Quantum Fluctuation Theorems beyond Two-Point Measurements

Probing coherent quantum thermodynamics using a trapped ion

Quantum thermodynamics is aimed at grasping thermodynamic laws as they apply to thermal machines operating in the deep quantum regime, where coherence and entanglement are expected to matter. Despite substantial progress, however, it has remained difficult to develop thermal machines in which such quantum effects are observed to be of pivotal importance. In this work, we demonstrate the possibility to experimentally measure and benchmark a genuine quantum correction, induced by quantum friction, to the classical work fluctuation-dissipation relation. This is achieved by combining laser-induced coherent Hamiltonian rotations and energy measurements on a trapped ion. Our results demonstrate that recent developments in stochastic quantum thermodynamics can be used to benchmark and unambiguously distinguish genuine quantum coherent signatures generated along driving protocols, even in presence of experimental SPAM errors and, most importantly, beyond the regimes for which theoretical predictions are available (e.g., in slow driving).

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.

Integral fluctuation theorems and trace-preserving map

The detailed fluctuation theorem implies symmetry in the generating function of entropy production probability. The integral fluctuation theorem directly follows from this symmetry and the normalization of the probability. In this paper, we rewrite the generating function by integrating measurements and evolution into a constructed mapping. This mapping is completely positive, and the original integral fluctuation theorem is determined by the trace-preserving property of these constructed maps. We illustrate the convenience of this method by discussing the eigenstate fluctuation theorem and heat exchange between two baths. This set of methods is also applicable to the generating functions of quasiprobability, where we observe the Petz recovery map arising naturally from this approach.

Atom-doped photon engine: Extracting mechanical work from a quantum system via radiation pressure

The possibility of efficiently converting heat into work at the microscale has triggered an intense research effort to understand quantum heat engines, driven by the hope of quantum superiority over classical counterparts. In this work, we introduce a model featuring an atom-doped optical quantum cavity propelling a classical piston through radiation pressure. The model, based on the Jaynes-Cummings Hamiltonian of quantum electrodynamics, demonstrates the generation of mechanical work through thermal energy injection. We establish the equivalence of the piston expansion work with Alicki's work definition, analytically for quasistatic transformations and numerically for finite-time protocols. We further employ the model to construct quantum Otto and Carnot engines, comparing their performance in terms of energetics, work output, efficiency, and power under various conditions. This model thus provides a platform to extract useful work from an open quantum system to generate net motion, and it sheds light on the quantum concepts of work and heat.

Improving the Cramér-Rao bound with the detailed fluctuation theorem

In some nonequilibrium systems, the distribution of entropy production p(Σ) satisfies the detailed fluctuation theorem (DFT) p(Σ)/p(-Σ)=exp(Σ). When the distribution p(Σ) shows a time dependence, the celebrated Cramér-Rao (CR) bound asserts that the mean entropy production rate is upper bounded in terms of the variance of Σ and the Fisher information with respect to time. In this paper we employ the DFT to derive an upper bound for the mean entropy production rate that improves the CR bound. We show that this new bound serves as an accurate approximation for the entropy production rate in the heat exchange problem mediated by a weakly coupled bosonic mode. The bound is saturated for the same setup when mediated by a weakly coupled qubit.

Thermodynamic variational relation

In systems far from equilibrium, the statistics of observables are connected to entropy production, leading to the thermodynamic uncertainty relation (TUR). However, the derivation of TURs often involves constraining the parity of observables, such as considering asymmetric currents, making it unsuitable for the general case. We propose a thermodynamic variational relation (TVR) between the statistics of general observables and entropy production, based on the variational representation of f divergences. From this result, we derive a universal TUR and other relations for higher-order statistics of observables.

Heat distribution in quantum Brownian motion

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

Stochastic Entropy Production: Fluctuation Relation and Irreversibility Mitigation in Non-unital Quantum Dynamics

In this work, we study the stochastic entropy production in open quantum systems whose time evolution is described by a class of non-unital quantum maps. In particular, as in Phys Rev E 92:032129 (2015), we consider Kraus operators that can be related to a nonequilibrium potential. This class accounts for both thermalization and equilibration to a non-thermal state. Unlike unital quantum maps, non-unitality is responsible for an unbalance of the forward and backward dynamics of the open quantum system under scrutiny. Here, concentrating on observables that commute with the invariant state of the evolution, we show how the non-equilibrium potential enters the statistics of the stochastic entropy production. In particular, we prove a fluctuation relation for the latter and we find a convenient way of expressing its average solely in terms of relative entropies. Then, the theoretical results are applied to the thermalization of a qubit with non-Markovian transient, and the phenomenon of irreversibility mitigation, introduced in Phys Rev Res 2:033250 (2020), is analyzed in this context.

