The Floquet theory of the two level system revisited

TL;DR

This paper revisits the Floquet theory applied to the driven two-level quantum system, especially the Rabi problem with linear polarization, providing a classical geometric perspective and analytical approximations.

Contribution

It offers a classical Hamiltonian approach to analyze the Floquet theory of the two-level system, deriving new analytical approximations and clarifying the resonance condition.

Findings

Resonance condition linked to vanishing average of a classical solution component.

Analytical approximations for physical quantities in the Rabi problem.

Asymptotic formulas for various limiting cases.

Abstract

We reconsider the periodically driven two level system and especially the Rabi problem with linear polarization. The Floquet theory of this problem can be reduced to its classical limit, i.e., to the investigation of periodic solutions of the classical Hamiltonian equations of motion in the Bloch sphere. The quasienergy is essentially the action integral over one period and the resonance condition due to J.H. Shirley is shown to be equivalent to the vanishing of the time average of a certain component of the classical solution. This geometrical approach is applied to obtain analytical approximations to physical quantities of the Rabi problem with linear polarization as well as asymptotic formulas for various limit cases.