Existence, uniqueness and approximation of a stochastic Schr\"odinger equation: the diffusive case

TL;DR

This paper proves the existence and uniqueness of a specific stochastic Schr"odinger equation in the diffusive case for a two-level quantum system and justifies its physical relevance as a continuous-time limit of a physical procedure.

Contribution

It establishes mathematical existence and uniqueness results for the diffusive stochastic Schr"odinger equation and links it to a physical quantum trajectory process.

Findings

Proves existence and uniqueness of the diffusive stochastic Schr"odinger equation.

Shows the equation is a continuous-time limit of a physical quantum measurement process.

Provides a rigorous mathematical foundation for quantum trajectory models.

Abstract

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.