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

TL;DR

This work clarifies why renormalising the system Hamiltonian by subtracting the reorganisation energy and removing Lamb-shift terms—practices common in weak-coupling master equations—often yields accurate dynamics and correct high-temperature equilibria. By analyzing a damped harmonic oscillator model, the authors show that in the adiabatic regime with a large environmental cutoff, the Lamb-shift contributions effectively cancel the coherent effects of the counter term, leaving a tractable second-order Redfield equation that converges to the mean-force Gibbs state $\pmb{\tau}_{MF}$ at high $T$. They derive precise conditions under which this artefact is rigorously justified, and demonstrate via exact solutions that the approach remains accurate beyond strict classical limits for a wide parameter range. The results support the conventional methodology for obtaining thermodynamically consistent steady states and offer guidance on when to rely on or deviate from the standard Lamb-shift removal and potential renormalisation. Practically, this informs modeling strategies for quantum thermodynamics and steady-state currents in harmonic networks.

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.

Figures (2)

• Figure 1: Dynamics of a damped harmonic oscillator. Exact thermalising dynamics of the position variance ${\langle \pmb{x}^2(t) \rangle}$ relative to $\langle\pmb{x}^2\rangle_\infty$, for an oscillator with Hamiltonian ${\pmb{H}_S = \pmb{H}_S^{(0)} + \frac{1}{2}\delta\omega^2\,\pmb{x}^2}$ (solid black) at an intermediate coupling of $\lambda = 0.1\,\omega_0$. The Redfield equation \ref{['eq:Redfield-explicit-combo']} derived from this Hamiltonian (dashed Blue) is accurate in the classical regime, as shown in panel (b), but breaks down at moderate and low temperatures (see (a)). Further performing the secular approximation yields even worse results (dot-dashed green). Specifically, the resulting GKLS equation forces the system into a thermal state with respect to ${\pmb{H}_S = \pmb{H}_S^{(0)} + \frac{1}{2}\delta\omega^2\,\pmb{x}^2}$, which is incompatible with the correct high-temperature limit from Eq. \ref{['eq:semi-classical-state']}. On the contrary, a Redfield equation derived from ${\pmb{H}_S = \pmb{H}_S^{(0)}}$, where the counter term is removed by an adding the reorganisation energy, yields excellent results provided that all Lamb-shift terms are eliminated (dashed yellow). The secular version of this modified equation also tracks the dynamics very reliably (dot-dashed orange). Both of these equations succeed in bringing the system close to the correct mean-force Gibbs state (red dotted line) over a very broad temperature range. The parameters are $\omega_0=1, \lambda=0.1$ and $\Lambda=100$. Hence, $\delta\omega^2 = 10$. The initial state chosen is $\langle\pmb x(0)\rangle = \langle\pmb{p}^2(0)\rangle = 1$, $\langle\pmb p(0)\rangle = \langle\pmb{x}^2(0)\rangle = 2$, $\frac{1}{2}\langle \{\pmb{x}(0),\pmb{p}(0)\} \rangle=\frac{1}{2}$. Recall that we work in units of $\hbar = k = 1$ . In panel (a) $T=1$ and in (b) $T=10$.
• Figure 2: Benchmarking the steady state of the master equations. Uhlmann fidelity between the exact steady state $\pmb{\tau}_{MF}$ and the asymptotic state resulting from different master equations, for varying temperature and system--bath coupling. Just like in Fig. \ref{['fig1']} the microscopic model includes an explicit counter term. In (a) the resulting full Redfield equation---Lamb shift included---is benchmarked against the mean-force Gibbs state. As we see, the agreement is good as long as the temperature is not too low, or the coupling too strong. In (b) we performed the secular approximation, which yields the thermal state corresponding to $\pmb{H}_S = \pmb{H}_S^{(0))} + Q\,\pmb{S}^2$ asymptotically. This is a very bad approximation to $\pmb{\tau}_{MF}$ due to the large reorganisation energy in our example. Finally, in (c) we cancel the counter term and vanish the Lamb shift. The resulting steady state is the high-temperature limit from Eq. \ref{['eq:semi-classical-state']}, which turns out to be an excellent approximation to the mean-force Gibbs, other than at very low temperatures.