On the Asymptotic Behaviour of Some Stochastic Differential Equations for Quantum States

Abstract

In this article we study the long time behaviour of a class of stochastic differential equations introduced in the theory of measurements continuous in time for quantum open systems. Such equations give the time evolution of the a posteriori states for a system underlying a continual measurement. First of all we give conditions for the equation to preserve pure states and then, in the case of a finite-dimensional Hilbert space, we obtain sufficient conditions from which the stochastic equation for a posteriori states is ensured to map, for t → +∞, mixed states into pure ones. Finally we study existence and uniqueness of an invariant measure for the equations which preserve pure states. We give a general theorem for the purely diffusive case, again in the finite-dimensional case; we then apply it to some physical examples. For the purely jump case, an example is discussed in which the invariant measure exists and is unique.