Exactly Solvable Models

Abstract

To understand the limit within which master equations are valid, it is quite instructive to compare the master equation results against exactly solvable models. Unfortunately, these models are quite rare. In this chapter, we will discuss two popular representatives of exactly solvable models: first, we investigate a pure dephasing spin-boson model, where the interaction Hamiltonian commutes with the system Hamiltonian. Such models obviously leave the system energy invariant but nevertheless may be used to investigate interesting features such as decoherence. Second, we consider a noninteracting model, where the Hamiltonian can be written as a quadratic form of fermionic annihilation and creation operators. Such models generally admit—at least formally—an exact solution, and can thus be used to study non-equilibrium setups and transport in a regime where the coupling between system and reservoir becomes strong. Furthermore, we note that the non-equilibrium stationary solution of these models may also define a non-equilibrium reservoir.