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arXiv:1910.02444

On the geometry of the classical Rabi problem

Heinz–Jürgen Schmidt

Published Oct 6, 2019• 0 citations
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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 rôle of spin and magnetic field can be interchanged, leading to the notion of "duality of loops" on the Bloch sphere.

DOI
10.48550/arxiv.1910.02444

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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 rôle of spin and magnetic field can be interchanged, leading to the notion of "duality of loops" on the Bloch sphere.
Abstract We consider the solution of the equation of motion of a classical/quantum spin subject to a monochromatical, elliptically polarized external field. The classical Rabi problem can be reduced to third-order differential equations with polynomial coefficients and hence solved in terms of power series in close analogy to the confluent Heun equation occurring for linear polarization. Application of Floquet theory yields physically interesting quantities like the quasienergy as a function of the problem’s parameters and expressions for the Bloch–Siegert shift of resonance frequencies. Various limit cases are thoroughly investigated.
Abstract In this article, we reconsider the periodically driven two-level system especially the Rabi problem with linear polarisation. 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 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 polarisation as well as asymptotic formulas for various limit cases.
The Bloch sphere concept is a useful tool to represent the actions of various quantum-mechanical operators on a spinor. Most importantly, it provides a link between the rather abstract concept of a spinor and the more intuitive way (although not rigorous) of thinking of the spin of an electron as being associated with a magnetic moment along the axis of self rotation. The Bloch sphere is particularly useful when describing the action of a spatially uniform external (including time-dependent) magnetic field on the spin [1]. This will be illustrated in the next chapter through the derivation of the Rabi formula. Bloch sphere concepts are also frequently invoked in discussions of spin-based quantum computing, so that a sound understanding of this notion is imperative to comprehend the literature in that field.