Quantum Stratonovich Stochastic Calculus and the Quantum Wong-Zakai Theorem

TL;DR

This paper develops a quantum Stratonovich calculus framework, extending stochastic differential equations to include scattering processes and establishing connections to classical perturbation theory and Wong-Zakai theorems.

Contribution

It introduces a quantum Stratonovich calculus as an algebraic modification of Ito calculus, including scattering processes, and proves its applicability to multi-dimensional quantum systems.

Findings

Defined Stratonovich calculus within quantum stochastic calculus.

Derived a conversion formula resembling Green's functions.

Extended Wong-Zakai type limit results to multiple dimensions.

Abstract

We extend the Ito -to- Stratonovich analysis or quantum stochastic differential equations, introduced by Gardiner and Collett for emission (creation), absorption (annihilation) processes, to include scattering (conservation) processes. Working within the framework of quantum stochastic calculus, we define Stratonovich calculus as an algebraic modification of the Ito one and give conditions for the existence of Stratonovich time-ordered exponentials. We show that conversion formula for the coefficients has a striking resemblance to Green's function formulae from standard perturbation theory. We show that the calculus conveniently describes the Markov limit of regular open quantum dynamical systemsin much the same way as in the Wong-Zakai approximation theorems of classical stochastic analysis. We extend previous limit results to multiple-dimensions with a proof that makes use of diagrammatic conventions.