Introduction

The dynamics of a "small" system coupled to heat and/or particle reservoirs can sometimes be described by an effective stochastic differential equation. This has been rigorously established by Eckmann, Pillet and Rey-Bellet in the case of classical Hamiltonian systems, rationally coupled to a classical field. In the quantum case, the development of a general theory is currently the object of active research. Some aspects of the theory have been investigated in a research project on Open systems.

Part of the PhD thesis of Jean-Philippe Aguilar concerned the study of a quantum spin coupled to classical noise. The results, which are contained in this paper, describe the system's invariant measure, the speed of convergence to it, and a one-dimensional diffusion approximation of the transition probability.

Transition probability from spin down to spin up as a function of time, for two different realisations of an Ornstein-Uhlenbeck noise. The relaxation time to the invariant measure is of the order of 2000.

Same as above figure, but for larger noise intensity. The relaxation time to the invariant measure is now of the order of 200.