Work measurement as a generalized quantum measurement

Operational Work Fluctuation Theorem for Open Quantum Systems

The classical Jarzynski equality establishes an exact relation between the stochastic work performed on a system driven out of thermal equilibrium and the free energy difference in a corresponding quasistatic process. This fluctuation theorem bears experimental relevance, as it enables the determination of the free energy difference through the measurement of externally applied work in a nonequilibrium process. In the quantum case, the Jarzynski equality only holds if the measurement procedure of the stochastic work is drastically changed: it is replaced by a so-called two-point measurement scheme that requires the knowledge of the initial and final Hamiltonian and therefore lacks the predictive power for the free energy difference that the classical Jarzynski equation is known for. Here, we propose a quantum fluctuation theorem that is valid for externally measurable quantum work determined during the driving protocol. In contrast to the two-point measurement case, the theorem also applies to open quantum systems and the scenario can be realized without knowing the system Hamiltonian. Our fluctuation theorem comes in the form of an inequality and therefore only yields bounds to the true free energy difference. The inequality is saturated in the quasiclassical case of vanishing energy coherences at the beginning and at the end of the protocol. Thus, there is a clear quantum disadvantage.

Consciousness and Energy Processing in Neural Systems

BACKGROUND

Our understanding of the relationship between neural activity and psychological states has advanced greatly in recent decades. But we are still unable to explain conscious experience in terms of physical processes occurring in our brains.

METHODS

This paper introduces a conceptual framework that may contribute to an explanation. All physical processes entail the transfer, transduction, and transformation of energy between portions of matter as work is performed in material systems. If the production of consciousness in nervous systems is a physical process, then it must entail the same. Here the nervous system, and the brain in particular, is considered as a material system that transfers, transduces, and transforms energy as it performs biophysical work.

CONCLUSIONS

Evidence from neuroscience suggests that conscious experience is produced in the organic matter of nervous systems when they perform biophysical work at classical and quantum scales with a certain level of dynamic complexity or organization. An empirically grounded, falsifiable, and testable hypothesis is offered to explain how energy processing in nervous systems may produce conscious experience at a fundamental physical level.

Work and efficiency fluctuations in a quantum Otto cycle with idle levels

We study the performance of a quantum Otto heat engine with two spins coupled by a Heisenberg interaction, taking into account not only the mean values of work and efficiency but also their fluctuations. We first show that, for this system, the output work and its fluctuations are directly related to the magnetization and magnetic susceptibility of the system at equilibrium with either heat bath. We analyze the regions where the work extraction can be done with low relative fluctuation for a given range of temperatures, while still achieving an efficiency higher than that of a single spin system heat engine. In particular, we find that, due to the presence of "idle" levels, an increase in the interspin coupling can either increase or decrease fluctuations, depending on the other parameters. In all cases, however, we find that the relative fluctuations in work or efficiency remain large, implying that this microscopic engine is not very reliable as a source of work.

Controlling work output and coherence in finite-time quantum Otto engines through monitoring

We examine the role of diagnostic quantum measurements on the work statistics of a finite-time quantum Otto heat engine operated in the steady state. We consider three pointer-based measurement schemes that differ in the number of system-pointer interactions and pointer measurements. We show that the coherence of the working substance and the work output of the engine can be controlled by tuning the monitoring measurements. Moreover, for a working substance consisting of a two-level system we show that while all three schemes reproduce the predictions of the cycle without any monitoring for the average work in the limit of infinitely weak measurement, only two of the schemes can reproduce the two-point projective measurement results in the limit of strong measurement.

A Wigner Quasiprobability Distribution of Work

In this article, we introduce a quasiprobability distribution of work that is based on the Wigner function. This proposal rests on the idea that the work conducted on an isolated system can be coherently measured by coupling the system to a quantum measurement apparatus. In this way, a quasiprobability distribution of work can be defined in terms of the Wigner function of the apparatus. This quasidistribution contains the information of the work statistics and also holds a clear operational definition that can be directly measured in a real experiment. Moreover, it is shown that the presence of quantum coherence in the energy eigenbasis is related with the appearance of features related to non-classicality in the Wigner function such as negativity and interference fringes. On the other hand, from this quasiprobability distribution, it is straightforward to obtain the standard two-point measurement probability distribution of work and also the difference in average energy for initial states with coherences.

