RETURN TO EQUILIBRIUM FOR SOME STOCHASTIC SCHRÖDINGER EQUATIONS

The principle of "indirect continuous measurement" in "open quantum system theory" is usually described by non-usual types of stochastic differential equations. These equations are called "stochastic Schrödinger equations" and their solutions are called "quantum trajectories". Physically, they describe the random evolution of the state of a quantum system undergoing indirect quantum measurement (such models are widely used in quantum optics, quantum computing and quantum information theory). In this chapter, we consider a physically realistic discrete-time setup for two-level quantum systems and we present the theory of "discrete quantum trajectories". These discrete trajectories are Markov chains which can be expressed as solutions of "discrete-time" stochastic differential equations". In particular, these equations appear as time discretization of "stochastic Schrödinger equations". Going to the continuous-time limit, we justify the stochastic Schrödinger equations associated to the two-level systems. Within this approach, we obtain two different types of behaviors described either by jump-type or diffusive-type stochastic differential equations. Finally we investigate the large time behavior of the solutions and we prove return to equilibrium properties for the associated physical models