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arXiv:2007.04255 · Journal of Statistical Mechanics Theory and Experiment

Heat exchange and fluctuation in Gaussian thermal states in the quantum realm

A. R. Usha Devi, Sudha, A. K. Rajagopal, A. M. Jayannavar

Published Feb 1, 2021• 0 citations
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Abstract

Abstract The celebrated exchange fluctuation theorem—proposed by Jarzynski and Wójcik (2004 Phys. Rev. Lett. 92 230602) for heat exchange between two systems in thermal equilibrium at different temperatures—is explored here for quantum Gaussian states in thermal equilibrium. We employ the Wigner distribution function formalism for quantum states, which exhibits a close resemblance to the classical phase-space trajectory description, to arrive at a formal Jarzynski–Wójcik result. For two Gaussian states in thermal equilibrium at two different temperatures kept in contact with each other for a fixed duration of time, we show that the Jarzynski–Wójcik- like relation reduces to the corresponding classical result in the limit ℏ → 0.

DOI
10.1088/1742-5468/abdd14

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Abstract The celebrated exchange fluctuation theorem—proposed by Jarzynski and Wójcik (2004 Phys. Rev. Lett. 92 230602) for heat exchange between two systems in thermal equilibrium at different temperatures—is explored here for quantum Gaussian states in thermal equilibrium. We employ the Wigner distribution function formalism for quantum states, which exhibits a close resemblance to the classical phase-space trajectory description, to arrive at a formal Jarzynski–Wójcik result. For two Gaussian states in thermal equilibrium at two different temperatures kept in contact with each other for a fixed duration of time, we show that the Jarzynski–Wójcik- like relation reduces to the corresponding classical result in the limit ℏ → 0.
Energy exchange statistics between two bodies at different thermal equilibria obey the Jarzynski-Wójcik fluctuation theorem. The corresponding energy scale factor is the difference of the inverse temperatures associated to the bodies at equilibrium. In this work, we consider a dissipative quantum dynamics leading the quantum system towards a possibly nonthermal, asymptotic state. To generalize the Jarzynski-Wójcik theorem to nonthermal states, we identify a sufficient condition I for the existence of an energy scale factor η^{*} that is unique, finite, and time independent, such that the characteristic function of the energy exchange distribution becomes identically equal to 1 for any time. This η^{*} plays the role of the difference of inverse temperatures. We discuss the physical interpretation of the condition I, showing that it amounts to an almost complete memory loss of the initial state. The robustness of our results against quantifiable deviations from the validity of I is evaluated by experimental studies on a single nitrogen-vacancy center subjected to a sequence of laser pulses and dissipation.
We study the heat transfer between two finite quantum systems initially at different temperatures. We find that a recently proposed fluctuation theorem for heat exchange, namely the exchange fluctuation theorem [C. Jarzynski and D. K. Wójcik, Phys. Rev. Lett. 92 (2004), 230602], does not generally hold in the presence of a finite heat transfer as in the original form proved for weak coupling. As the coupling is weakened, the deviation from the theorem and the heat transfer vanish in the same order of the coupling. We then discover a condition for the exchange fluctuation theorem to hold in the presence of a finite heat transfer, namely the commutable-coupling condition. We explicitly calculate the deviation from the exchange fluctuation theorem as well as the heat transfer for simple models. We confirm for the models that the deviation indeed has a finite value as far as the coupling between the two systems is finite except for the special point of the commutable-coupling condition. We also confirm analytically that the commutable-coupling condition indeed lets the exchange fluctuation theorem hold exactly under a finite heat transfer.
Two fundamental ingredients play a decisive role in the foundations of fluctuation relations-the principle of micro-reversibility and the fact that thermal equilibrium is described by Gibbs canonical ensemble. Here we investigate the statistics of heat exchange between quantum finite system coupled to two reservoirs at temperatures Th and Tc in the transient regime. We have obtained an analytical result for quantum heat fluctuation theorem in the transient regime. We consider a composite system consisting of two different heat baths at different temperatures with an intervening medium(system of interest). The system of interest is driven by an external protocol. For our study we invoke von Neumann two point projective measurement in quantum mechanics. Our result is a generalisation of Jarzynski-Wojcik heat fluctuation theorem.
We study the heat transfer between two finite quantum systems initially at different temperatures. We find that a recently proposed fluctuation theorem for heat exchange, namely the exchange fluctuation theorem [C. Jarzynski and D. K. Wójcik, Phys. Rev. Lett. 92, 230602 (2004)], does not generally hold in the presence of a finite heat transfer as in the original form proved for weak coupling. As the coupling is weakened, the deviation from the theorem and the heat transfer vanish in the same order of the coupling. We then discover a condition for the exchange fluctuation theorem to hold in the presence of a finite heat transfer, namely the commutable-coupling condition. We explicitly calculate the deviation from the exchange fluctuation theorem as well as the heat transfer for simple models. We confirm for the models that the deviation indeed has a finite value as far as the coupling between the two systems is finite except for the special point of the commutable-coupling condition. We also confirm analytically that the commutable-coupling condition indeed lets the exchange fluctuation theorem hold exactly under a finite heat transfer.
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In this work, we investigate the heat exchange between two quantum systems whose initial equilibrium states are described by the generalized Gibbs ensemble. First, we generalize the fluctuation relations for heat exchange discovered by Jarzynski and Wójcik to quantum systems prepared in the equilibrium states described by the generalized Gibbs ensemble at various generalized temperatures. Secondly, we extend the connections between heat exchange and the Rényi divergences to quantum systems under generic initial conditions. These relations are applicable for quantum systems with conserved quantities and universally valid for quantum systems in the integrable and chaotic regimes.