Single-energy-measurement integral fluctuation theorem and nonprojective measurements

We study a Jarzysnki-type equality for work in systems that are monitored using nonprojective unsharp measurements. The information acquired by the observer from the outcome f of an energy measurement and the subsequent conditioned normalized state ρ[over ̂](t,f) evolved up to a final time t are used to define work, as the difference between the final expectation value of the energy and the result f of the measurement. The Jarzynski equality obtained depends on the coherences that the state develops during the process, the characteristics of the meter used to measure the energy, and the noise it induces into the system. We analyze those contributions in some detail to unveil their role. We show that in very particular cases, but not in general, the effect of such noise gives a factor multiplying the result that would be obtained if projective measurements were used instead of nonprojective ones. The unsharp character of the measurements used to monitor the energy of the system, which defines the resolution of the meter, leads to different scenarios of interest. In particular, if the distance between neighboring elements in the energy spectrum is much larger than the resolution of the meter, then a similar result to the projective measurement case is obtained, up to a multiplicative factor that depends on the meter. A more subtle situation arises in the opposite case in which measurements may be noninformative, i.e., they may not contribute to update the information about the system. In this case a correction to the relation obtained in the nonoverlapping case appears. We analyze the conditions in which such a correction becomes negligible. We also study the coherences, in terms of the relative entropy of coherence developed by the evolved post-measurement state. We illustrate the results by analyzing a two-level system monitored by a simple meter.

Enantiomer detection via quantum Otto cycle

Enantiomers are chiral molecules that exist in right-handed and left-handed conformations. Optical techniques of enantiomers' detection are widely employed to discriminate between left- and right-handed molecules. However, identical spectra of enantiomers make enantiomer detection a very challenging task. Here, we investigate the possibility of exploiting thermodynamic processes for enantiomer detection. In particular, we employ a quantum Otto cycle in which a chiral molecule described by a three-level system with cyclic optical transitions is considered a working medium. Each energy transition of the three-level system is coupled with an external laser drive. We find that the left- and right-handed enantiomers operate as a quantum heat engine and a thermal accelerator, respectively, when the overall phase is the control parameter. In addition, both enantiomers act as heat engines by keeping the overall phase constant and using the laser drives' detuning as the control parameter during the cycle. However, the molecules can still be distinguished because both cases' extracted work and efficiency are quantitatively very different. Accordingly, the left- and right-handed molecules can be distinguished by evaluating the work distribution in the Otto cycle.

Energy dynamics, heat production and heat-work conversion with qubits: toward the development of quantum machines

We present an overview of recent advances in the study of energy dynamics and mechanisms for energy conversion in qubit systems with special focus on realizations in superconducting quantum circuits. We briefly introduce the relevant theoretical framework to analyze heat generation, energy transport and energy conversion in these systems with and without time-dependent driving considering the effect of equilibrium and non-equilibrium environments. We analyze specific problems and mechanisms under current investigation in the context of qubit systems. These include the problem of energy dissipation and possible routes for its control, energy pumping between driving sources and heat pumping between reservoirs, implementation of thermal machines and mechanisms for energy storage. We highlight the underlying fundamental phenomena related to geometrical and topological properties, as well as many-body correlations. We also present an overview of recent experimental activity in this field.

Joint measurability in nonequilibrium quantum thermodynamics

In this Letter we investigate the concept of quantum work and its measurability from the viewpoint of quantum measurement theory. Very often, quantum work and fluctuation theorems are discussed in the framework of projective two-point measurement (TPM) schemes. According to a well-known no-go theorem, there is no work observable which satisfies both (i) an average work condition and (ii) the TPM statistics for diagonal input states. Such projective measurements represent a restrictive class among all possible measurements. It is desirable, both from a theoretical and experimental point of view, to extend the scheme to the general case including suitably designed unsharp measurements. This shifts the focus to the question of what information about work and its fluctuations one is able to extract from such generalized measurements. We show that the no-go theorem no longer holds if the observables in a TPM scheme are jointly measurable for any intermediate unitary evolution. We explicitly construct a model with unsharp energy measurements and derive bounds for the visibility that ensure joint measurability. In such an unsharp scenario a single work measurement apparatus can be constructed that allows us to determine the correct average work and to obtain free energy differences with the help of a Jarzynski equality.

