Thermalization of noninteracting quantum systems coupled to blackbody radiation: A Lindblad-based analysis

TL;DR

This paper investigates the thermalization process of noninteracting quantum systems coupled to blackbody radiation, highlighting issues with standard master equations and proposing a Lindblad-based approach that accounts for degeneracies and ensemble size.

Contribution

It introduces a Lindblad-based thermalization framework that addresses degeneracies and basis arbitrariness, providing explicit formulas for dissipation and decoherence times.

Findings

Standard quantum optical master equations can exhibit pathologies due to degeneracies.

A Lindblad approach with fixed amplitudes and detailed balance yields consistent thermalization.

Thermalization times depend explicitly on system energy levels, ensemble size, and temperature.

Abstract

We study the thermalization of an ensemble of $N$ elementary, arbitrarily-complex, quantum systems, mutually noninteracting but coupled as electric or magnetic dipoles to a blackbody radiation. The elementary systems can be all the same or belong to different species, distinguishable or indistinguishable, located at fixed positions or having translational degrees of freedom. Even if the energy spectra of the constituent systems are nondegenerate, as we suppose, the ensemble unavoidably presents degeneracies of the energy levels and/or of the energy gaps. We show that, due to these degeneracies, a thermalization analysis performed by the popular quantum optical master equation reveals a number of serious pathologies, possibly including a lack of ergodicity. On the other hand, a consistent thermalization scenario is obtained by introducing a Lindblad-based approach, in which the Lindblad operators, instead of being derived from a microscopic calculation, are established as the elements of an operatorial basis with squared amplitudes fixed by imposing a detailed balance condition and requiring their correspondence with the dipole transition rates evaluated under the first-order perturbation theory. Due to the above-mentioned degeneracies, this procedure suffers a basis arbitrariness which, however, can be removed by exploiting the fact that the thermalization of an ensemble of noninteracting systems cannot depend on the ensemble size. As a result, we provide a clear-cut partitioning of the thermalization time into dissipation and decoherence times, for which we derive formulas giving the dependence on the energy levels of the elementary systems, the size $N$ of the ensemble, and the temperature of the blackbody radiation.