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arXiv:1805.00773 · Physical review. E

Nonequilibrium quantum-heat statistics under stochastic projective measurements

Stefano Gherardini, Lorenzo Buffoni, Matthias M. Müller, Filippo Caruso, Michele Campisi, Andrea Trombettoni, Stefano Ruffo

Published Sep 7, 2018• 27 citations
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Abstract

In this paper we aim at characterizing the effect of stochastic fluctuations on the distribution of the energy exchanged by a quantum system with the 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 Jarzynski equality is obeyed. Therefore, the fluctuation relation is robust against the presence of randomness in the times intervals between measurements. Then, for the paradigmatic case of a two-level system, we analytically characterize the quantum-heat transfer. Particular attention is devoted to the limit of large number of measurements and to the effects caused by the stochastic fluctuations. The relation with the stochastic Zeno regime is also discussed.

DOI
10.1103/physreve.98.032108

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In this paper we aim at characterizing the effect of stochastic fluctuations on the distribution of the energy exchanged by a quantum system with the 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 Jarzynski equality is obeyed. Therefore, the fluctuation relation is robust against the presence of randomness in the times intervals between measurements. Then, for the paradigmatic case of a two-level system, we analytically characterize the quantum-heat transfer. Particular attention is devoted to the limit of large number of measurements and to the effects caused by the stochastic fluctuations. The relation with the stochastic Zeno regime is also discussed.
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 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 eff for a large number of measurements and study its dependence on the initial state. Our predictions could be checked experimentally in quantum optics.
We investigate the statistics of heat exchange between a finite system coupled to reservoir(s). We have obtained analytical results for heat fluctuation theorems in the transient regime considering the Hamiltonian dynamics of the composite system consisting of the system of interest and the heat bath(s). The system of interest is driven by an external protocol. We first derive it in the context of a single heat bath. The result is in exact agreement with known result. We then generalize the treatment to two heat baths. We further extend the study to quantum systems and show that relations similar to the classical case hold in the quantum regime. For our study we invoke von Neumann two-point projective measurement in quantum mechanics in the transient regime. The study of quantum systems follows the same lines of argument as that of the classical system, and as a result the treatment used in the latter complements that used in the former. Our result is a generalization of Jarzynski-Wòjcik heat fluctuation theorem.
We study the thermodynamic notion of quantum projective measurements, using a framework for the fluctuation theorem of nonequilibrium work. The energy change induced by measurements satisfies the Jarzynski equality, leading us to the interpretation that the quantum projective measurements perform nonequilibrium work on the measured system. The work average exhibits intriguing limiting behaviors due to the heat-up effect caused by repeated measurements and the quantum Zeno effect caused by measurements of an infinite frequency. If the measured system relaxes back to its initial equilibrium state, the work is completely dissipated in the form of heat into a reservoir. The corresponding entropy increase in the reservoir is shown to be not less than the von Neumann entropy change generated during the course of the measurements, proving Landauer's principle.
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.
The manifestation of measurements, randomly distributed in time, on the evolution of quantum systems are analyzed in detail. The set of randomly distributed measurements (RDM) is modeled within the renewal theory, in which the distribution is characterized by the probability density function (PDF) W(t) of times t between successive events (measurements). The evolution of the quantum system affected by the RDM is shown to be described by the density matrix satisfying the stochastic Liouville equation. This equation is applied to the analysis of the RDM effect on the evolution of a two level systems for different types of RDM statistics, corresponding to different PDFs W(t). Obtained general results are illustrated as applied to the cases of the Poissonian [W(t) ~ e^{-w_r t}] and anomalous [W(t) ~ 1/t^{1+α}, (αis smaller or equal to 1)], RDM statistics. In particular, specific features of the quantum and inverse Zeno effects, resulting from the RDM, are thoroughly discussed.