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

We study the thermalization of an ensemble ofNelementary, arbitrarily complex, quantum systems, mutuallynoninteracting but coupled as electric or magnetic dipoles to a blackbody radiation. The elementary systems canbe all the same or belong to different species, distinguishable or indistinguishable, located at fixed positions orhaving 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 opticalmaster equation reveals a number of serious pathologies, possibly including a lack of ergodicity. On the otherhand, a consistent thermalization scenario is obtained by introducing a Lindblad-based approach, in which theLindblad operators, instead of being derived from a microscopic calculation, are established as the elementsof an operatorial basis with squared amplitudes fixed by imposing a detailed balance condition and requiringtheir correspondence with the dipole transition rates evaluated under the first-order perturbation theory. Due tothe above-mentioned degeneracies, this procedure suffers a basis arbitrariness, which, however, can be removedby exploiting the fact that the thermalization of an ensemble of noninteracting systems cannot depend on theensemble size. As a result, we provide a clear-cut partitioning of the thermalization time into dissipation anddecoherence times, for which we derive formulas giving the dependence on the energy levels of the elementarysystems, the sizeNof the ensemble, and the temperature of the blackbody radiation