Quantum-heat fluctuation relations in $3$-level systems under projective measurements
Guido Giachetti, Stefano Gherardini, Andrea Trombettoni, Stefano Ruffo
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12 Pages, 6 Figures. Contribution to the Special Issue "Many Body Quantum Chaos" in honour of Shmuel Fishman, published in the journal "Condensed Matter"
We study the statistics of energy fluctuations in a three-level quantum system subject to a sequence of projective quantum measurements. We check that, as expected, the quantum Jarzynski equality holds provided that the initial state is thermal. The latter condition is trivially satisfied for two-level systems, while this is generally no longer true for $N$-level systems, with $N > 2$. Focusing on three-level systems, we discuss the occurrence of a unique energy scale factor $β_{\rm eff}$ that formally plays the role of an effective inverse temperature in the Jarzynski equality. To this aim, we introduce a suitable parametrization of the initial state in terms of a thermal and a non-thermal component. We determine the value of $β_{\rm eff}$ for a large number of measurements and study its dependence on the initial state. Our predictions could be checked experimentally in quantum optics.
In this paper we aim at characterizing the effect of stochastic fluctuations on the distribution of the energy exchanged by a quantum system with an external environment under sequences of quantum measurements performed at random times. Both quenched and annealed averages are considered. The information about fluctuations is encoded in the quantum-heat probability density function, or equivalently in its characteristic function, whose general expression for a quantum system with arbitrary Hamiltonian is derived. We prove that, when a stochastic protocol of measurements is applied, the quantum …
The probability densities of work that can be exerted on a quantum system initially staying in thermal equilibrium are constrained by the fluctuation relations of Jarzynski and Crooks, when the work is determined by two projective energy measurements. We investigate the question whether these fluctuation relations may still hold if one employs generalized energy measurements rather than projective ones. Restricting ourselves to a class of universal measurements which are independent of several details of the system on which the work is done, we find sets of necessary and sufficient conditions …
We study the statistics of heat exchange of a quantum system that collides sequentially with an arbitrary number of ancillas. This can describe, for instance, an accelerated particle going through a bubble chamber. Unlike other approaches in the literature, our focus is on the \emph{joint} probability distribution that heat $Q_1$ is exchanged with ancilla 1, heat $Q_2$ is exchanged with ancilla 2, and so on. This allows one to address questions concerning the correlations between the collisional events. The joint distribution is found to satisfy a Fluctuation theorem of the Jarzynski-Wójcik ty…
The thermodynamic properties of quantum heat engines are stochastic owing to the presence of thermal and quantum fluctuations. We here experimentally investigate the efficiency and nonequilibrium entropy production statistics of a spin-1/2 quantum Otto cycle. We first study the correlations between work and heat within a cycle by extracting their joint distribution for different driving times. We show that near perfect anticorrelation, corresponding to the tight-coupling condition, can be achieved. In this limit, the reconstructed efficiency distribution is peaked at the macroscopic efficiency…
In this work, we investigate the heat exchange between two quantum systems whose initial equilibrium states are described by the generalized Gibbs ensemble. First, we generalize the fluctuation relations for heat exchange discovered by Jarzynski and Wójcik to quantum systems prepared in the equilibrium states described by the generalized Gibbs ensemble at different generalized temperatures. Second, we extend the connections between heat exchange and Rényi divergences to quantum systems with very general initial conditions.These relations are applicable for quantum systems with conserved quanti…
Fluctuations of thermodynamic quantities become non-negligible and play an important role when the system size is small. We develop finite-time thermodynamics of fluctuations in microscopic heat engines whose environmental temperature and mechanical parameter are driven periodically in time. Within the slow-driving regime, this formalism universally holds in a coarse-grained time scale whose resolution is much longer than the correlation time of the fluctuations, and is shown to be consistent with the relation analogous to the fluctuation-dissipation relation. Employing a geometric argument, a…
In a generalized framework for the Landauer erasure protocol, we study bounds on the heat dissipated in typical nonequilibrium quantum processes. In contrast to thermodynamic processes, quantum fluctuations are not suppressed in the nonequilibrium regime and cannot be ignored, making such processes difficult to understand and treat. Here we derive an emergent fluctuation relation that virtually guarantees the average heat produced to be dissipated into the reservoir either when the system or reservoir is large (or both) or when the temperature is high. The implication of our result is that for…
Reducing work fluctuation and dissipation in heat engines or, more generally, information heat engines that perform feedback control is vital to maximize their efficiency. The same problem arises when we attempt to maximize the efficiency of a given thermodynamic task that undergoes nonequilibrium processes for arbitrary initial and final states. We find that the most general trade-off relation between work fluctuation and dissipation applicable to arbitrary nonequilibrium processes is bounded from below by the information distance characterizing how far the system is from thermal equilibrium.…
The Casimir force between arbitrary objects in equilibrium is related to scattering from individual bodies. We extend this approach to heat transfer and Casimir forces in non-equilibrium cases where each body, and the environment, is at a different temperature. The formalism tracks the radiation from each body and its scatterings by the other objects. We discuss the radiation from a cylinder, emphasizing its polarized nature, and obtain the heat transfer between a sphere and a plate, demonstrating the validity of proximity transfer approximation at close separations and arbitrary temperatures.
