Basic properties of nonlinear stochastic Schr\"{o}dinger equations driven by Brownian motions

TL;DR

This paper investigates the mathematical properties of nonlinear stochastic Schr"{o}dinger equations driven by Brownian motions, focusing on existence, uniqueness, and invariant measures, with applications to quantum oscillators.

Contribution

It establishes new criteria for the existence of regular solutions and invariant measures for NSSEs, advancing understanding of quantum stochastic dynamics.

Findings

Proved existence and uniqueness of regular solutions.

Derived criteria for invariant measures.

Applied results to a quantum oscillator model.

Abstract

The paper is devoted to the study of nonlinear stochastic Schr\"{o}dinger equations driven by standard cylindrical Brownian motions (NSSEs) arising from the unraveling of quantum master equations. Under the Born--Markov approximations, this class of stochastic evolutions equations on Hilbert spaces provides characterizations of both continuous quantum measurement processes and the evolution of quantum systems. First, we deal with the existence and uniqueness of regular solutions to NSSEs. Second, we provide two general criteria for the existence of regular invariant measures for NSSEs. We apply our results to a forced and damped quantum oscillator.