Non-Markovian Dynamics of Open Quantum Systems: Stochastic Equations and their Perturbative Solutions

TL;DR

This paper develops a formalism for solving non-Markovian open quantum system dynamics using perturbation theory without common approximations, ensuring physical validity and expanding understanding of non-Markovian effects.

Contribution

It introduces a perturbative approach to non-Markovian quantum dynamics that preserves complete positivity and extends the quantum regression theorem.

Findings

Validates late-time perturbative master equations

Shows preservation of complete positivity without Lindblad form

Provides methods for analyzing near-resonant systems

Abstract

We treat several key stochastic equations for non-Markovian open quantum system dynamics and present a formalism for finding solutions to them via canonical perturbation theory, without making the Born-Markov or rotating wave approximations (RWA). This includes master equations of the (asymptotically) stationary, periodic, and time-nonlocal type. We provide proofs on the validity and meaningfulness of the late-time perturbative master equation and on the preservation of complete positivity despite a general lack of Lindblad form. More specifically, we show how the algebraic generators satisfy the theorem of Lindblad and Gorini, Kossakowski and Sudarshan, even though the dynamical generators do not. These proofs ensure the mathematical viability and physical soundness of solutions to non-Markovian processes. Within the same formalism we also expand upon known results for non-Markovian corrections to the quantum regression theorem. Several directions where these results can be usefully applied to are also described, including the analysis of near-resonant systems where the RWA is inapplicable and the calculation of the reduced equilibrium state of open systems.