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arXiv:1807.04500 · Physical Review A

Entropy production and asymptotic factorization via thermalization: A collisional model approach

Stefano Cusumano, Vasco Cavina, Maximilian Keck, Antonella De Pasquale, Vittorio Giovannetti

Published Sep 26, 2018• 49 citations
arXivHTMLPDF

Abstract

The Markovian evolution of an open quantum system is characterized by a positive entropy production, while the global entropy gets redistributed between the system and the environment degrees of freedom. Starting from these premises, we analyze the entropy variation of an open quantum system in terms of two distinct relations: the Clausius inequality, that provides an intrinsic bound for the entropy variation in terms of the heat absorbed by the system, and an extrinsic inequality, which instead relates the former to the corresponding entropy increment of the environment. By modeling the thermalization process with a Markovian collisional model, we compare and discuss the two bounds, showing that the latter is asymptotically saturated in the limit of large interaction time. In this regime not only the reduced density matrix of the system reaches an equilibrium configuration, but it also factorizes from the environment degrees of freedom. This last result is proven analytically when the system-bath coupling is sufficiently strong and through numerical analysis in the weak-coupling regime.

DOI
10.1103/physreva.98.032119

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The Markovian evolution of an open quantum system is characterized by a positive entropy production, while the global entropy gets redistributed between the system and the environment degrees of freedom. Starting from these premises, we analyze the entropy variation of an open quantum system in terms of two distinct relations: the Clausius inequality, that provides an intrinsic bound for the entropy variation in terms of the heat absorbed by the system, and an extrinsic inequality, which instead relates the former to the corresponding entropy increment of the environment. By modeling the thermalization process with a Markovian collisional model, we compare and discuss the two bounds, showing that the latter is asymptotically saturated in the limit of large interaction time. In this regime not only the reduced density matrix of the system reaches an equilibrium configuration, but it also factorizes from the environment degrees of freedom. This last result is proven analytically when the system-bath coupling is sufficiently strong and through numerical analysis in the weak-coupling regime.
Accessing the physical mechanisms behind non-Markovian phenomena in open quantum dynamics requires the study of the statistical properties of the joint system-environment dynamics. This is impossible at the level of the reduced dynamics of the open system alone as the latter is obtained by suitably eliminating the environment. The task is instead made possible by considering multi-time correlation functions involving observables of the open system, only: the open system-environment interactions turn them into global ones thus building up correlations between the two systems. Multi-time correlations form the basis of both the theory of quantum stochastic processes and of the Alicki-Lindblad-Fannes dynamical entropy (ALF entropy for short). This latter quantity provides for quantum systems a measure of the dynamical entropy production as the Kolmogorov-Sinai entropy does for classical systems. In the case of a collisional model whereby the dissipative dynamics of a finite-level system is obtained by its coupling to an infinite classical spin chain, the ALF entropy can be explicitly computed. It turns out to depend on the parameters characterizing the statistical properties of the environment and can be related to the activation and super-activation of memory effects in the open quantum system.
Thermalization in open systems coupled to macroscopic environments is usually analyzed from the perspective of relaxation of the reduced state of the system to the equilibrium state. Less emphasis is given to the change of the state of the bath. However, as previously shown for some specific models, during the thermalization the environment may undergo a nontrivial dynamics, indicated by the the change of its von Neumann entropy, at timescales even longer than the relaxation time of the system; here such a behavior is called post-thermalization. We provide a more detailed analysis of this phenomenon by simulating the full dynamics of a variety of systems together with their environment. In particular, the post-thermalization is qualitatively explained as a result of reconversion of the initially built up correlation between the system and the bath into the correlation between the degrees of freedom in the environment. We also present exemplary systems in which such a reconversion is suppressed due to non-Markovian dynamics or the presence of interactions.
In the weak-coupling limit approach to open quantum systems, the presence of the bath is eliminated and accounted for by a master equation that introduces dissipative contributions to the system reduced dynamics: within this framework, there are no bath entropy contributions to the entropy balance. We show that, as a consequence, the entropy production fails to be positive for a class of physically legitimate, that is completely positive and trace preserving, non-Markovian dynamical maps. Moreover, in absence of the semigroup property, if the reduced dynamics has a thermal asymptotic state, this need not be stationary. Then even the integrated entropy production becomes negative. These observations imply that, when the conditions leading to reduced dynamics of semigroup type are relaxed, a consistent formulation of the second law of thermodynamics requires that the environment contribution to the entropy balance be explicitly taken into account.
We investigate the irreversible entropy production of a qubit in contact with an environment modelled by a microscopic collision model in both Markovian and non-Markovian regimes. Our main goal is to contribute to the discussions on the relationship between non-Markovian dynamics and negative entropy production rates. We employ two different types of collision models that do or do not keep the correlations established between the system and the incoming environmental particle, while both of them pertain to their non-Markovian nature through information backflow from the environment to the system. We observe that as the former model, where the correlations between the system and environment are preserved, gives rise to negative entropy production rates in the transient dynamics, the latter one always maintains positive rates, even though the convergence to the steady-state value is slower as compared to the corresponding Markovian dynamics. Our results suggest that the mechanism underpinning the negative entropy production rates is not solely non-Markovianity through information backflow, but rather the contribution to it through established system-environment correlations.
The asymmetry between heating and cooling in open quantum systems is a hallmark of nonequilibrium dynamics, yet its thermodynamic origin has remained unclear. Here, we investigate the thermalization of a quantum system weakly coupled to a thermal bath, focusing on the entropy production rate and the quantum thermokinetic uncertainty relation (TKUR). We derive an analytical expression for the entropy production rate, showing that heating begins with a higher entropy production, which drives faster thermalization than cooling. The quantum TKUR links this asymmetry to heat-current fluctuations, demonstrating that larger entropy production suppresses fluctuations, making heating more stable than cooling. Our results reveal the thermodynamic basis of asymmetric thermalization and highlight uncertainty relations as key to nonequilibrium quantum dynamics.
We examine the elements of the balance equation of entropy in open quantum evolutions and their response as we go from a Markovian to a non-Markovian situation. In particular, we look at the heat current and entropy production rate in the non-Markovian reduced evolution, as well as a Markovian limit of the same, experienced by one of two interacting systems immersed in a Markovian bath. The analysis naturally leads us to define a heat current deficit and an entropy production rate deficit, which are differences between the global and local versions of the corresponding quantities. The investigation leads, in certain cases, to a complementarity of the time-integrated heat current deficit and the relative entropy of entanglement between the two systems.