Spectral Filtering Induced by Non-Hermitian Evolution with Balanced Gain and Loss: Enhancing Quantum Chaos

The dynamical signatures of quantum chaos in an isolated system are captured by the spectral form factor, which exhibits as a function of time a dip, a ramp, and a plateau, with the ramp being governed by the correlations in the level spacing distribution. While decoherence generally suppresses these dynamical signatures, the nonlinear non-Hermitian evolution with balanced gain and loss (BGL) in an energy-dephasing scenario can enhance manifestations of quantum chaos. In the Sachdev-Ye-Kitaev model and random matrix Hamiltonians, BGL increases the span of the ramp, lowering the dip as well as the value of the plateau, providing an experimentally realizable physical mechanism for spectral filtering. The chaos enhancement due to BGL is optimal over a family of filter functions that can be engineered with fluctuating Hamiltonians.

Class of quasiprobability distributions of work with initial quantum coherence

The work is a concept of fundamental importance in thermodynamics. An open question is how to describe the work fluctuation for quantum coherent processes in the presence of initial quantum coherence in the energy basis. With the aim of giving a unified description, here we introduce and study a class of quasiprobability distributions of work, which give an average work equal to the average energy change of the system and reduce to the two-projective-measurement scheme for an initial incoherent state. Moreover, we characterize the work with the help of fluctuation relations. In particular, by considering the joint distribution of work and initial quantum coherence, we find a fluctuation theorem involving quantum coherence, from which follows a second law of thermodynamics in the case of initial thermal populations.

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.

Experimental Validation of Fully Quantum Fluctuation Theorems Using Dynamic Bayesian Networks

Fluctuation theorems are fundamental extensions of the second law of thermodynamics for small systems. Their general validity arbitrarily far from equilibrium makes them invaluable in nonequilibrium physics. So far, experimental studies of quantum fluctuation relations do not account for quantum correlations and quantum coherence, two essential quantum properties. We here apply a novel dynamic Bayesian network approach to experimentally test detailed and integral fully quantum fluctuation theorems for heat exchange between two quantum-correlated thermal spins-1/2 in a nuclear magnetic resonance setup. We concretely verify individual integral fluctuation relations for quantum correlations and quantum coherence, as well as for the sum of all quantum contributions. We further investigate the thermodynamic cost of creating correlations and coherence.

Quantum and Classical Ergotropy from Relative Entropies

The quantum ergotropy quantifies the maximal amount of work that can be extracted from a quantum state without changing its entropy. Given that the ergotropy can be expressed as the difference of quantum and classical relative entropies of the quantum state with respect to the thermal state, we define the classical ergotropy, which quantifies how much work can be extracted from distributions that are inhomogeneous on the energy surfaces. A unified approach to treat both quantum as well as classical scenarios is provided by geometric quantum mechanics, for which we define the geometric relative entropy. The analysis is concluded with an application of the conceptual insight to conditional thermal states, and the correspondingly tightened maximum work theorem.

Stiffness of probability distributions of work and Jarzynski relation for initial microcanonical and energy eigenstates

We consider closed quantum systems which are driven such that only negligible heating occurs. If driving only affects small parts of the system, it may nonetheless be strong. Our analysis aims at clarifying under which conditions the Jarzynski relation (JR) holds in such setups, if the initial states are microcanonical or even energy eigenstates. We find that the validity of the JR for the microcanonical initial state hinges on an exponential density of states and on stiffness. The latter indicates an independence of the probability density functions (PDFs) of work of the energy of the respective microcanonical initial state. The validity of the JR for initial energy eigenstates is found to additionally require smoothness. The latter indicates an independence of the work PDFs of the specific energy eigenstates within a microcanonical energy shell. As the validity of the JR for pure initial energy eigenstates has no analog in classical systems, we consider it a genuine quantum phenomenon.

