Location of crossings in the Floquet spectrum of a driven two-level system

TL;DR

This paper investigates how the positions of crossings in the Floquet quasi-energy spectrum of a driven two-level system change with varying field parameters, providing analytical relations and an empirical formula for the drift of these crossings.

Contribution

It introduces a perturbational approach linking driving field forms to quasi-energies and derives an empirical formula describing crossing drift at lower frequencies.

Findings

Derived relations for quasi-energies in high-frequency limit.

Established an empirical formula for crossing drift.

Found the formula to be exact for square-wave driving.

Abstract

Calculation of the Floquet quasi-energies of a system driven by a time-periodic field is an efficient way to understand its dynamics. In particular, the phenomenon of dynamical localization can be related to the presence of close approaches between quasi-energies (either crossings or avoided crossings). We consider here a driven two-level system, and study how the locations of crossings in the quasi-energy spectrum alter as the field parameters are changed. A perturbational scheme provides a direct connection between the form of the driving field and the quasi-energies which is exact in the limit of high frequencies. We firstly obtain relations for the quasi-energies for some common types of applied field in the high-frequency limit. We then show how the locations of the crossings drift as the frequency is reduced, and find a simple empirical formula which describes this drift extremely well in general, and appears to be exact for the specific case of square-wave driving.