Mathematics – Probability
Scientific paper
2007-09-11
Annals of Probability 2008, Vol. 36, No. 6, 2332-2353
Mathematics
Probability
Published in at http://dx.doi.org/10.1214/08-AOP391 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Scientific paper
10.1214/08-AOP391
Recent developments in quantum physics make heavy use of so-called "quantum trajectories." Mathematically, this theory gives rise to "stochastic Schr\"odinger equations", that is, perturbation of Schr\"odinger-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 Schr\"odinger 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.
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