Bound for the moment generating function from the detailed fluctuation theorem

A famous consequence of the detailed fluctuation theorem (FT), p(Σ)/p(-Σ)=exp(Σ), is the integral FT 〈exp(-Σ)〉=1 for a random variable Σ and a distribution p(Σ). When Σ represents the entropy production in thermodynamics, the main outcome of the integral FT is the second law, 〈Σ〉≥0. However, a full description of the fluctuations of Σ might require knowledge of the moment generating function (MGF), G(α):=〈exp(αΣ)〉. In the context of the detailed FT, we show the MGF is lower bounded in the form G(α)≥B(α,〈Σ〉) for a given mean 〈Σ〉. As applications, we verify that the bound is satisfied for the entropy produced in the heat exchange problem between two reservoirs mediated by a weakly coupled bosonic mode and a qubit swap engine.

Stochastic Thermodynamics of a Finite Quantum System Coupled to Two Heat Baths

We consider a situation where an N-level system (NLS) is coupled successively to two heat baths with different temperatures without being necessarily thermalized and approaches a steady state. For this situation we apply a general Jarzynski-type equation and conclude that heat and entropy is flowing from the hot bath to the cold one. The Clausius relation between increase of entropy and transfer of heat divided by a suitable temperature assumes the form of two inequalities. Our approach is illustrated by an analytical example. For the linear regime, i.e., for small temperature differences between the two heat baths, we derive an expression for the heat conduction coefficient.

Demonstrating Quantum Microscopic Reversibility Using Coherent States of Light

The principle of microscopic reversibility lies at the core of fluctuation theorems, which have extended our understanding of the second law of thermodynamics to the statistical level. In the quantum regime, however, this elementary principle should be amended as the system energy cannot be sharply determined at a given quantum phase space point. In this Letter, we propose and experimentally test a quantum generalization of the microscopic reversibility when a quantum system interacts with a heat bath through energy-preserving unitary dynamics. Quantum effects can be identified by noting that the backward process is less likely to happen in the existence of quantum coherence between the system's energy eigenstates. The experimental demonstration has been realized by mixing coherent and thermal states in a beam splitter, followed by heterodyne detection in an optical setup. We verify that the quantum modification for the principle of microscopic reversibility is critical in the low-temperature limit, while the quantum-to-classical transition is observed as the temperature of the thermal field gets higher.

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.

Verification of Information Thermodynamics in a Trapped Ion System

Information thermodynamics has developed rapidly over past years, and the trapped ions, as a controllable quantum system, have demonstrated feasibility to experimentally verify the theoretical predictions in the information thermodynamics. Here, we address some representative theories of information thermodynamics, such as the quantum Landauer principle, information equality based on the two-point measurement, information-theoretical bound of irreversibility, and speed limit restrained by the entropy production of system, and review their experimental demonstration in the trapped ion system. In these schemes, the typical physical processes, such as the entropy flow, energy transfer, and information flow, build the connection between thermodynamic processes and information variation. We then elucidate the concrete quantum control strategies to simulate these processes by using quantum operators and the decay paths in the trapped-ion system. Based on them, some significantly dynamical processes in the trapped ion system to realize the newly proposed information-thermodynamic models is reviewed. Although only some latest experimental results of information thermodynamics with a single trapped-ion quantum system are reviewed here, we expect to find more exploration in the future with more ions involved in the experimental systems.

Spin Quantum Heat Engine Quantified by Quantum Steering

Following the rising interest in quantum information science, the extension of a heat engine to the quantum regime by exploring microscopic quantum systems has seen a boon of interest in the last decade. Although quantum coherence in the quantum system of the working medium has been investigated to play a nontrivial role, a complete understanding of the intrinsic quantum advantage of quantum heat engines remains elusive. We experimentally demonstrate that the quantum correlation between the working medium and the thermal bath is critical for the quantum advantage of a quantum Szilárd engine, where quantum coherence in the working medium is naturally excluded. By quantifying the nonclassical correlation through quantum steering, we reveal that the heat engine is quantum when the demon can truly steer the working medium. The average work obtained by taking different ways of work extraction on the working medium can be used to verify the real quantum Szilárd engine.

