Time Evolution of Two-Level Systems Driven by Periodic Fields

Abstract

In this paper we study the time evolution of a class of two-level systems driven by periodic fields in terms of new convergent perturbative expansions for the associated propagator U(t). The main virtue of these expansions is that they do not contain secular terms, leading to a very convenient method for quantitatively studying the long-time behaviour of that systems. We present a complete description of an algorithm to numerically compute the perturbative expansions. In particular, we applied the algorithm to study the case of an ac-dc field (monochromatic interaction), exploring various situations and showing results on (time-dependent) observable quantities, like transition probabilities. For a simple ac field, we analised particular situations where an approximate effect of dynamical localisation is exhibited by the driven system. The accuracy of our calculations was tested measuring the unitarity of the propagator U(t), resulting in very small deviations, even for very long times compared to the cycle of the driving field.