Steady state in ultrastrong coupling regime: perturbative expansion and first orders

TL;DR

This paper derives analytical first-order corrections to the mean force Gibbs state in the ultrastrong coupling regime, confirming their validity through numerical comparisons and linking steady states to dynamical approaches.

Contribution

It provides the first analytical expressions for first-order corrections to the mean force Gibbs state in the ultrastrong coupling regime, connecting static and dynamical descriptions.

Findings

First-order correction matches dynamical approach results.

Good numerical agreement at high temperatures and ultrastrong coupling.

Results illustrate the transition from ultrastrong to weak coupling in coherences.

Abstract

Understanding better the dynamics and steady states of systems strongly coupled to thermal baths is a great theoretical challenge with promising applications in several fields of quantum technologies. Among several strategies to gain access to the steady state, one consists in obtaining approximate expressions of the mean force Gibbs state, the reduced state of the global system-bath thermal state, largely credited to be the steady state. Here, we present analytical expressions of corrective terms to the ultrastrong coupling limit of the mean force Gibbs state, which has been recently derived. We find that the first order term precisely coincides with the first order correction obtained from a dynamical approach -- master equation in the strong-decoherence regime. This strengthens the identification of the reduced steady state with the mean force Gibbs state. Additionally, we also compare our expressions with another recent result obtained from a high temperature expansion of the mean force Gibbs state. We observe numerically a good agreement for ultra strong coupling as well as for high temperatures. This confirms the validity of all these results. In particular, we show that, in term of coherences, all three results allow one to sketch the transition from ultrastrong coupling to weak coupling.