Recent developments in quantum physics make heavy use of so-called “quantum trajectories.” Mathematically, this theory gives rise to “stochastic Schrödinger equations,” that is, perturbation of Schrödinger-type equations under the form of stochastic differential equations. But such equations are in general not of the usual type as considered in the literature. They pose a serious problem in terms of justifying the existence and uniqueness of a solution, justifying the physical pertinence of the equations. In this article we concentrate on a particular case: the diffusive case, for a two-level system. We prove existence and uniqueness of the associated stochastic Schrödinger equation. We physically justify the equations by proving that they are a continuous-time limit of a concrete physical procedure for obtaining a quantum trajectory.
"Existence, uniqueness and approximation of a stochastic Schrödinger equation: The diffusive case." Ann. Probab. 36 (6) 2332 - 2353, November 2008. https://doi.org/10.1214/08-AOP391