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arXiv:2007.14445 · Physical Review A

Entropy production dynamics in quench protocols of a driven-dissipative critical system

Bruno O. Goes, Gabriel Teixeira Landi

Published Nov 3, 2020• 7 citations
arXivHTMLPDF

Abstract

Driven-dissipative phase transitions are currently a topic of intense research due to the prospect of experimental realizations in quantum optical setups. The most paradigmatic model presenting such a transition is the Kerr model, which predicts the phenomenon of optical bistability, where the system may relax to two different steady states for the same driving condition. These states, however, are inherently out-of-equilibrium and are thus characterized by the continuous production of irreversible entropy, a key quantifier in thermodynamics. In this paper, we study the dynamics of the entropy production rate in a quench scenario of the Kerr model, where the external pump is abruptly changed. This is accomplished using a recently developed formalism, based on the Husimi $Q$-function, which is particularly tailored for driven-dissipative and non-Gaussian bosonic systems, Goes et al. [Phys. Rev. Res. 2, 013136 (2020)]. Within this framework, the entropy production can be split into two contributions, one being extensive with the drive and describing classical irreversibility, and the other being intensive and directly related to quantum fluctuations. The latter, in particular, is found to reveal the high degree of nonadiabaticity for quenches between different metastable states.

DOI
10.1103/physreva.102.052202

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Driven-dissipative phase transitions are currently a topic of intense research due to the prospect of experimental realizations in quantum optical setups. The most paradigmatic model presenting such a transition is the Kerr model, which predicts the phenomenon of optical bistability, where the system may relax to two different steady states for the same driving condition. These states, however, are inherently out-of-equilibrium and are thus characterized by the continuous production of irreversible entropy, a key quantifier in thermodynamics. In this paper, we study the dynamics of the entropy production rate in a quench scenario of the Kerr model, where the external pump is abruptly changed. This is accomplished using a recently developed formalism, based on the Husimi QQQ-function, which is particularly tailored for driven-dissipative and non-Gaussian bosonic systems, Goes et al. [Phys. Rev. Res. 2, 013136 (2020)]. Within this framework, the entropy production can be split into two contributions, one being extensive with the drive and describing classical irreversibility, and the other being intensive and directly related to quantum fluctuations. The latter, in particular, is found to reveal the high degree of nonadiabaticity for quenches between different metastable states.
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.
We study finite-time driving across second-order dissipative quantum phase transitions described by Lindblad dynamics. We show that the nonadiabatic entropy production, which quantifies deviations from the instantaneous nonequilibrium steady state, exhibits universal power-law scaling with the ramp duration in analogy to the Kibble-Zurek mechanism for closed systems. This establishes the universality of irreversible dissipation induced by driving an open quantum system near criticality. Furthermore, in systems described by bosonic Gaussian states, our scaling laws predict that the nonadiabatic entropy production is independent of driving speed to leading order, revealing a distinctive feature of Gaussian dissipative quantum phase transitions. We validate these analytical predictions in the thermodynamic limit of the driven-dissipative Dicke model and via finite-size scaling in the open Kerr model. Our results establish a general framework for understanding universal nonequilibrium response and thermodynamic irreversibility in critical open quantum systems.
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The irreversible work during a driving protocol constitutes one of the most widely studied measures in nonequilibrium thermodynamics, as it constitutes a proxy for entropy production. In quantum systems, it has been shown that the irreversible work has an additional, genuinely quantum mechanical contribution, due to coherence produced by the driving protocol. The goal of this paper is to explore this contribution in systems that undergo a quantum phase transition. Substantial effort has been dedicated in recent years to understanding the role of quantum criticality in work protocols. However, practically nothing is known about how coherence contributes to it. To shed light on this issue, we study the entropy production in infinitesimal quenches of the one-dimensional XY model. We find that coherence plays a significant role in the entropy production and can even account for most of it in certain situations. Moreover, at low temperatures, the coherence presents a finite cusp at the critical point, whereas the entropy production diverges logarithmically. For high temperatures, however, the coherence presents a kink at the critical point indicating the quantum phase transition that occurs only at T=0. Alternatively, if the quench is performed in the anisotropy parameter, then we find that there are situations where all of the entropy produced is due to quantum coherences.
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.