Perturbative Steady States of Completely Positive Quantum Master Equations

TL;DR

This paper investigates the steady states of Lindbladian quantum master equations derived without the secular approximation, revealing differences from the mean force Gibbs state and conditions under which they coincide.

Contribution

It explicitly calculates the steady states of these Lindbladian QMEs and compares them with the mean force Gibbs state, highlighting the impact of enforcing complete positivity.

Findings

Steady states of Lindbladian QMEs differ from the MFG state.

Enforcing complete positivity alters the steady state away from MFG.

High-temperature regime leads to convergence to the Gibbs state under certain conditions.

Abstract

The Lindblad form guarantees complete positivity of a Markovian quantum master equation (QME). However, its microscopic derivation for a quantum system weakly interacting with a thermal bath requires several approximations, which may result in inaccuracies in the QME. Recently, various Lindbladian QMEs were derived without resorting to the secular approximation from the Redfield equation which does not guarantee the complete positivity. Here we explicitly calculate, in a perturbative manner, the equilibrium steady states of these Lindbladian QMEs. We compare the results with the steady state of the Redfield equation obtained from an analytic continuation method, which coincides with the so-called mean force Gibbs (MFG) state. The MFG state is obtained by integrating out the bath degrees of freedom for the Gibbs state of the total Hamiltonian. We explicitly show that the steady states of the Lindbladian QMEs are different from the MFG state. Our results indicate that manipulations of the Redfield equation needed to enforce complete positivity of a QME drives its steady state away from the MFG state. We also find that, in the high-temperature regime, both the steady states of the Lindbladian QMEs and MFG state reduce to the same Gibbs state of a system Hamiltonian under certain conditions.