Potential renormalisation, Lamb shift and mean-force Gibbs state -- to shift or not to shift?

TL;DR

This paper examines the roles of counter terms and Lamb shift in open quantum systems, showing that neglecting them can sometimes improve accuracy and that under certain conditions they do not affect the dynamics.

Contribution

It provides a rigorous analysis of when counter terms and Lamb shifts can be neglected in master equations, clarifying their impact on system dynamics and steady states.

Findings

Counter term does not influence dissipative processes to second order in coupling.

Lamb-shift terms can cancel coherent effects of the counter term at large environmental cutoff.

Neglecting both terms can yield more accurate dynamics under specific conditions.

Abstract

Often, the microscopic interaction mechanism of an open quantum system gives rise to a `counter term' which renormalises the system Hamiltonian. Such term compensates for the distortion of the system's potential due to the finite coupling to the environment. Even if the coupling is weak, the counter term is, in general, not negligible. Similarly, weak-coupling master equations feature a number of `Lamb-shift terms' which, contrary to popular belief, cannot be neglected. Yet, the practice of vanishing both counter term and Lamb shift when dealing with master equations is almost universal; and, surprisingly, it can yield better results. By accepting the conventional wisdom, one may approximate the dynamics more accurately and, importantly, the resulting master equation is guaranteed to equilibrate to the correct steady state in the high-temperature limit. In this paper we discuss why is this the case. Specifically, we show that, if the potential distortion is small -- but non-negligible -- the counter term does not influence any dissipative processes to second order in the coupling. Furthermore, we show that, for large environmental cutoff, the Lamb-shift terms approximately cancel any coherent effects due to the counter term -- this renders the combination of both contributions irrelevant in practice. We thus provide precise conditions under which the open-system folklore regarding Lamb shift and counter terms is rigorously justified.