Geometric phases in quantum control disturbed by classical stochastic processes

TL;DR

This paper investigates how classical stochastic processes, like noise, influence geometric (Berry) phases in quantum systems, introducing a composite bundle framework to analyze the combined effects of control parameters and stochastic disturbances.

Contribution

It introduces a novel composite bundle structure to describe geometric phases affected by classical stochastic processes in quantum control systems.

Findings

Decomposition of total geometric phase into control and stochastic components

Explicit construction of the composite bundle framework

Calculation of average geometric phase under Gaussian white noise

Abstract

We describe the geometric (Berry) phases arising when some quantum systems are driven by control classical parameters but also by outer classical stochastic processes (as for example classical noises). The total geometric phase is then divided into an usual geometric phase associated with the control parameters and a second geometric phase associated with the stochastic processes. The geometric structure in which these geometric phases take place is a composite bundle (and not an usual principal bundle), which is explicitely built in this paper. We explain why the composite bundle structure is the more natural framework to study this problem. Finally we treat a very simple example of a two level atom driven by a phase modulated laser field with a phase instability described by a gaussian white noise. In particular we compute the average geometric phase issued from the noise.