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arXiv:1907.07743 · Entropy

Quantum Thermodynamics in the Refined Weak Coupling Limit

Ángel Rivas

Published Jul 25, 2019• 17 citations
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Abstract

We present a thermodynamic framework for the refined weak coupling limit. In this limit, the interaction between system and environment is weak, but not negligible. As a result, the system dynamics becomes non-Markovian breaking divisibility conditions. Nevertheless, we propose a derivation of the first and second law just in terms of the reduced system dynamics. To this end, we extend the refined weak coupling limit for allowing slowly-varying external drivings and reconsider the definition of internal energy due to the non-negligible interaction.

DOI
10.3390/e21080725

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We present a thermodynamic framework for the refined weak coupling limit. In this limit, the interaction between system and environment is weak, but not negligible. As a result, the system dynamics becomes non-Markovian breaking divisibility conditions. Nevertheless, we propose a derivation of the first and second law just in terms of the reduced system dynamics. To this end, we extend the refined weak coupling limit for allowing slowly-varying external drivings and reconsider the definition of internal energy due to the non-negligible interaction.
We present a thermodynamic framework for the refined weak coupling limit. In this limit, the interaction between system and environment is weak, but not negligible. As a result, the system dynamics becomes non-Markovian breaking divisibility conditions. Nevertheless, we propose a derivation of the first and second law just in terms of the reduced system dynamics. To this end, we extend the refined weak coupling limit for allowing slowly-varying external drivings and reconsider the definition of internal energy due to the non-negligible interaction.
We study the properties of a refined weak-coupling limit that preserves complete positivity in order to describe non-Markovian dynamics in the spin-boson model. With this tool, we show the system presents a rich non-Markovian phenomenology. This implies a dynamical difference between entanglement and coherence: the latter undergoes revivals, whereas the former not, despite the induced dynamics being fully incoherent. In addition, the evolution presents ``quasieternal'' non-Markovianity, becoming nondivisible at any time period where the system evolves qualitatively. Furthermore, the method allows for an exact derivation of a master equation that accounts for a reversible energy exchange between system and environment. Specifically, this is obtained in the form of a time-dependent Lamb shift term.
We present a consistent thermodynamic theory for the resonant level model in the wide-band limit, whose level energy is driven slowly by an external force. The problem of defining ``system'' and ``bath'' in the strong-coupling regime is circumvented by considering as the system everything that is influenced by the externally driven level. The thermodynamic functions that are obtained to first order beyond the quasistatic limit fulfill the first and second law with a positive entropy production, successfully connect to the forces experienced by the external driving, and reproduce the correct weak-coupling limit of stochastic thermodynamics.
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.
Two non-directly interacting qubits with equal frequencies can become entangled via a Markovian, dissipative dynamics through the action of a weakly coupled Ohmic heat bath. In the standard weak-coupling limit derivation, this purely dissipative effect disappears if the frequencies are different because of the “ergodic average” used by this approach. However, there are physical situations where this technique is too rough to capture all the relevant aspects of the dissipative dynamics. In these cases, in order to better describe the physical behavior of the open system, it is necessary to go beyond the “ergodic average”. We show that, in this more refined framework, the entanglement capability of the environment persists also in the case of different frequencies.
Quantum systems strongly coupled to many-body systems equilibrate to the reduced state of a global thermal state, deviating from the local thermal state of the system as it occurs in the weak-coupling limit. Taking this insight as a starting point, we study the thermodynamics of systems strongly coupled to thermal baths. First, we provide strong-coupling corrections to the second law applicable to general systems in three of its different readings: As a statement of maximal extractable work, on heat dissipation, and bound to the Carnot efficiency. These corrections become relevant for small quantum systems and vanish in first order in the interaction strength. We then move to the question of power of heat engines, obtaining a bound on the power enhancement due to strong coupling. Our results are exemplified on the paradigmatic non-Markovian quantum Brownian motion.