Kossakowski, Andrzej, et al. ("Quantum detailed balance and KMS condition." Communications in Mathematical Physics 57.2 (1977): 97-110) gave a proof that the stationary state of a quantum dynamical semigroup arising from the coupling of a system Hamiltonian with non-degenerate spectrum and at least 6 states has a gibbs state as fixed point.
Spohn, Herbert, and Joel L. Lebowitz. ("Irreversible thermodynamics for quantum systems weakly coupled to thermal reservoirs." Advances in Chemical Physics: For Ilya Prigogine, Volume 38 (2007): 109-142.) remarked, in section III, that for systems with degenerate spectrum, one may have transitions between diagonal and off-diagonal elements.
Indeed, one can find examples in the literature which show precisely this phenomenon- some contingency of the quantum biology community likes to reference Agarwal, Girish S. Quantum statistical theories of spontaneous emission and their relation to other approaches. Springer Berlin Heidelberg, 1974, page 94, for instance. But this example requires fine tuning of the parameters. When one relaxes the fine tuning, the coherences go away.
For a time, there was interest in driving degenerate states into an entangled state via an effective interaction through a common bath- this effective interaction arises through the lamb shift and damping terms in the Lindblad generator. But the arguments presented in these papers tend to show some production of entanglement in the short time dynamics. Arguments about the long time dynamics are markedly absent.
Indeed, there seems to be a prevailing intuition that the effective Hamiltonian can't change the late time behavior, and one should just expect thermalization to the system hamiltonian's Gibbs state (basically because, as in Kossakowski et al, the transition rates in the detailed balance equation are set by the original system Hamiltonian's spectrum). Yet the literature on thermalization is adamant about requiring non-degenerate spectrum, and their is no GKLS type result about equilibration to Gibbs states save in the non-degenerate case.
My questions are: Is there an literature about the late time dynamics of degenerate systems with bath-mediated interactions? Is there any physics literature on the late time dynamics of a degenerate systems? Are there any examples of degenerate systems with non-gibbsian late time dynamics that don't require fine tuning of the bath parameters (as in Agarwal's example)?