Easy Access to Energy Fluctuations in Nonequilibrium Quantum Many-Body Systems

We combine theoretical and experimental efforts to propose a method for studying energy fluctuations, in particular, to obtain the related bistochastic matrix of transition probabilities by means of simple measurements at the end of a protocol that drives a many-body quantum system out of equilibrium. This scheme is integrated with numerical optimizations in order to ensure a proper analysis of the experimental data, leading to physical probabilities. The method is experimentally evaluated employing a two interacting spin-1/2 system in a nuclear magnetic resonance setup. We show how to recover the transition probabilities using only local measures, which enables an experimental verification of the detailed fluctuation theorem in a many-body system driven out of equilibrium.

Quantum Fluctuation Theorems beyond Two-Point Measurements

We derive detailed and integral quantum fluctuation theorems for heat exchange in a quantum correlated bipartite thermal system using the framework of dynamic Bayesian networks. Contrary to the usual two-projective-measurement scheme that is known to destroy quantum features, these fluctuation relations fully capture quantum correlations and quantum coherence at arbitrary times. We further obtain individual integral fluctuation theorems for classical and quantum correlations, as well as for local and global quantum coherences.

Work statistics for sudden quenches in interacting quantum many-body systems

Work in isolated quantum systems is a random variable and its probability distribution function obeys the celebrated fluctuation theorems of Crooks and Jarzynski. In this study, we provide a simple way to describe the work probability distribution function for sudden quench processes in quantum systems with large Hilbert spaces. This description can be constructed from two elements: the level density of the initial Hamiltonian, and a smoothed strength function that provides information about the influence of the perturbation over the eigenvectors in the quench process, and is especially suited to describe quantum many-body interacting systems. We also show how random models can be used to find such smoothed work probability distribution and apply this approach to different one-dimensional spin-1/2 chain models. Our findings provide an accurate description of the work distribution of such systems in the cases of intermediate and high temperatures in both chaotic and integrable regimes.

Characterizing a Statistical Arrow of Time in Quantum Measurement Dynamics

In both thermodynamics and quantum mechanics, the arrow of time is characterized by the statistical likelihood of physical processes. We characterize this arrow of time for the continuous quantum measurement dynamics of a superconducting qubit. By experimentally tracking individual weak measurement trajectories, we compare the path probabilities of forward and backward-in-time evolution to develop an arrow of time statistic associated with measurement dynamics. We compare the statistics of individual trajectories to ensemble properties showing that the measurement dynamics obeys both detailed and integral fluctuation theorems, thus establishing the consistency between microscopic and macroscopic measurement dynamics.

Probing the Full Distribution of Many-Body Observables By Single-Qubit Interferometry

We present an experimental scheme to measure the full distribution of many-body observables in spin systems, both in and out of equilibrium, using an auxiliary qubit as a probe. We focus on the determination of the magnetization and the kink number statistics at thermal equilibrium. The corresponding characteristic functions are related to the analytically continued partition function. Thus, both distributions can be directly extracted from experimental measurements of the coherence of a probe qubit that is coupled to an Ising-type bath, as reported in [X. Peng et al., Phys. Rev. Lett. 114, 010601 (2015)PRLTAO0031-900710.1103/PhysRevLett.114.010601] for the detection of Lee-Yang zeros.

Quantum fluctuation theorem for error diagnostics in quantum annealers

Near term quantum hardware promises unprecedented computational advantage. Crucial in its development is the characterization and minimization of computational errors. We propose the use of the quantum fluctuation theorem to benchmark the accuracy of quantum annealers. This versatile tool provides simple means to determine whether the quantum dynamics are unital, unitary, and adiabatic, or whether the system is prone to thermal noise. Our proposal is experimentally tested on two generations of the D-Wave machine, which illustrates the sensitivity of the fluctuation theorem to the smallest aberrations from ideal annealing. In addition, for the optimally operating D-Wave machine, our experiment provides the first experimental verification of the integral fluctuation in an interacting, many-body quantum system.

