Existence, uniqueness and approximation of stochastic Schrodinger equation: the diffusive case

Clement Pellegrini (1)

1 ICJ - Institut Camille Jordan

Recent developments in quantum physics make heavy use of so-called ``quantum trajectories''. Mathematically, this theory gives rise to ``stochastic Schrödinger equations'', that is, pertubations of Schrodinger-type equations under the form of stochastic differential equations. But such equations are in general not of the usual type as considered in the litterature. 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 Schrodinger equation. We physically justify the equations by proving that they are continuous time limit of a concrete physical procedure for obtainig quantum trajectory.

Mots clés: stochastic differential equation; diffusion approximation; quantum trajectory; stochastic Schrodinger equation

Domaines: Probabilités [math.PR]; Physique Quantique [quant-ph]

Clement Pellegrini. Existence, uniqueness and approximation of stochastic Schrodinger equation: the diffusive case. 2007. ⟨hal-00171129⟩