Quantum White Noises and The Master Equation for Gaussian Reference States
John Gough
TL;DR
This work develops a minimal quantum white-noise framework that reproduces quantum stochastic calculus when proper normal or time ordering is enforced, and extends the analysis to Gaussian reference states through a generalized Araki–Woods construction. By formulating normal- and time-ordered QSDEs and providing explicit conversion rules between these forms, the paper derives the associated quantum Langevin-type equations and master equations for Gaussian states, including thermal and squeezed cases. It further constructs Gaussian noise via a two-copy scheme, derives the Hudson–Parthasarathy QSDE, and presents the corresponding Itô table and stochastic Heisenberg equation, culminating in a master equation for reduced dynamics. The result is a comprehensive, order-aware framework for open quantum systems with Gaussian references, with potential applications in quantum optics and quantum statistical mechanics.
Abstract
We show that a basic quantum white noise process formally reproduces quantum stochastic calculus when the appropriate normal / chronological orderings are prescribed. By normal ordering techniques for integral equations and a generalization of the Araki-Woods representation, we derive the master and random Heisenberg equations for an arbitrary Gaussian state: this includes thermal and squeezed states.