Semigroup representations, site couplings, and relaxation in quantum systems

The semigroup formalism permits a general description of relaxation process es in quantum mechanical systems coupled to Markovian baths. From its defin ition, the semigroup formalism should be expressed in terms of the dissipat ive and relaxation coupling of the eigenstates of the quantum system to the bath modes. Often, for convenience both in calculation and in physical int erpretation, it is more straightforward to couple local states of the quant um mechanical system to the bath. Within a simple two-site system, we exami ne the generalities of these two different coupling schemes, noting situati ons in which they are the same and situations for which substantial differe nces occur.