Adiabatic theorem for a class of quantum stochastic equations

TL;DR

This paper develops an adiabatic theory for a class of quantum stochastic differential equations, linking stationary states of different operators and applying it to analyze tunneling statistics in a dephasing quantum system.

Contribution

It introduces a novel adiabatic framework for quantum stochastic equations where stationary states coincide, and applies it to derive tunneling statistics in a driven dephasing quantum system.

Findings

Derived an adiabatic theory for quantum stochastic equations.

Established conditions linking stationary states of different operators.

Analyzed tunneling statistics in a driven stochastic Schrödinger equation.

Abstract

We derive an adiabatic theory for a stochastic differential equation, $ \varepsilon\, \mathrm{d} X(s) = L_1(s) X(s)\, \mathrm{d} s + \sqrt{\varepsilon} L_2(s) X(s) \, \mathrm{d} B_s, $ under a condition that instantaneous stationary states of $L_1(s)$ are also stationary states of $L_2(s)$. We use our results to derive the full statistics of tunneling for a driven stochastic Schr\"{o}dinger equation describing a dephasing process.