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arXiv:2006.16948 · Zeitschrift für Naturforschung A

The Rabi problem with elliptical polarization

Heinz–Jürgen Schmidt

Published Sep 24, 2020• 1 citation
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Abstract

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.

DOI
10.1515/zna-2020-0181

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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.
We studied the two-level spin resonance in a new perspective. Using the Floquet theory, the periodic interaction Hamiltonians were transfromed into a time-independent interaction. Using the Brillouin-Wigner perturbation method, a degenerated subspace is constructed, where the effective Hamiltonian is given in a perturbation expansion. In this framework, we found that the upper triangular element ⟨α∣H1∣β⟩\langle α| H^1 | β\rangle⟨α∣H1∣β⟩, determines whether the resonance happens. The generalized Rabi frequency and the Bloch-Siegert shift were solved straightforwardly as the first order and the second order solution, proving the benefit of the developed method.
The PT -symmetric semiclassical Rabi model explores the fundamental interaction between a two-level atom and a classical field, revealing novel phenomena in open systems through the inclusion of non-Hermitian terms. We propose a single similarity transformation that yields an effective Hamiltonian in rotating-wave approximation, enabling an analytical solution. The phase boundary of the PT -broken phase, derived from the analytical eigenvalues, closely matches the numerically exact one over a wide range of atomic frequencies, demonstrating the effectiveness of the analytical approach, especially at the main resonance. The Floquet parity operator is also introduced, providing a deeper physical understanding of the emergence of the PT -broken phase. Furthermore, by analyzing the dynamics of excited-state population, we observe several stable oscillations in the Fourier spectrum, demonstrating the applicability of the analytical method beyond the single-photon resonance region. The Bloch-Siegert shift is also discussed, and surprisingly, resembles its Hermitian counterpart, except for the higher-order terms in the coupling strength. The present analytical treatment provides a concise and accurate description of the main physics of this non-Hermitian atom-field interaction system.
Abstract The problem in question is to find the motion of an atom having arbitrary spin J in a strong oscillating magnetic field and a static field whose component perpendicular to the oscillating field is small (case B). It is shown how to relate this problem to that of ordinary magnetic resonance–an atom in a small oscillating field perpendicular to a strong static field (case A)–by applying to each case a suitable transformation. In the transformed frames the dominant terms in the equations of motion are equivalent and subsidiary terms are similar, so that results well-known in case A may be taken over to case B. The dominant terms in case B describe a set of resonances whose line-shapes are given explicitly for a particular example. Subsidiary terms lead to shifts of the resonances analogous to the Bloch-Siegert shift. The leading term s in these shifts are given. Terms of higher order may be calculated by employing a technique due to Shirley which is described.