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arXiv:1806.04463 · Physical Review A

Spin-phase-space-entropy production

Jader P. Santos, Lucas C. Céleri, Frederico Brito, Gabriel Teixeira Landi, Mauro Paternostro

Published May 22, 2018• 25 citations
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Abstract

Quantifying the degree of irreversibility of an open system dynamics represents a problem of both fundamental and applied relevance. Even though a well-known framework exists for thermal baths, the results give diverging results in the limit of zero temperature and are also not readily extended to nonequilibrium reservoirs, such as dephasing baths. Aimed at filling this gap, in this paper we introduce a phase-space-entropy production framework for quantifying the irreversibility of spin systems undergoing Lindblad dynamics. The theory is based on the spin Husimi-Q function and its corresponding phase-space entropy, known as Wehrl entropy. Unlike the von Neumann entropy production rate, we show that in our framework, the Wehrl entropy production rate remains valid at any temperature and is also readily extended to arbitrary nonequilibrium baths. As an application, we discuss the irreversibility associated with the interaction of a two-level system with a single-photon pulse, a problem which cannot be treated using the conventional approach.

DOI
10.1103/physreva.97.052123

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Quantifying the degree of irreversibility of an open system dynamics represents a problem of both fundamental and applied relevance. Even though a well-known framework exists for thermal baths, the results give diverging results in the limit of zero temperature and are also not readily extended to nonequilibrium reservoirs, such as dephasing baths. Aimed at filling this gap, in this paper we introduce a phase-space-entropy production framework for quantifying the irreversibility of spin systems undergoing Lindblad dynamics. The theory is based on the spin Husimi-Q function and its corresponding phase-space entropy, known as Wehrl entropy. Unlike the von Neumann entropy production rate, we show that in our framework, the Wehrl entropy production rate remains valid at any temperature and is also readily extended to arbitrary nonequilibrium baths. As an application, we discuss the irreversibility associated with the interaction of a two-level system with a single-photon pulse, a problem which cannot be treated using the conventional approach.
The second law of thermodynamics can be expressed in terms of entropy production, which can be used to quantify the degree of irreversibility of a process. In this Chapter, we consider the standard scenario of open quantum systems, where a system irreversibly interacts with an external environment. We show that the standard approach, based on von Neumann entropy, can be replaced by a phase-space formulation of the problem. In particular, we focus on spin systems that can be described using the so-called spin coherent states. We deploy this formalism to study the interplay between the entropy production rate and the initial quantum coherence available to the system.
The characterization of irreversibility in general quantum processes is an open problem of increasing technological relevance. Yet, the tools currently available to this aim are mostly limited to the assessment of dynamics induced by equilibrium environments, a situation that often does not match the reality of experiments at the microscopic and mesoscopic scale. We propose a theory of irreversible entropy production that is suited for quantum systems exposed to general, nonequilibrium reservoirs. We illustrate our framework by addressing a set of physically relevant situations that clarify both the features and the potential of our proposal.
The physics of systems out of equilibrium is a topic of great interest, mainly due to the possibility of exploring phenomena that can not be observed in equilibrium systems. Driven-dissipative phase transitions open the opportunity of studying phases with no classical counterparts, and these can be experimentally realized in quantum optical platforms. Since these transitions occur in systems kept out of equilibrium, they are characterized by a finite entropy production rate. However, due to technical difficulties regarding the zero temperature limit and the non-gaussianity of such models, very little is known about how entropy production behaves around criticality. Using a quantum phase-space method, based on the Husimi Q-function, we put forth a framework that allows for the complete characterization of the entropy production in driven-dissipative transitions. This new theoretical framework is tailored specifically to describe photon loss dissipation, which is effectively a zero temperature process for which the standard theory of entropy production breaks down. It makes no assumptions about Gaussianity about the model or the state. It works for both, steady-states as well as the dynamics and as an application, we study both situations in the paradigmatic driven-dissipative Kerr model, which presents a discontinuous phase transition. For general driven-dissipative critical systems, where one can define a thermodynamic limit, we find that the entropy production rate and flux naturally split into two contributions: an extensive one and a contribution due to quantum fluctuations only. Moreover, we identify a contribution to the entropy production due to unitary dynamics, and we find that the behavior of this contribution at the non-equilibrium steady-state (NESS) matches the behavior of entropy production rate observed in classical systems. The quantum contributions are found to diverge at the critical point.
Entropy production plays a fundamental role in nonequilibrium thermodynamics to quantify the irreversibility of open systems. Its positivity can be ensured for a wide class of setups, but the entropy production rate can become negative sometimes. This is often taken as an indicator of non-Markovianity. We make this link precise by showing under which conditions a negative entropy production rate implies non-Markovianity and when it does not. For a system coupled to a single heat bath, this can be established within a unified language for two setups: (i) the dynamics resulting from a coarse-grained description of a Markovian master equation and (ii) the classical Hamiltonian dynamics of a system coupled to a bath. The quantum version of the latter result is shown not to hold despite the fact that the integrated thermodynamic description is formally equivalent to the classical case. The instantaneous fixed point of a non-Markovian dynamics plays an important role in our study. Our key contribution is to provide a consistent theoretical framework to study the finite-time thermodynamics of a large class of dynamics with a precise link to its non-Markovianity.
We discuss thermalization of isolated quantum systems by using the Husimi–Wehrl entropy evaluated in the semiclassical treatment. The Husimi–Wehrl entropy is the Wehrl entropy obtained by using the Husimi function for the phase space distribution. The time evolution of the Husimi function is given by smearing the Wigner function, whose time evolution is obtained in the semiclassical approximation. We show the efficiency and usefulness of this semiclassical treatment in describing the entropy production of a couple of quantum-mechanical systems, whose classical counter-systems are known to be chaotic. We propose two methods to evaluate the time evolution of the Husimi–Wehrl entropy, the test-particle method and the two-step Monte Carlo method. We demonstrate the characteristics of the two methods by numerical calculations, and show that simultaneous application of the two methods ensures the reliability of the results of the Husimi–Wehrl entropy at a given time.
We discuss thermalization of isolated quantum systems by using the Husimi–Wehrl entropy evaluated in the semiclassical treatment. The Husimi–Wehrl entropy is the Wehrl entropy obtained by using the Husimi function for the phase space distribution. The time evolution of the Husimi function is given by smearing the Wigner function, whose time evolution is obtained in the semiclassical approximation. We show the efficiency and usefulness of this semiclassical treatment in describing the entropy production of a couple of quantum-mechanical systems, whose classical counter-systems are known to be chaotic. We propose two methods to evaluate the time evolution of the Husimi–Wehrl entropy, the test-particle method and the two-step Monte Carlo method. We demonstrate the characteristics of the two methods by numerical calculations, and show that simultaneous application of the two methods ensures the reliability of the results of the Husimi–Wehrl entropy at a given time.