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arXiv:1507.05763 · Physical Review X

Quantum-Classical Correspondence Principle for Work Distributions

Christopher Jarzynski, H. T. Quan, Saar Rahav

Published Sep 17, 2015• 101 citations
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Abstract

The definition of quantum work differs markedly from the definition found in classical mechanics textbooks. Now, by showing that quantum work distributions can be understood as interference patterns between classical trajectories, researchers bridge the gap between classical and quantum notions of work.
DOI
10.1103/physrevx.5.031038

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A definition of quantum mechanical work is introduced in this dissertation, preserving the mathematical structure of the Classical Mechanics concept of work without, however, in any way invoking the notion of trajectory. By use of Gaussian states and the Caldirola-Kanai model, a case study is conducted through which the proposed quantum work is compared with another Quantum Thermodynamics well-known definition, both in quantum and semiclassical regimes, showing promising results. Finally the new definition is investigated in comparison with a classical-statistical approach for superposition and mixed states.
We numerically study the work distributions in a chaotic system and examine the relationship between quantum work and classical work. Our numerical results suggest that there exists a correspondence principle between quantum and classical work distributions in a chaotic system. This correspondence was proved for one-dimensional integrable systems in a recent work [Jarzynski, Quan, and Rahav, Phys. Rev. X 5, 031038 (2015)1063-651X10.1103/PhysRevX.5.031038]. Our investigation further justifies the definition of quantum work via two-point energy measurements.
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The standard definition of quantum fluctuating work is based on the two-projective energy measurement, which however does not apply to systems with initial quantum coherence because the first projective energy measurement destroys the initial coherence, and affects the subsequent evolution of the system. To address this issue, several alternative definitions, such as those based on the full counting statistics and the Margenau-Hill distribution, have been proposed recently. These definitions seem ad hoc because justifications for them are still lacking. In the current study, by utilizing the quantum Feynman-Kac formula and the phase space formulation of quantum mechanics, we prove that the leading order of work distributions is equal to the classical work distribution. Thus we prove the validity of the quantum-classical correspondence of work distributions for initial states with quantum coherence, and provide some justification for those definitions of work. We use an exactly solvable model of the linearly dragged harmonic oscillator to demonstrate our main results.
Work in closed quantum systems is usually defined by a two-point measurement. This definition of work is compatible with quantum fluctuation theorems but it fundamentally differs from its classical counterpart. In this paper, we study the correspondence principle in quantum chaotic systems. We derive a semiclassical expression of the work distribution for chaotic systems undergoing a general, finite time, process. This semiclassical distribution converges to the classical distribution in the usual classical limit. We show numerically that, for a particle inside a chaotic cavity, the semiclassical distribution provides a good approximation to quantum distribution.
Work in closed quantum systems is usually defined by a two-point measurement. This definition of work is compatible with quantum fluctuation theorems but it fundamentally differs from its classical counterpart. In this paper, we study the correspondence principle in quantum chaotic systems. We derive a semiclassical expression of the work distribution for chaotic systems undergoing a general, finite time, process. This semiclassical distribution converges to the classical distribution in the usual classical limit. We show numerically that, for a particle inside a chaotic cavity, the semiclassical distribution provides a good approximation to quantum distribution.