On the Thermal Symmetry of the Markovian Master Equation

TL;DR

This paper demonstrates that the quantum Markovian master equation for a harmonic oscillator exhibits a thermal symmetry, revealing deep connections between temperature, quantum effects, and the dynamics of open quantum systems.

Contribution

It introduces a thermal symmetry in the master equation via a Bogoliubov transformation, linking states at different temperatures and extending the understanding of quantum thermodynamics.

Findings

Thermal symmetry relates states at different temperatures.

The symmetry involves a hyperbolic rotation in Liouville space.

Classical and quantum equations share this symmetry.

Abstract

The quantum Markovian master equation of the reduced dynamics of a harmonic oscillator coupled to a thermal reservoir is shown to possess thermal symmetry. This symmetry is revealed by a Bogoliubov transformation that can be represented by a hyperbolic rotation acting on the Liouville space of the reduced dynamics. The Liouville space is obtained as an extension of the Hilbert space through the introduction of tilde variables used in the thermofield dynamics formalism. The angle of rotation depends on the temperature of the reservoir, as well as the value of Planck's constant. This symmetry relates the thermal states of the system at any two temperatures. This includes absolute zero, at which purely quantum effects are revealed. The Caldeira-Leggett equation and the classical Fokker-Planck equation also possess thermal symmetry. We compare the thermal symmetry obtained from the Bogoliubov transformation in related fields and discuss the effects of the symmetry on the shape of a Gaussian wave packet.