Corrections to the Hamiltonian induced by finite-strength coupling to the environment

TL;DR

This paper derives analytical formulas for second-order Hamiltonian corrections induced by finite system-environment coupling, including Lamb-shift, mean-force, and quasi-steady state corrections, across various master equations and models.

Contribution

It provides a unified framework for calculating second-order Hamiltonian corrections for different master equations and introduces formulas for the quasi-steady state correction.

Findings

Derived general formulas for mean-force, Lamb-shift, and quasi-steady state corrections.

Applied formulas to Davies, Bloch-Redfield, and cumulant equations.

Analyzed spin-boson model and reaction-coordinate approach for corrections.

Abstract

If a quantum system interacts with the environment, then the Hamiltonian acquires a correction known as the Lamb-shift term. There are two other corrections to the Hamiltonian, related to the stationary state. Namely, the stationary state is to first approximation a Gibbs state with respect to original Hamiltonian. However, if we have finite coupling, then the true stationary state will be different, and regarding it as a Gibbs state to some effective Hamiltonian, one can extract a correction, which is called "steady-state" correction. Alternatively, one can take a static point of view, and consider the reduced state of total equilibrium state, i.e., system plus bath Gibbs state. The extracted Hamiltonian correction is called the "mean-force" correction. This paper presents several analytical results on second-order corrections (in coupling strength) of the three types mentioned above. Instead of the steady state, we focus on a state annihilated by the Liouvillian of the master equation, labeling it as the "quasi-steady state." Specifically, we derive a general formula for the mean-force correction as well as the quasi-steady state and Lamb-shift correction for a general class of master equations. Furthermore, specific formulas for corrections are obtained for the Davies, Bloch-Redfield, and cumulant equation (refined weak coupling). In particular, the cumulant equation serves as a case study of the Liouvillian, featuring a nontrivial fourth-order generator. This generator forms the basis for calculating the diagonal quasi-steady-state correction. We consider spin-boson model as an example, and in addition to using our formulas for corrections, we consider mean-force correction from the reaction-coordinate approach.