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arXiv:1807.03572 · Physical review. E

Heat distribution of a quantum harmonic oscillator

Tobias Denzler, Eric Lutz

Published Nov 6, 2018• 31 citations
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Abstract

We consider a thermal quantum harmonic oscillator weakly coupled to a heat bath at a different temperature. We analytically study the quantum heat exchange statistics between the two systems using the quantum-optical master equation. We exactly compute the characteristic function of the heat distribution and show that it verifies the Jarzynski-W\'ojcik fluctuation theorem. We further evaluate the heat probability density in the limit of long thermalization times, both in the low- and the high-temperature regimes, and investigate its time evolution by calculating its first two cumulants.

DOI
10.1103/physreve.98.052106

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We consider a thermal quantum harmonic oscillator weakly coupled to a heat bath at a different temperature. We analytically study the quantum heat exchange statistics between the two systems using the quantum-optical master equation. We exactly compute the characteristic function of the heat distribution and show that it verifies the Jarzynski-W'ojcik fluctuation theorem. We further evaluate the heat probability density in the limit of long thermalization times, both in the low- and the high-temperature regimes, and investigate its time evolution by calculating its first two cumulants.
The nonequilibrium fluctuation relation is a cornerstone of quantum thermodynamics. It is widely believed that the system-bath heat exchange obeys the famous Jarzynski-Wójcik fluctuation theorem. However, this theorem is established in the Born-Markovian approximation under the weak-coupling condition. Via studying the energy exchange between a harmonic oscillator and its coupled bath in the non-Markovian dynamics, we establish a generalized quantum fluctuation theorem for energy exchange being valid for arbitrary coupling strength. The Jarzynski-Wójcik fluctuation theorem is recovered in the weak-coupling limit. We also find the average energy exchange exhibits rich nonequilibrium characteristics when different numbers of system-bath bound states are formed, which suggests a useful way to control the quantum heat. Deepening our understanding of the fluctuation relation in quantum thermodynamics, our result lays the foundation to design high-efficiency quantum heat engines.
Exact calculations for a quantum oscillator linearly coupled to a thermal reservoir are presented. Explicit results (variances and correlations) for the response and the spontaneous fluctuations are evaluated for two different models of the heat bath; a reservoir with a spectrum which is similar to that of acoustic phonons and another which is analogous to optic phonons. The low-temperature behavior is shown to depend sensitively on the spectral density of the thermal reservoir. The mass and the frequency of the oscillator become renormalized through the coupling to the bath. These renormalizations are intrinsically frequency dependent. Moreover, the commonly used form of Ehrenfest's equation for the dynamics of the linearly damped oscillator is shown to only be a reasonable approximation within certain frequency regimes.
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.
As the dimensions of physical systems approach the nanoscale, the laws of thermodynamics must be reconsidered due to the increased importance of fluctuations and quantum effects. While the statistical mechanics of small classical systems is relatively well understood, the quantum case still poses challenges. Here, we set up a formalism that allows us to calculate the full probability distribution of energy exchanges between a periodically driven quantum system and a thermalized heat reservoir. The formalism combines Floquet theory with a generalized master equation approach. For a driven two-level system and in the long-time limit, we obtain a universal expression for the distribution, providing clear physical insight into the exchanged energy quanta. We illustrate our approach in two analytically solvable cases and discuss the differences in the corresponding distributions. Our predictions could be directly tested in a variety of systems, including optical cavities and solid-state devices.
Nonequilibrium fluctuation relation lies at the heart of the quantum thermodynamics. Many previous studies have demonstrated that the heat exchange between a quantum system and a thermal bath initially prepared in their own Gibbs states at different temperatures obeys the famous Jarzynski-Wójcik fluctuation theorem. However, this conclusion is obtained under the assumption of Born-Markovian approximation. In this paper, going beyond the Born-Markovian limitation, we investigate the statistics of quantum heat in an exactly non-Markovian relaxation process described by the well-known Caldeira-Leggett model. It is revealed that the Jarzynski-Wójcik fluctuation theorem breaks down in the strongly non-Markovian regime. Moreover, we find the steady-state quantum heat within the non-Markovian framework can be widely tunable by using the quantum reservoir-engineering technique. These results are sharply contrary to the common Born-Markovian predictions. Our results presented in this paper may update the understanding of the quantum thermodynamics in strongly coupled and low-temperature systems. Moreover, the controllable heat may have some potential applications in improving the performance of a quantum heat engine.