Quantum work statistics, Loschmidt echo and information scrambling

A universal relation is established between the quantum work probability distribution of an isolated driven quantum system and the Loschmidt echo dynamics of a two-mode squeezed state. When the initial density matrix is canonical, the Loschmidt echo of the purified double thermofield state provides a direct measure of information scrambling and can be related to the analytic continuation of the partition function. Information scrambling is then described by the quantum work statistics associated with the time-reversal operation on a single copy, associated with the sudden negation of the system Hamiltonian.

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.

Leggett-Garg Inequalities for Quantum Fluctuating Work

The Leggett-Garg inequalities serve to test whether or not quantum correlations in time can be explained within a classical macrorealistic framework. We apply this test to thermodynamics and derive a set of Leggett-Garg inequalities for the statistics of fluctuating work done on a quantum system unitarily driven in time. It is shown that these inequalities can be violated in a driven two-level system, thereby demonstrating that there exists no general macrorealistic description of quantum work. These violations are shown to emerge within the standard Two-Projective-Measurement scheme as well as for alternative definitions of fluctuating work that are based on weak measurement. Our results elucidate the influences of temporal correlations on work extraction in the quantum regime and highlight a key difference between quantum and classical thermodynamics.

Quantum Fluctuation Theorems, Contextuality, and Work Quasiprobabilities

We discuss the role of contextuality within quantum fluctuation theorems, in the light of a recent no-go result by Perarnau-Llobet et al. We show that any fluctuation theorem reproducing the two-point-measurement scheme for classical states either admits a notion of work quasiprobability or fails to describe protocols exhibiting contextuality. Conversely, we describe a protocol that smoothly interpolates between the two-point-measurement work distribution for projective measurements and Allahverdyan's work quasiprobability for weak measurements, and show that the negativity of the latter is a direct signature of contextuality.

Using a quantum work meter to test non-equilibrium fluctuation theorems

Work is an essential concept in classical thermodynamics, and in the quantum regime, where the notion of a trajectory is not available, its definition is not trivial. For driven (but otherwise isolated) quantum systems, work can be defined as a random variable, associated with the change in the internal energy. The probability for the different values of work captures essential information describing the behaviour of the system, both in and out of thermal equilibrium. In fact, the work probability distribution is at the core of “fluctuation theorems” in quantum thermodynamics. Here we present the design and implementation of a quantum work meter operating on an ensemble of cold atoms, which are controlled by an atom chip. Our device not only directly measures work but also directly samples its probability distribution. We demonstrate the operation of this new tool and use it to verify the validity of the quantum Jarzynksi identity.

Defining and measuring work and heat are non-trivial tasks in the quantum regime. Here, the authors design a scheme to directly sample from the work probability distribution, and use it to verify the validity of the quantum version of the Jarzynksi identity using cold atoms on an atomic chip.

Quantum-to-classical transition in the work distribution for chaotic systems

The work distribution is a fundamental quantity in nonequilibrium thermodynamics mainly due to its connection with fluctuation theorems. Here, we develop a semiclassical approximation to the work distribution for a quench process in chaotic systems that provides a link between the quantum and classical work distributions. The approach is based on the dephasing representation of the quantum Loschmidt echo and on the quantum ergodic conjecture, which states that the Wigner function of a typical eigenstate of a classically chaotic Hamiltonian is equidistributed on the energy shell. Using numerical simulations, we show that our semiclassical approximation accurately describes the quantum distribution as the temperature is increased.

Autonomous rotor heat engine

The triumph of heat engines is their ability to convert the disordered energy of thermal sources into useful mechanical motion. In recent years, much effort has been devoted to generalizing thermodynamic notions to the quantum regime, partly motivated by the promise of surpassing classical heat engines. Here, we instead adopt a bottom-up approach: we propose a realistic autonomous heat engine that can serve as a test bed for quantum effects in the context of thermodynamics. Our model draws inspiration from actual piston engines and is built from closed-system Hamiltonians and weak bath coupling terms. We analytically derive the performance of the engine in the classical regime via a set of nonlinear Langevin equations. In the quantum case, we perform numerical simulations of the master equation. Finally, we perform a dynamic and thermodynamic analysis of the engine's behavior for several parameter regimes in both the classical and quantum case and find that the latter exhibits a consistently lower efficiency due to additional noise.

