On the geometry of the classical Rabi problem

TL;DR

This paper explores the geometric and dynamical properties of a classical spin in a periodic magnetic field, introducing the concept of duality of loops on the Bloch sphere through Floquet theory and differential geometry.

Contribution

It introduces the notion of duality of loops on the Bloch sphere and applies Floquet theory and differential geometry to analyze classical spin dynamics.

Findings

Duality of loops on the Bloch sphere is established.

Conditions for interchangeability of spin and magnetic field roles are identified.

Examples illustrate the geometric properties of the classical Rabi problem.

Abstract

We investigate the motion of a classical spin precessing around a periodic magnetic field using Floquet theory as well as elementary differential geometry and considering a couple of examples. Under certain conditions the r\^{o}le of spin and magnetic field can be interchanged, leading to the notion of "duality of loops" on the Bloch sphere.