the free-energy difference when the Jarzynski nonequilibrium work relation or the Crooks fluctuation relation are used. The nature of the convergence depends on the regularization method. For the Fourier method, the convergence of the free-energy difference and work distribution go as 1/M , while both quantities converge as 1/M(2) when the bead regularization procedure is used. The implications of these results to more general systems are discussed.

Fluctuation-dissipation theorem for the microcanonical ensemble

A derivation of the fluctuation-dissipation theorem for the microcanonical ensemble is presented using linear response theory. The theorem is stated as a relation between the frequency spectra of the symmetric correlation and response functions. When the system is not in the thermodynamic limit, this result can be viewed as an extension of the fluctuation-dissipation relations to a situation where dynamical fluctuations determine the response. Therefore, the relation presented here between equilibrium fluctuations and response can have a very different physical nature from the usual one in the canonical ensemble. These considerations imply that the fluctuation-dissipation theorem is not restricted to the context of the canonical ensemble, where it is usually derived. Dispersion relations and sum rules are also obtained and discussed in the present case. Although analogous to the Kramers-Kronig relations, they are not related to the frequency spectrum but to the energy dependence of the response function.

Statistics of work performed on a forced quantum oscillator

Various aspects of the statistics of work performed by an external classical force on a quantum mechanical system are elucidated for a driven harmonic oscillator. In this special case two parameters are introduced that are sufficient to completely characterize the force protocol. Explicit results for the characteristic function of work and the corresponding probability distribution are provided and discussed for three different types of initial states of the oscillator: microcanonical, canonical, and coherent states. Depending on the choice of the initial state the probability distributions of the performed work may greatly differ. This result in particular also holds true for identical force protocols. General fluctuation and work theorems holding for microcanonical and canonical initial states are confirmed.

Complementary expressions for the entropy-from-work theorem

We establish an expression of the entropy-from-work theorem that is complementary to the one originally proposed by Talkner, Hanggi, and Morillo [Phys. Rev. E 77, 051131 (2008)]. In the original expression the final energy is fixed, whereas in the present expression the initial energy is fixed.