No-Go Theorem for the Characterization of Work Fluctuations in Coherent Quantum Systems

An open question of fundamental importance in thermodynamics is how to describe the fluctuations of work for quantum coherent processes. In the standard approach, based on a projective energy measurement both at the beginning and at the end of the process, the first measurement destroys any initial coherence in the energy basis. Here we seek extensions of this approach which can possibly account for initially coherent states. We consider all measurement schemes to estimate work and require that (i) the difference of average energy corresponds to average work for closed quantum systems and that (ii) the work statistics agree with the standard two-measurement scheme for states with no coherence in the energy basis. We first show that such a scheme cannot exist. Next, we consider the possibility of performing collective measurements on several copies of the state and prove that it is still impossible to simultaneously satisfy requirements (i) and (ii). Nevertheless, improvements do appear, and in particular, we develop a measurement scheme that acts simultaneously on two copies of the state and allows us to describe a whole class of coherent transformations.

Comparison between thermodynamic work and heat in autonomous quantum systems

One of the most important problems in quantum thermodynamics is how to distinguish work and heat in autonomous quantum systems. In this paper, work and heat are defined through the following criterion, i.e., work is the energy that cannot change the entropy of the energy resource, and satisfies the Jarzynski equality, while heat does not. Two kinds of definitions satisfying the two corresponding requirements are proposed and demonstrated, and the consistency condition of the two kinds is given. Through the first definition, the problem of entropy production is investigated. A model study is also presented to verify the proposal.

Quantum work and the thermodynamic cost of quantum measurements

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 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 thermodynamic quantity of only the system. As main results we derive a modified quantum Jarzynski equality and a sharpened maximum work theorem in terms of the information free energy. A comparison of our results with the standard approach allows one to quantify the informational cost of projective measurements.

Optimal processes for probabilistic work extraction beyond the second law

According to the second law of thermodynamics, for every transformation performed on a system which is in contact with an environment of fixed temperature, the average extracted work is bounded by the decrease of the free energy of the system. However, in a single realization of a generic process, the extracted work is subject to statistical fluctuations which may allow for probabilistic violations of the previous bound. We are interested in enhancing this effect, i.e. we look for thermodynamic processes that maximize the probability of extracting work above a given arbitrary threshold. For any process obeying the Jarzynski identity, we determine an upper bound for the work extraction probability that depends also on the minimum amount of work that we are willing to extract in case of failure, or on the average work we wish to extract from the system. Then we show that this bound can be saturated within the thermodynamic formalism of quantum discrete processes composed by sequences of unitary quenches and complete thermalizations. We explicitly determine the optimal protocol which is given by two quasi-static isothermal transformations separated by a finite unitary quench.

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.

Fluctuations of the heat exchanged between two quantum spin chains

The statistics of the heat exchanged between two quantum XX spin chains prepared at different temperatures is studied within the assumption of weak coupling. This provides simple formulas for the average heat and its corresponding characteristic function, from which the probability distribution may be computed numerically. These formulas are valid for arbitrary sizes and therefore allow us to analyze the role of the thermodynamic limit in this nonequilibrium setting. It is found that all thermodynamic quantities are extremely sensitive to the quantum phase transition of the XX chain.

Thermodynamics of Weakly Measured Quantum Systems

We consider continuously monitored quantum systems and introduce definitions of work and heat along individual quantum trajectories that are valid for coherent superposition of energy eigenstates. We use these quantities to extend the first and second laws of stochastic thermodynamics to the quantum domain. We illustrate our results with the case of a weakly measured driven two-level system and show how to distinguish between quantum work and heat contributions. We finally employ quantum feedback control to suppress detector backaction and determine the work statistics.

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.

Full distribution of work done on a quantum system for arbitrary initial states

We propose an approach to define and measure the statistics of work, internal energy and dissipated heat in a driven quantum system. In our framework the presence of a physical detector arises naturally and work and its statistics can be investigated in the most general case. In particular, we show that the quantum coherence of the initial state can lead to measurable effects on the moments of the work done on the system. At the same time, we recover the known results if the initial state is a statistical mixture of energy eigenstates. Our method can also be applied to measure the dissipated heat in an open quantum system. By sequentially coupling the system to a detector, we can track the energy dissipated in the environment while accessing only the system degrees of freedom.

