Asymptotic Stochastic Transformations for Nonlinear Quantum Dynamical Systems

TL;DR

This paper extends classical stochastic calculus to quantum systems, introducing a Stratonovich form for quantum SDEs driven by Poisson processes and establishing asymptotic relations between nonlinear quantum ODEs and SDEs.

Contribution

It introduces the Stratonovich form for quantum SDEs with Poisson processes and generalizes classical stochastic calculus transformations to the quantum domain.

Findings

Stratonovich form for quantum SDEs with Poisson processes introduced

Relation between nonlinear quantum ODEs and asymptotic quantum SDEs established

Extension of Khasminskii theorem to quantum stochastic systems

Abstract

The Ito and Stratonovich approaches are carried over to quantum stochastic systems. Here the white noise representation is shown to be the most appropriate as here the two approaches appear as Wick and Weyl orderings, respectively. This introduces for the first time the Stratonovich form for SDEs driven by Poisson processes or quantum SDEs including the conservation process. The relation of the nonlinear Heisenberg ODES to asymptotic quantum SDEs is established extending previous work on linear (Schrodinger) equations. This is shown to generalize the classical integral transformations between the various forms of stochastic calculi and to extend the Khasminskii theorem to the quantum setting.