On Generated Dynamics for Open Quantum Systems: Spectral Analysis of Effective Liouville

TL;DR

This paper analyzes the spectral properties of an effective Liouville operator that generates the dynamics of open quantum systems, revealing how it encodes memory effects, decoherence, and relaxation processes.

Contribution

It introduces a spectral analysis framework for the effective Liouville, linking eigenvalues to physical phenomena like memory effects and relaxation times in open quantum systems.

Findings

Effective Liouville's spectrum has negative imaginary parts indicating irreversibility.

Spectral analysis represents quantum dynamics using complex eigenvalues and bi-orthonormal modes.

Relaxation time scale is determined by the inverse of the smallest negative imaginary part.

Abstract

We point out that the quantum dynamical map of an open quantum system can be generated by an effective Liouville operator. The effective Liouville shows the dynamical breaking of time reversibility. This breaking of reversibility is expressed by the effective Liouville's discrete spectrum having negative imaginary parts. This generated dynamics of open quantum systems is capable of memory effects described by a frequency dependence of the spectrum. When memory effects can be neglected or smoothed out, the effective Liouville generates the well known semi-group dynamics of open systems in the Markov approximation. The spectral analysis of the effective Liouville -with or without memory- allows to represent the quantum dynamical map and expectation values of physical quantities by metric expressions using complex eigenvalues and bi-orthonormal eigenmodes. The long time dynamics is dominated by the zero-mode, which forms the stationary state of the open system. The relaxation time scale $\tau$ is set by the inverse of the smallest negative imaginary part of the non-vanishing effective eigenvalues of the effective Liouville. In generic cases of system-environment-coupling this time scale $\tau$ is the relevant scale for both, the relaxation of diagonal elements (populations) and the decay of off-diagonal elements (coherences) of an initial density matrix in the stationary state's diagonal representation. It also sets the time scale after which the entropy tends to its stationary value. The effective Liouville and its spectral content thus give a framework to study the dynamic breaking of time reversibility, memory effects, decoherence, relaxation to a stationary state and the role of entropy in open quantum systems.