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Asymptotic relaxation in quantum Markovian dynamics

Giovanni Di Meglio, Dariusz Chruściński, Koenraad Audenaert, Martin B. Plenio, Susana F. Huelga

TL;DR

The paper develops a framework to study the long-time behavior of time-dependent GKLS dynamics, introducing the notion of weak relaxation to handle driving and control. It extends the Spohn–Frigerio results to time-dependent generators and derives explicit contraction bounds on the traceless sector in terms of the instantaneous steady state and time-integrated dissipation rates. A graph-theoretic construction links the structure of jump operators to relaxation properties, providing practical bounds via graph Laplacians and spectral gaps, and the approach is illustrated with driven N‑level systems, including a detailed 3‑level example. The results also apply to non-Markovian time-local master equations that become GKLS at long times, paving the way for a broader theory of relaxation beyond Markovian dynamics.

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.

Asymptotic relaxation in quantum Markovian dynamics

TL;DR

The paper develops a framework to study the long-time behavior of time-dependent GKLS dynamics, introducing the notion of weak relaxation to handle driving and control. It extends the Spohn–Frigerio results to time-dependent generators and derives explicit contraction bounds on the traceless sector in terms of the instantaneous steady state and time-integrated dissipation rates. A graph-theoretic construction links the structure of jump operators to relaxation properties, providing practical bounds via graph Laplacians and spectral gaps, and the approach is illustrated with driven N‑level systems, including a detailed 3‑level example. The results also apply to non-Markovian time-local master equations that become GKLS at long times, paving the way for a broader theory of relaxation beyond Markovian dynamics.

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.

Paper Structure

This paper contains 14 sections, 18 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. General theory
  3. Preliminaries and notation
  4. Main results
  5. Proof of the main results
  6. Application to driven N-level systems
  7. Instantaneous steady state
  8. Graph-theoretic formulation of the jump operators
  9. Spectral bounds
  10. Example: 3-level system
  11. Conclusions
  12. Heisenberg picture analysis
  13. Graph theory
  14. Master equation driven 3-level system

Key Result

Theorem 2.1

Let $\mathcal{L}$ be the generator of a CPTP semigroup and let $\mathcal{J}=\{L_{\alpha}\}_{\alpha}$ be the set of jump operators. If then there exists a faithful (i.e. full-rank) steady state $\omega$ and the semigroup is relaxing, namely $\lim\limits_{t\rightarrow\infty}e^{t\mathcal{L}}(\rho_0)=\omega$, $\forall\rho_0 \in\mathscr{S}_1^+$.

Theorems & Definitions (48)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.1: Spohn-Frigerio
  • Theorem 2.2
  • Theorem 2.3
  • Corollary 2.1
  • Corollary 2.2
  • ...and 38 more