Adiabatic elimination for open quantum systems with effective Lindblad master equations

TL;DR

This paper develops a geometric asymptotic expansion method to derive effective Lindblad master equations for open quantum systems with two timescales, ensuring the reduced model preserves physical properties like complete positivity.

Contribution

It introduces a novel geometric approach for adiabatic elimination in open quantum systems, providing a systematic way to obtain effective Lindblad equations at any order.

Findings

Derived first-order effective Lindblad equations with simple formulas.

Showed second-order corrections for Hamiltonian perturbations.

Illustrated the method on a dissipative harmonic oscillator system.

Abstract

We consider an open quantum system described by a Lindblad-type master equation with two times-scales. The fast time-scale is strongly dissipative and drives the system towards a low-dimensional decoherence-free space. To perform the adiabatic elimination of this fast relaxation, we propose a geometric asymptotic expansion based on the small positive parameter describing the time-scale separation. This expansion exploits geometric singular perturbation theory and center-manifold techniques. We conjecture that, at any order, it provides an effective slow Lindblad master equation and a completely positive parameterization of the slow invariant sub-manifold associated to the low-dimensional decoherence-free space. By preserving complete positivity and trace, two important structural properties attached to open quantum dynamics, we obtain a reduced-order model that directly conveys a physical interpretation since it relies on effective Lindbladian descriptions of the slow evolution. At the first order, we derive simple formulae for the effective Lindblad master equation. For a specific type of fast dissipation, we show how any Hamiltonian perturbation yields Lindbladian second-order corrections to the first-order slow evolution governed by the Zeno-Hamiltonian. These results are illustrated on a composite system made of a strongly dissipative harmonic oscillator, the ancilla, weakly coupled to another quantum system.