We study the heat current flowing between two baths consisting of harmonic oscillators interacting with a qubit through a spin-boson coupling. An explicit expression for the generating function of the total heat flowing between the hot and cold baths is derived by evaluating the corresponding Feynman-Vernon path integral under the non-interacting blip approximation (NIBA). This generating function satisfies the Gallavotti-Cohen fluctuation theorem, both before and after performing the NIBA. We also verify that the heat conductivity is proportional to the variance of the heat current, retrievin…
We propose a machine learning approach based on artificial neural network to gain faster insights on the role of geometric contributions to the nonequilibrium fluctuations of an adiabatically temperature-driven quantum heat engine coupled to a cavity. Using the artificial neural network we have explored the interplay between bunched and antibunched photon exchange statistics for different engine parameters. We report that beyond a pivotal cavity temperature, the Fano factor oscillates between giant and low values as a function of phase difference between the driving protocols. We further obser…
The celebrated exchange fluctuation theorem -- proposed by Jarzynski and Wózcik, (Phys Rev. Lett. 92, 230602 (2004)) for heat exchange between two systems in thermal equilibrium at different temperatures -- is explored here for quantum Gaussian states in thermal equilibrium. We employ Wigner distribution function formalism for quantum states, which exhibits close resemblance with the classcial phase-space trajectory description, to arrive at this theorem. For two Gaussian states in thermal equilibrium at two different temperatures kept in contact with each other for a fixed duration of time we…
We study the stochastic energetic exchanges in quantum heat engines. Due to microreversibility, these obey a fluctuation relation, called the heat engine fluctuation relation, which implies the Carnot bound: no machine can have an efficiency larger than Carnot's efficiency. The stochastic thermodynamics of a quantum heat engine (including the joint statistics of heat and work and the statistics of efficiency) is illustrated by means of an optimal two-qubit heat engine, where each qubit is coupled to a thermal bath and a two-qubit gate determines energy exchanges between the two qubits. We disc…
Thermodynamic behaviors in a quantum Brownian motion coupled to a classical heat bath is studied. We then define a heat operator by generalizing the stochastic energetics and show the energy balance (first law) and the upper bound of the expectation value of the heat operator (second law). We further find that this upper bound depends on the memory effect induced by quantum fluctuations and hence the maximum extractable work can be qualitatively modified in quantum thermodynamics.