Experimental Verification of Dissipation-Time Uncertainty Relation

Dissipation is vital to any cyclic process in realistic systems. Recent research focus on nonequilibrium processes in stochastic systems has revealed a fundamental trade-off, called dissipation-time uncertainty relation, that entropy production rate associated with dissipation bounds the evolution pace of physical processes [Phys. Rev. Lett. 125, 120604 (2020)PRLTAO0031-900710.1103/PhysRevLett.125.120604]. Following the dissipative two-level model exemplified in the same Letter, we experimentally verify this fundamental trade-off in a single trapped ultracold ^{40}Ca^{+} ion using elaborately designed dissipative channels, along with a postprocessing method developed in the data analysis, to build the effective nonequilibrium stochastic evolutions for the energy transfer between two heat baths mediated by a qubit. Since the dissipation-time uncertainty relation imposes a constraint on the quantum speed regarding entropy flux, our observation provides the first experimental evidence confirming such a speed restriction from thermodynamics on quantum operations due to dissipation, which helps us further understand the role of thermodynamical characteristics played in quantum information processing.

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

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

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.

Thermalization processes induced by quantum monitoring in multilevel systems

We study the heat statistics of a multilevel N-dimensional quantum system monitored by a sequence of projective measurements. The late-time, asymptotic properties of the heat characteristic function are analyzed in the thermodynamic limit of a high, ideally infinite, number M of measurements (M→∞). In this context, the conditions allowing for an infinite-temperature thermalization (ITT), induced by the repeated monitoring of the quantum system, are discussed. We show that ITT is identified by the fixed point of a symmetric random matrix that models the stochastic process originated by the sequence of measurements. Such fixed point is independent on the nonequilibrium evolution of the system and its initial state. Exceptions to ITT, which we refer to as partial thermalization, take place when the observable of the intermediate measurements is commuting (or quasicommuting) with the Hamiltonian of the quantum system or when the time interval between measurements is smaller or comparable with the system energy scale (quantum Zeno regime). Results on the limit of infinite-dimensional Hilbert spaces (N→∞), describing continuous systems with a discrete spectrum, are also presented. We show that the order of the limits M→∞ and N→∞ matters: When N is fixed and M diverges, then ITT occurs. In the opposite case, the system becomes classical, so that the measurements are no longer effective in changing the state of the system. A nontrivial result is obtained fixing M/N^{2} where instead partial ITT occurs. Finally, an example of partial thermalization applicable to rotating two-dimensional gases is presented.

Initial-state dependence of thermodynamic dissipation for any quantum process

Exact results about the nonequilibrium thermodynamics of open quantum systems at arbitrary timescales are obtained by considering all possible variations of initial conditions of a system. First we obtain a quantum-information theoretic equality for entropy production, valid for an arbitrary initial joint state of system and environment. For any finite-time process with a fixed initial environment, we then show that the system's loss of distinction-relative to the minimally dissipative state-exactly quantifies its thermodynamic dissipation. The quantum component of this dissipation is the change in coherence relative to the minimally dissipative state. Implications for quantum state preparation and local control are explored. For nonunitary processes-like the preparation of any particular quantum state-we find that mismatched expectations lead to divergent dissipation as the actual initial state becomes orthogonal to the anticipated one.

Quantum Work Statistics with Initial Coherence

The two-point measurement scheme for computing the thermodynamic work performed on a system requires it to be initially in equilibrium. The Margenau-Hill scheme, among others, extends the previous approach to allow for a non-equilibrium initial state. We establish a quantitative comparison between both schemes in terms of the amount of coherence present in the initial state of the system, as quantified by the l1-coherence measure. We show that the difference between the two first moments of work, the variances of work, and the average entropy production obtained in both schemes can be cast in terms of such initial coherence. Moreover, we prove that the average entropy production can take negative values in the Margenau-Hill framework.

Quantum Jarzynski Equality in Open Quantum Systems from the One-Time Measurement Scheme

In open quantum systems, a clear distinction between work and heat is often challenging, and extending the quantum Jarzynski equality to systems evolving under general quantum channels beyond unitality remains an open problem in quantum thermodynamics. In this Letter, we introduce well-defined notions of guessed quantum heat and guessed quantum work, by exploiting the one-time measurement scheme, which only requires an initial energy measurement on the system alone. We derive a modified quantum Jarzynski equality and the principle of maximum work with respect to the guessed quantum work, which requires the knowledge of the system only. We further show the significance of guessed quantum heat and work by linking them to the problem of quantum hypothesis testing.

Landauer's Principle at Zero Temperature

Landauer's bound relates changes in the entropy of a system with the inevitable dissipation of heat to the environment. The bound, however, becomes trivial in the limit of zero temperature. Here we show that it is possible to derive a tighter bound which remains nontrivial even as T→0. As in the original case, the only assumption we make is that the environment is in a thermal state. Nothing is said about the state of the system or the kind of system-environment interaction. Our bound is valid for all temperatures and is always tighter than the original one, tending to it in the limit of high temperatures.