Asymptotic stochastic transformations for nonlinear quantum dynamical 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 (Schrödinger) 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.