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Journal of Physics Communications

Crooks fluctuation theorem in ${\mathscr{PT}}$-symmetric quantum mechanics

Meng Zeng, Ee Hou Yong

Published Sep 26, 2017• 21 citations
Publisher

Abstract

Following the recent work by Deffner and Saxena (2015 Phys. Rev. Lett. 114 150601), where the Jarzynski equality is generalised to non-Hermitian quantum mechanics, we prove in this work a stronger form of Jarzynski equality, the Crooks fluctuation theorem, also in the non-Hermitian formalism when the system is in the unbroken -symmetric phase.

DOI
10.1088/2399-6528/aa8f26

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Crooks fluctuation theorem in PT{\mathscr{PT}}PT-symmetric quantum mechanics
Following the recent work by Deffner and Saxena (2015 Phys. Rev. Lett. 114 150601), where the Jarzynski equality is generalised to non-Hermitian quantum mechanics, we prove in this work a stronger form of Jarzynski equality, the Crooks fluctuation theorem, also in the non-Hermitian formalism when the system is in the unbroken -symmetric phase.
In a recent paper, Deffner and Saxena (2015 Phys. Rev. Lett. 114 150601) showed that quantum Jarzynski equality generalizes to PT- symmetric quantum mechanics with unbroken symmetry. later Zeng and Yong (2017 Journal of Phys. Commun. 1 031001) extended this work to Crook's fluctuation theorem. In another recent paper, Andrew Smith et al 2018 discusses non-equilibrium work relation in open quantum system. In this paper, we will discuss the validity of non-Hermitian fluctuation theorem in open quantum system, in a region of unbroken PT- symmetry.
We consider the quantum mechanical generalization of Crooks Fluctuation Theorem and Jarzynski Equality for an open quantum system. The explicit expression for microscopic work for an arbitrary prescribed protocol is obtained, and the relation between quantum Crooks Fluctuation Theorem, quantum Jarzynski Equality and their classical counterparts are clarified. Numerical simulations based on a two-level toy model are used to demonstrate the validity of the quantum version of the two theorems beyond linear response theory regime.
There are two related theorems which hold even in far from equilibrium, namely fluctuation theorem and Jarzynski equality. Fluctuation theorem states the existence of symmetry of fluctuation of entropy production, while the Jarzynski equality enables us to estimate the free energy change between two states by using irreversible processes. On the other hand, the relationship between these theorems was investigated by Crooks [Phys. Rev. E 60, 2721 (1999)] for the classical stochastic systems. In this paper, we derive quantum analogues of fluctuation theorem and Jarzynski equality in terms of microscopic reversibility. In other words, the quantum analog of the work by Crooks is presented. Also, for the quasiclassical Langevin system, microscopically reversible condition is confirmed.
We show that the quantum Jarzynski equality generalizes to PT\mathcal{P}\mathcal{T}PT-symmetric quantum mechanics with unbroken PT\mathcal{P}\mathcal{T}PT symmetry. In the regime of broken PT\mathcal{P}\mathcal{T}PT symmetry, the Jarzynski equality does not hold as also the CPT\mathcal{C}\mathcal{P}\mathcal{T}CPT norm is not preserved during the dynamics. These findings are illustrated for an experimentally relevant system---two coupled optical waveguides. It turns out that for these systems the phase transition between the regimes of unbroken and broken PT\mathcal{P}\mathcal{T}PT symmetry is thermodynamically inhibited as the irreversible work diverges at the critical point.
Based on the observation that the thermodynamic equilibrium free energy of an open quantum system in contact with a thermal environment is the difference between the free energy of the total system and that of the bare environment, the validity of the Crooks theorem and of the Jarzynski equality is extended to open quantum systems. No restrictions on the nature of the environment or on the strength of the coupling between system and environment need to be imposed. This free energy entering the Crooks theorem and the Jarzynski equality is closely related to the Hamiltonian of mean force that generalizes the classical statistical mechanical concept of the potential of mean force.