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Breaking of the Similarity Principle in Markov Generators of Low Density Limit Type and the Role of Degeneracies in the Landscape of Invariant States

The similarity principle is an extension of the principle of thermal relaxation that naturally arises in the stochastic limit of quantum theory. We construct examples of Low Density Limit (LDL) generators, associated to an environment state in equilibrium at inverse temperature β, which admit non-(β, HS)-equilibrium states. We prove that in some cases, the attraction domain of the (β, HS)-equilibrium state is empty. This means that the similarity principle, in its original thermodynamical formulation, can be broken in the LDL limit. This result is obtained as a consequence of a more general phenomenon: the role of degeneracies in the spectrum of the Liouvillian of the system Hamiltonian associated to the generator. We start from the definition of LDL type generators given in [5] and we introduce a finer classification of these generators based on the above degeneracies. The simplest subclass, called 2-generic, is a nontrivial extension of the generators associated to the so-called Λ and V configurations, widely used in quantum optics and involving 2 levels of the system Hamiltonian. Since each 2-generic block involves 3 or 4 levels of the system Hamiltonian we expect that they can reveal some interesting new physical phenomenon, as it happened in the 2-level case. In the last section, we restrict our attention to a 3-level system with a Hamiltonian that is associated to a class of 2-generic LDL generators. Finally, we prove that, for some LDL generators in this class the statement formulated at the beginning holds true.