Asymptotic relaxation in quantum Markovian dynamics

TL;DR

This paper studies the long-term behavior of quantum Markovian systems, extending existing theorems to time-dependent cases and providing conditions for relaxation that depend on system structure and dissipation rates.

Contribution

It introduces a notion of weak relaxation, extends the Spohn-Frigerio theorem to time-dependent generators, and links relaxation properties to the structure of jump operators.

Findings

Derived explicit contraction bounds for quantum dynamics.

Provided a graph-theoretic characterization of relaxation conditions.

Extended the theory to non-Markovian dynamics at long times.

Abstract

We investigate the long-time behavior of quantum Markovian dynamics generated by time-dependent Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) master equations. We introduce a notion of weak relaxation and derive sufficient conditions guaranteeing asymptotic independence from the initial state. Our results provide a quantitative extension of the Spohn-Frigerio theorem to time-dependent generators, yielding explicit contraction bounds in terms of the instantaneous steady state and time-integrated dissipation rates. For a class of microscopically derived master equations, we further obtain a graph-theoretic characterization of the aforementioned conditions that directly links the structure of the jump operators to the relaxation properties. The general theory is illustrated by applications to driven finite-level systems, including a detailed three-level example, and is extended to a non-Markovian setting by means of time-local master equations that become of GKLS form at long times. These findings pave the way for the development of a more general theory of relaxation beyond the Markovian case.