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

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.

Correlation-enabled energy exchange in quantum systems without external driving

“…where γ A,B mn are jump rates. We consider here the case where the dissipators have thermal steady states [29]. To this end, we assume that the transition rates satisfy the detailed-balance condition…”

Fluctuation relation for heat exchange in Markovian open quantum systems

“…Conditions under which a Lindbladian dynamics can yield a stationary state which is a thermal equilibrium state in a Gibbsian form have been studied in the literature extensively [3,4,[20][21][22]. Here we follow the formalism and context laid out recently in Ref.…”

“…Here we follow the formalism and context laid out recently in Ref. [20]. Here the conditions for the evolution (4)-with L α s chosen as L mn = l mn |m n| (m = n)-to have the stationary solution of the thermal form…”

“…where C mn = C nm > 0 and C mn s depend on the interaction of the system and the heat bath. From these relations it follows that the diagonal elements of have an evolution decoupled from that of off-diagonal terms [20]. Defining the vector |v(τ ) = m mm (τ )|m , we have the following evolution:…”

Correlation-Enabled Energy Exchange in Quantum Systems without External Driving

“…where 𝛾 A,B 𝑚𝑛 are jump rates. We consider here the case where the dissipators have thermal steady states [29]. To this end, we assume that the transition rates satisfy the detailed-balance condition…”