Sochastic Schrödinger equations and memory

By starting from the stochastic Schröödinger equation and quantum trajectory theory, we introduce memory effects by considering stochastic adapted coefficients. As an example of a natural non-Markovian extension of the theory of white noise quantum trajectories we use an Ornstein-Uhlenbeck coloured noise as the output driving process. Under certain conditions a random Hamiltonian evolution is recovered. Moreover, we show that our non-Markovian stochastic Schröödinger equations unravel some master equations with memory kernels.