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

Luis A. Correa; Jonas Glatthard

doi:10.22331/q-2025-09-30-1868

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

Luis A. Correa^1,2,3 and Jonas Glatthard³

¹Instituto Universitario de Estudios Avanzados (IUdEA)
²Sección de Física, Facultad de Ciencias, Universidad de La Laguna, La Laguna 38203, Spain
³Department of Physics and Astronomy, University of Exeter, Exeter EX4 4QL, United Kingdom

Published:	2025-09-30, volume 9, page 1868
Editor:	Abhinav Deshpande
Eprint:	arXiv:2305.08941v4
Doi:	https://doi.org/10.22331/q-2025-09-30-1868
Citation:	Quantum 9, 1868 (2025).

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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.

Featured image: Time evolution of the position variance of a quantum Brownian particle according the Redfield and secular master equations derived from a Hamiltonian with and without counter term, as compared with the exact dynamics.

► BibTeX data

@article{Correa2025potential,
  doi = {10.22331/q-2025-09-30-1868},
  url = {https://doi.org/10.22331/q-2025-09-30-1868},
  title = {Potential renormalisation, {L}amb shift and mean-force {G}ibbs state – to shift or not to shift?},
  author = {Correa, Luis A. and Glatthard, Jonas},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1868},
  month = sep,
  year = {2025}
}

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