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Zeitschrift für Naturforschung A

Geometry of the Rabi Problem and Duality of Loops

Heinz–Jürgen Schmidt

Published Feb 14, 2020• 2 citations
Publisher

Abstract

Abstract We investigate the motion of a classical spin processing around a periodic magnetic field using Floquet theory, as well as elementary differential geometry and considering a couple of examples. Under certain conditions, the role of spin and magnetic field can be interchanged, leading to the notion of “duality of loops” on the Bloch sphere.

DOI
10.1515/zna-2019-0352

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Abstract We investigate the motion of a classical spin processing around a periodic magnetic field using Floquet theory, as well as elementary differential geometry and considering a couple of examples. Under certain conditions, the role 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.
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.
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 important, 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.
In this chapter, we will consider the general approach used to calculate the time evolution of the probability of a spinor reaching a particular location on the Bloch sphere from another location, when subjected to the simultaneous actions of two spatially uniform magnetic fields. One of these fields is static (time-independent) and directed along a specific axis, while the other is timedependent, has a different amplitude, and rotates in the plane perpendicular to the first magnetic field. This problem was originally considered by I. I. Rabi who calculated the probability of a “spin flip,” i.e., the probability for a spinor, originally located at the north pole on the Bloch sphere, to ultimately reach the south pole [1, 2, 3]. Rabi calculated the time evolution of the probability in the case of a constant (dc) magnetic field along the z-axis and a circularly polarized time-varying (ac) magnetic field of different amplitude, with a specific frequency, rotating in the (x-y) plane. This led to the derivation of what is known as the Rabi formula which is one of the key ingredients in Nuclear Magnetic Resonance (NMR). In most textbooks, the Rabi formula is derived by solving the time-dependent Pauli equation [Equation (2.31)] with a judicious choice of the reference frame to describe the time-evolution of the 2-component spinor wavefunction. Here, we use a different approach based on the concept of the Bloch sphere described in the previous chapter. This approach is more instructive since it directly relates to NMR concepts and is more synergistic with concepts used in spin-based quantum computing.