Symmetrized Liouvillian Gap in Markovian Open Quantum Systems

TL;DR

This paper introduces the symmetrized Liouvillian gap as a new measure to accurately bound the transient decay in Markovian open quantum systems, improving understanding of relaxation dynamics beyond the standard spectral gap.

Contribution

The paper defines the symmetrized Liouvillian gap and demonstrates it provides a rigorous upper bound on auto-correlation decay, especially when detailed balance is absent.

Findings

Symmetrized Liouvillian gap always bounds auto-correlation decay.

Standard Liouvillian gap may not accurately bound decay without detailed balance.

Numerical results confirm the effectiveness of the symmetrized gap.

Abstract

Markovian open quantum systems display complicated relaxation dynamics. The spectral gap of the Liouvillian characterizes the asymptotic decay rate towards the steady state, but it does not necessarily give a correct estimate of the relaxation time because the crossover time to the asymptotic regime may be too long. We here give a rigorous upper bound on the transient decay of auto-correlation functions in the steady state by introducing the symmetrized Liouvillian gap. The standard Liouvillian gap and the symmetrized one are identical in an equilibrium situation but differ from each other in the absence of the detailed balance condition. It is numerically shown that the symmetrized Liouvillian gap always give a correct upper bound on the decay of the auto-correlation function, but the standard Liouvillian gap does not.