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Journal of Physics B Atomic and Molecular Physics

Theory of the Bloch-Siegert shift

F Ahmad, R. Bullough

Published Jun 1, 1974• 37 citations
Publisher

Abstract

The method recently developed for treating Bloch-Siegert shift of the main resonance in a magnetic resonance experiment is successfully applied to the higher order resonances due to multiquantum transitions. A general analytical expression for the (2n+1)th resonances is given. Results for the main and third order resonance in the low or vanishingly low range of static magnetic fields are also reported.

DOI
10.1088/0022-3700/7/9/001

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Proceedings of the Royal Society of London A Mathematical and Physical Sciences•Mar 27, 1973•28 citations
The method recently developed for treating Bloch-Siegert shift of the main resonance in a magnetic resonance experiment is successfully applied to the higher order resonances due to multiquantum transitions. A general analytical expression for the (2n+1)th resonances is given. Results for the main and third order resonance in the low or vanishingly low range of static magnetic fields are also reported.
A semi-classical approach developed by Pegg and Series (1970, 1973) is used to study higher order effects in magnetic resonance produced by an oscillating magnetic field perpendicular to a static field as the oscillating field strength is increased. It is shown that the well-known 'multiple quantum' transitions arise from a semi-classical theory, being transformed into 'single quantum' transitions induced by field harmonics in the most appropriate reference frame for treating the problem. Higher order terms in the Bloch-Siegert shift support those derived by Shirley (1965) by means of Floquet theory, and are in gross disagreement with those derived by Chang and Stehle (1971) from quantum electrodynamics, particularly at very high field intensities where the results are not even qualitatively similar. It should be possible to decide easily between the two approaches by experiment.
The Bloch-Siegert-type shifts and multiple-quantum resonance shifts of a spin-one system subjected to an oscillating radio-frequency applied at right angles to a DC field have been calculated to sixth order in V, the amplitude of the oscillating field, for the single-quantum resonance, and to fourth order in V for the higher resonances. The level crossing criterion appropriate to two-level systems cannot be used here, and a resonance condition based on a previously derived analogue of Shirley's equation for spin-one systems is used instead. The calculation uses degenerate continued-fraction perturbation theory. Numerical calculations were performed in order to obtain the shape of the resonances, and partially check the analytic results. The atomic levels were assumed to be completely stable; the corrections due to finite atomic lifetimes are expected to be small.
In a recent publication [ J. Opt. Soc. Am. B5, 2015 ( 1988)], Chakmakjian et al. gave an empirical expression for the positions of subharmonic resonances of a two-level system driven by a fully amplitude modulated laser. Here we calculate exact positions of the resonances and adapt a dressed-atom analysis from the corresponding magnetic resonance problem. Thus subharmonic resonances are explained as the limiting behavior of multiphoton resonances, and, in the limit of vanishing damping, the positions of the resonances are given by analytical expressions for the generalized Bloch–Siegert shifts.
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.