We analyze an autonomous thermoelectric engine composed of two superconducting qubits coupled to separate heat baths and connected by a Josephson junction. Work and heat are process quantities and not observables of the engine quantum system, but their rates can be derived from the steady state expectation value of appropriate system observables and their fluctuations are given by correlation functions determined with the GKLS master equation and quantum regression theorem. The correlation functions also reveal a cyclical, dynamical transfer of energy - the strokes of the engine - hiding under…
Determining the work which is supplied to a system by an external agent, provides a crucial step in any experimental realization of transient fluctuation relations. This, however, poses a problem for quantum systems, where the standard procedure requires the projective measurement of energy at the beginning and the end of the protocol. Unfortunately, projective measurements, which are preferable from the point of view of theory, seem to be difficult to implement experimentally. We demonstrate that, when using a particular type of generalized energy measurements, the resulting work statistics i…
A fluctuation relation for the heat exchange of an open quantum system under a thermalizing Markovian dynamics is derived. We show that the probability of that the system absorbs an amount of heat from its bath, at a given time interval, divided by the probability of the reverse process (releasing the same amount of heat to the bath) is given by an exponential factor which depends on the amount of heat and the difference between the temperatures of the system and the bath. We also argue that the probability of the violation of the second law of thermodynamics (here in the form of net heat tran…
The standard approach to deriving fluctuation theorems fails to capture the effect of quantum correlation and coherence in the initial state of the system. Here we overcome this difficulty and derive heat exchange fluctuation theorem in the full quantum regime by showing that the energy exchange between two locally thermal states in the presence of initial quantum correlations is faithfully captured by a quasiprobability distribution. Its negativities, being associated with proofs of contextuality, are proxys of non-classicality. We discuss the thermodynamic interpretation of negative probabil…
We study heat current and the full statistics of heat fluctuations in a capacitively-coupled double quantum dot system. This work is motivated by recent theoretical studies and experimental works on heat currents in quantum dot circuits. As expected intuitively, within the (static) mean-field approximation, the system at steady-state decouples into two single-dot equilibrium systems with renormalized dot energies, leading to zero average heat flux and fluctuations. This reveals that dynamic correlations induced between electrons on the dots is solely responsible for the heat transport between …
This work concerns the statistics of the Two-Time Measurement definition of heat variation in each reservoir of a thermodynamic quantum system. We study the cumulant generating function of the heat flows in the thermodynamic and large-time limits. It is well-known that, if the system is time-reversal invariant, this cumulant generating function satisfies the celebrated Evans--Searles symmetry. We show in addition that, under appropriate ultraviolet regularity assumptions on the local interaction between the reservoirs, it satisfies a translation-invariance property, as proposed in [Andrieux et…
Fluctuations of thermodynamic observables, such as heat and work, contain relevant information on the underlying physical process. These fluctuations are however not taken into account in the traditional laws of thermodynamics. While the second law is extended to fluctuating systems by the celebrated fluctuation theorems, the first law is generally believed to hold even in the presence of fluctuations. Here we show that in the presence of quantum fluctuations, also the first law of thermodynamics may break down. This happens because quantum mechanics imposes constraints on the knowledge of hea…
We study heat fluctuations in the two-time measurement framework. For bounded perturbations, we give sufficient ultraviolet regularity conditions on the perturbation for the moments of the heat variation to be uniformly bounded in time, and for the Fourier transform of the heat variation distribution to be analytic and uniformly bounded in time in a complex neighborhood of 0. On a set of canonical examples, with bounded and unbounded perturbations, we show that our ultraviolet conditions are essentially necessary. If the form factor of the perturbation does not meet our assumptions, the heat v…
We discuss the non-zero frequency noise of heat current with the explicit example of energy carried by thermal photons in a circuit. Instead of the standard circuit modelling that gives a convenient way of predicting time-averaged heat current, we describe a setup composed of two resistors forming the heat baths by collections of bosonic oscillators. In terms of average heat transport this model leads to identical results with the conventional one, but besides this, it yields a convenient way of dealing with noise as well. The non-zero frequency heat current noise does not vanish in equilibriu…
Quantum and thermal fluctuations of electromagnetic waves are the cornerstone of quantum and statistical physics, and inherent to such phenomena as thermal radiation and van der Waals forces. While the basic principles are the material of elementary texts, recent experimental and technological advances have made it necessary to come to terms with counterintuitive consequences of electromagnetic fluctuations at short scales -- in the so called {\it near-field} regime. We focus on three manifestations of such behavior: {\bf (i)} The Stefan--Boltzmann law describes thermal radiation from macrosco…
We develop an efficient machine learning protocol to predict the noise-induced coherence from the nonequilibrium fluctuations of photon exchange statistics in a quantum heat engine. The engine is a four-level quantum system coupled to a unimodal quantum cavity. The nonequilibrium fluctuations correspond to the work done during the photon exchange process between the four-level system and the cavity mode. We specifically evaluate the mean, variance, skewness, and kurtosis for a range of engine parameters using a full counting statistical approach combined with a quantum master equation techniqu…
