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arXiv:2303.05935 · Physical Review A

Quantum stochastic thermodynamics: A semiclassical theory in phase space

Zhaoyu Fei

Published Dec 7, 2023• 2 citations
arXivHTMLPDF

Abstract

A formalism for quantum many-body systems is proposed through a semiclassical treatment in phase space, allowing us to establish stochastic thermodynamics incorporating quantum statistics. Specifically, we utilize a stochastic Fokker-Planck equation as the dynamics at the mesoscopic level. Here, the noise term characterizing the fluctuation of the flux density accounts for the finite-$N$ effects arising from random collisions between the system and the reservoir. Accordingly, the stationary solution is a quasiequilibrium state in a canonical system. We define stochastic thermodynamic quantities based on the trajectories of the phase-space distribution. The conservation law of energy, the $H$ theorem, and fluctuation theorems are therefore obtained. Our work sets an alternative formalism of quantum stochastic thermodynamics that is independent of the two-point measurement scheme. The numerous projective measurements of quantum systems are replaced by the sampling of the phase-space distribution, offering hope for experimental verifications in the future.

DOI
10.1103/physreva.108.062207

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A formalism for quantum many-body systems is proposed through a semiclassical treatment in phase space, allowing us to establish stochastic thermodynamics incorporating quantum statistics. Specifically, we utilize a stochastic Fokker-Planck equation as the dynamics at the mesoscopic level. Here, the noise term characterizing the fluctuation of the flux density accounts for the finite-NNN effects arising from random collisions between the system and the reservoir. Accordingly, the stationary solution is a quasiequilibrium state in a canonical system. We define stochastic thermodynamic quantities based on the trajectories of the phase-space distribution. The conservation law of energy, the HHH theorem, and fluctuation theorems are therefore obtained. Our work sets an alternative formalism of quantum stochastic thermodynamics that is independent of the two-point measurement scheme. The numerous projective measurements of quantum systems are replaced by the sampling of the phase-space distribution, offering hope for experimental verifications in the future.
This article sets up a formalism to describe stochastic thermodynamics for driven out-of-equilibrium open quantum systems. A stochastic Schrödinger equation allows to construct quantum trajectories describing the dynamics of the system state vector in presence of an eventually monitored environment. Thermodynamic quantities are defined at the single quantum trajectory level, independently of any energy measurement, at any time of the protocol. We thereby identify coherent contributions, without classical counterparts, leading to quantum fluctuations of thermodynamic quantities. This formalism eventually leads to central fluctuation theorems for entropy, extending in the quantum regime results obtained in classical stochastic thermodynamics. The thermal imprint of coherences on a simple implementation of Jarzynski equality is investigated, opening avenues for a thermodynamic approach to decoherence.
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We build a framework for the thermodynamics of macroscopic quantum systems. In contrast with approaches requiring access to the full density matrix, our framework relies on a coarse-grained description, based on measurement statistics of a few observables. When these observables commute, the outcomes define classical macrostates whose entropy is quantified by observational entropy, accounting for uncertainty about both the macrostate and the microstate within it. We extend this notion to non-commuting observables forming a subalgebra of the operator space, and use Jaynes' principle to define an algebra-dependent entropy interpolating between von Neumann and observational entropies. Given initial and final measurement sets, connected by internal and/or environment-induced dynamics, we derive a second law for the coarse-grained dynamics. Unlike formulations based on von Neumann entropy, our inequality captures irreversibility from both non-unitary environment-induced dynamics and internal equilibration. It takes the usual form of a positive entropy production when the system is initially at internal equilibrium, while correction terms capture nonequilibrium resources ignored by the coarse-graining. We also derive fluctuation theorems for coarse-grained thermodynamic quantities. Along a quasi-static path of measurement schemes, we identify quantum macroscopic notions of work and heat fulfilling the first and second laws, including an additional work contribution from manipulating the algebra to which the system is confined, through external constraints or quantum measurement backaction. Finally, we apply our framework to examples illustrating the impact of varying the coarse-graining scheme. Our approach unifies macroscopic and stochastic thermodynamics in a genuinely quantum framework, laying the basis for a versatile, experimentally friendly toolbox to analyze complex quantum dynamics.
Abstract The theory of open quantum systems plays a fundamental role in several scientific and technological disciplines, from quantum computing and information science to molecular electronics and quantum thermodynamics. Despite its widespread relevance, a rigorous formulation of quantum dissipation in conjunction with thermal noise remains a topic of active research. In this work, we establish a formal correspondence between classical stochastic thermodynamics, in particular the Fokker–Planck and Klein–Kramers equations, and the quantum master equation. Building on prior studies of multiplicative noise in classical stochastic differential equations, we demonstrate that thermal noise at the quantum level manifests as a multidimensional geometric stochastic process. By applying canonical quantization, we introduce a novel Hermitian dissipation operator that serves as a quantum analogue of classical viscous friction. This operator allows for a well-defined expression of heat exchange between a system and its environment, enabling the formulation of an alternative quantum equipartition theorem. Our framework ensures a precise energy balance that aligns with the first law of thermodynamics and an entropy balance consistent with the second law. The theoretical formalism is applied to two prototypical quantum systems, the harmonic oscillator and a particle in an infinite potential well, for which it provides new insights into nonequilibrium thermodynamics at the quantum scale. Our results advance the understanding of dissipation in quantum systems and establish a foundation for future studies on stochastic thermodynamics in the quantum domain.
Operational quantum stochastic thermodynamics is a recently proposed theory to study the thermodynamics of open systems based on the rigorous notion of a quantum stochastic process or quantum causal model. In there, a stochastic trajectory is defined solely in terms of experimentally accessible measurement results, which serve as the basis to define the corresponding thermodynamic quantities. In contrast to this observer-dependent point of view, a `black box', which evolves unitarily and can simulate a quantum causal model, is constructed here. The quantum thermodynamics of this big isolated system can then be studied using widely accepted arguments from statistical mechanics. It is shown that the resulting definitions of internal energy, heat, work, and entropy have a natural extension to the trajectory level. The canonical choice of them coincides with the proclaimed definitions of operational quantum stochastic thermodynamics, thereby providing strong support in favour of that novel framework. However, a few remaining ambiguities in the definition of stochastic work and heat are also discovered and in light of these findings some other proposals are reconsidered. Finally, it is demonstrated that the first and second law hold for an even wider range of scenarios than previously thought, covering a large class of quantum causal models based solely on a single assumption about the initial system-bath state.
Abstract We introduce a theoretical technique known as the two-point measurement scheme, which allows us to derive a variety of quantum fluctuation theorems with the same form as in classical stochastic thermodynamics. Alternatively, we show how to study fluctuations based on a quantum master equation approach combined with counting field or quantum jump trajectory methods. Although quantum fluctuation theorems have the same form as classical fluctuation theorems, they are more limited in their scope and practical applicability. Nevertheless, we study one important theoretical consequence of them, known as the thermodynamic uncertainty relation. This relation is used to discuss the impossibility of Carnot efficiency at finite power. The latter point also naturally connects to the third law of thermodynamics.