Geometric phases in quantum control disturbed by classical stochastic processes

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 explicitly 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.