Symmetry and block structure of the Liouvillian superoperator in partial secular approximation

TL;DR

This paper reveals how the partial secular approximation induces symmetries in the Liouvillian superoperator, enabling block decomposition that simplifies the analysis of steady states in open quantum systems.

Contribution

It demonstrates that the partial secular approximation naturally induces symmetry and block structure in the Liouvillian, simplifying master equations and steady-state analysis.

Findings

Symmetry arises from partial secular approximation in the Liouvillian.

Block decomposition reduces complexity of solving master equations.

Unique steady state constrains steady-state coherences.

Abstract

We address the structure of the Liouvillian superoperator for a broad class of bosonic and fermionic Markovian open systems interacting with stationary environments. We show that the accurate application of the partial secular approximation in the derivation of the Bloch-Redfield master equation naturally induces a symmetry on the superoperator level, which may greatly reduce the complexity of the master equation by decomposing the Liouvillian superoperator into independent blocks. Moreover, we prove that, if the steady state of the system is unique, one single block contains all the information about it, and that this imposes a constraint on the possible steady-state coherences of the unique state, ruling out some of them. To provide some examples, we show how the symmetry appears for two coupled spins interacting with separate baths, as well as for two harmonic oscillators immersed in a common environment. In both cases the standard derivation and solution of the master equation is simplified, as well as the search for the steady state. The block-diagonalization does not appear when a local master equation is chosen.