Ultrastrong coupling limit to quantum mean force Gibbs state for anharmonic environment

TL;DR

This paper derives analytical expressions for the mean force Gibbs state in the ultrastrong coupling regime for generalized quantum system-environment models, extending beyond the harmonic environment assumption of the Caldeira-Leggett model.

Contribution

It provides analytical formulas for the USC mean force Gibbs state for more general SE models, revealing conditions under which the CL results remain valid.

Findings

USC-MFGS is diagonal in the SE interaction basis.

The USC-MFGS differs from CL results in generic models.

Numerical verification supports the analytical expressions.

Abstract

The equilibrium state of a quantum system can deviate from the Gibbs state if the system-environment (SE) coupling is not weak. An analytical expression for this mean force Gibbs state (MFGS) is known in the ultrastrong coupling (USC) regime for the Caldeira-Leggett (CL) model that assumes a harmonic environment. Here, we derive analytical expressions for the MFGS in the USC regime for more general SE models. For all the generalized models considered here, we find the USC state to be diagonal in the basis set by the SE interaction, just like in the CL case. While for the generic model considered, the corresponding USC-MFGS is found to alter from the CL-result, we do identify a class of models more general than the CL model for which the CL-USC result remains unchanged. We also provide numerical verification for our results. These results provide key tools for the study of strong coupling quantum thermodynamics and several quantum chemistry and biology problems under more realistic SE models, going beyond the CL model.