Perturbative Analysis of Dynamical Localisation

Abstract

In this paper we extend previous results on convergent perturbative solutions of the Schroedinger equation of a class of periodically time-dependent two-level systems. The situation treated here is particularly suited for the investigation of two-level systems exhibiting the phenomenon of (approximate) dynamical localisation. We also present a convergent perturbative expansion for the secular frequency and discuss in detail the particular case of monochromatic interactions (ac-dc fields), providing a complete perturbative solution for that case. Our method is based on a ``renormalisation'' procedure, which we develop in a more systematic way here. For being free of secular terms and uniformly convergent in time, our expansions allow a rigorous study of the long-time behaviour of such systems and are also well-suited for numerical computations, as we briefly discuss, leading to very accurate calculations of quantities like transition probabilities for very long times compared to the cycles of the external field.