Semigroup representations, site couplings, and relaxation in quantum systems

The semigroup formalism permits a general description of relaxation processes in quantum mechanical systems coupled to Markovian baths. From its definition, the semigroup formalism should be expressed in terms of the dissipative and relaxation coupling of the eigenstates of the quantum system to the bath modes. Often, for convenience both in calculation and in physical interpretation, it is more straightforward to couple local states of the quantum mechanical system to the bath. Within a simple two-site system, we examine the generalities of these two different coupling schemes, noting situations in which they are the same and situations for which substantial differences occur.