Perturbative Analysis of Dynamical Localisation

TL;DR

This paper develops a systematic, convergent perturbative method to analyze dynamical localisation in periodically driven two-level quantum systems, enabling accurate long-time predictions and numerical computations.

Contribution

It introduces a renormalisation-based perturbative approach that is free of secular terms and convergent, specifically applied to systems exhibiting dynamical localisation.

Findings

Provides a convergent perturbative expansion for secular frequency.

Develops a complete solution for monochromatic (ac-dc) fields.

Enables precise long-time numerical calculations of transition probabilities.

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.