Generalized Schrieffer-Wolff Formalism for Dissipative Systems

TL;DR

This paper develops a generalized Schrieffer-Wolff perturbation theory for Markovian open quantum systems, enabling systematic adiabatic elimination of fast degrees of freedom to arbitrary order.

Contribution

It introduces a non-unitary transformation formalism to derive effective Liouvillians for dissipative systems, extending the Schrieffer-Wolff method beyond closed systems.

Findings

Effective Liouvillian reproduces low excitation spectrum

Perturbative expansion valid to arbitrary order

Demonstrated on two generic open systems

Abstract

We present a formalized perturbation theory for Markovian open systems in the language of a generalized Schrieffer-Wolff (SW) transformation. A non-unitary rotation decouples the unper- turbed steady states from all fast degrees of freedom, in order to obtain an effective Liouvillian, that reproduces the exact low excitation spectrum of the system. The transformation is derived in a constructive way, yielding a perturbative expansion of the effective Liouville operator. The presented formalism realizes an adiabatic elimination of fast degrees of freedom to arbitrary orders in the perturbation. We exemplarily employ the SW formalism to two generic open systems and discuss general properties of the different orders of the perturbation.