On a class of stochastic differential equations used in quantum optics

TL;DR

This paper introduces a class of stochastic differential equations in Hilbert spaces used in quantum optics, explaining their derivation from quantum models and their relation to measurement theory, utilizing an isomorphism between Fock and Wiener spaces.

Contribution

It presents a new class of stochastic differential equations for quantum systems and details their derivation from quantum mechanical models and measurement processes.

Findings

Establishes properties of the proposed stochastic equations.

Connects quantum objects with probabilistic models via Fock-Wiener isomorphism.

Clarifies the relationship between quantum dynamics and stochastic calculus.

Abstract

Stochastic differential equations for processes with values in Hilbert spaces are now largely used in the quantum theory of open systems. In this work we present a class of such equations and discuss their main properties; moreover, we explain how they are derived from purely quantum mechanical models, where the dynamics is represented by a unitary evolution in a Hilbert space, and how they are related to the theory of continual measurements. An essential tool is an isomorphism between the bosonic Fock space and the Wiener space, which allow to connect certain quantum objects with probabilistic ones.