Landauer's Principle in Multipartite Open Quantum System Dynamics

We investigate the link between information and thermodynamics embodied by Landauer's principle in the open dynamics of a multipartite quantum system. Such irreversible dynamics is described in terms of a collisional model with a finite temperature reservoir. We demonstrate that Landauer's principle holds, for such a configuration, in a form that involves the flow of heat dissipated into the environment and the rate of change of the entropy of the system. Quite remarkably, such a principle for heat and entropy power can be explicitly linked to the rate of creation of correlations among the elements of the multipartite system and, in turn, the non-Markovian nature of their reduced evolution. Such features are illustrated in two exemplary cases.

Exact work statistics of quantum quenches in the anisotropic XY model

We derive exact analytic expressions for the average work done and work fluctuations in instantaneous quenches of the ground and thermal states of a one-dimensional anisotropic XY model. The average work and a quantum fluctuation relation is used to determine the amount of irreversible entropy produced during the quench, eventually revealing how the closing of the excitation gap leads to increased dissipated work. The work fluctuation is calculated and shown to exhibit nonanalytic behavior as the prequench anisotropy parameter and transverse field are tuned across quantum critical points. Exact compact formulas for the average work and work fluctuation in ground state quenches of the transverse field Ising model allow us to calculate the first singular field derivative at the critical field values.

Quantum work statistics of charged Dirac particles in time-dependent fields

The quantum Jarzynski equality is an important theorem of modern quantum thermodynamics. We show that the Jarzynski equality readily generalizes to relativistic quantum mechanics described by the Dirac equation. After establishing the conceptual framework we solve a pedagogical, yet experimentally relevant, system analytically. As a main result we obtain the exact quantum work distributions for charged particles traveling through a time-dependent vector potential evolving under Schrödinger as well as under Dirac dynamics, and for which the Jarzynski equality is verified. Special emphasis is put on the conceptual and technical subtleties arising from relativistic quantum mechanics.

General formalism of local thermodynamics with an example: Quantum Otto engine with a spin-1/2 coupled to an arbitrary spin

We investigate a quantum heat engine with a working substance of two particles, one with a spin-1/2 and the other with an arbitrary spin (spin s), coupled by Heisenberg exchange interaction, and subject to an external magnetic field. The engine operates in a quantum Otto cycle. Work harvested in the cycle and its efficiency are calculated using quantum thermodynamical definitions. It is found that the engine has higher efficiencies at higher spins and can harvest work at higher exchange interaction strengths. The role of exchange coupling and spin s on the work output and the thermal efficiency is studied in detail. In addition, the engine operation is analyzed from the perspective of local work and efficiency. We develop a general formalism to explore local thermodynamics applicable to any coupled bipartite system. Our general framework allows for examination of local thermodynamics even when global parameters of the system are varied in thermodynamic cycles. The generalized definitions of local and cooperative work are introduced by using mean field Hamiltonians. The general conditions for which the global work is not equal to the sum of the local works are given in terms of the covariance of the subsystems. Our coupled spin quantum Otto engine is used as an example of the general formalism.

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.

Statistics of work distribution in periodically driven closed quantum systems

We study the statistics of the work distribution P(w) in a d-dimensional closed quantum system with linear dimension L subjected to a periodic drive with frequency ω(0). We show that the corresponding rate function I(w)=-ln[P(w)/Λ(0)]/L^{d} after a drive period satisfies a universal lower bound I(0)≥n(d) and has a zero at w=QL(d)/N, where n(d) and Q are the excitation and the residual energy densities generated during the drive, Λ(0) is a constant fixed by the normalization of P(w), and N is the total number of constituent particles or spins in the system. We supplement our results by calculating I(w) for a class of d-dimensional integrable models and show that I(w) has an oscillatory dependence on ω(0) originating from Stuckelberg interference generated due to double passage through the critical point or region during the drive. We suggest experiments to test our theory.