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Journal of Physics B Atomic and Molecular Physics

Magnetic resonance in arbitrarily oriented fields

David Thomas Pegg

Published Apr 21, 1977• 6 citations
Publisher

Abstract

The magnetic resonance of a spin-half system in a static field and an arbitrarily oriented oscillating field has been studied recently by McClean and Swain (1976) by a continued-fraction technique in quantum-electrodynamical perturbation theory, a difficult procedure involving twenty fourth-order processes and some rather heavy analysis. It is shown how the semiclassical approach of Pegg and Series (1970, 1973) yields the same results more simply. This approach also has the advantage of being applicable to systems of arbitrary spin.

DOI
10.1088/0022-3700/10/6/013

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The magnetic resonance of a spin-half system in a static field and an arbitrarily oriented oscillating field has been studied recently by McClean and Swain (1976) by a continued-fraction technique in quantum-electrodynamical perturbation theory, a difficult procedure involving twenty fourth-order processes and some rather heavy analysis. It is shown how the semiclassical approach of Pegg and Series (1970, 1973) yields the same results more simply. This approach also has the advantage of being applicable to systems of arbitrary spin.
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 resonance behavior of magnetic and optical two-level systems under the influence of strong modulated fields is studied in detail by solving the Bloch equations in the limit of vanishing damping, namely, in terms of matrix continued fractions. Thus both analytical and numerical results are obtained for the positions of the various multiple quantum resonances in the time-averaged as well as modulated interaction components. Specifically, the role of the orientation of the optical pump beam in magnetic-resonance configurations is investigated, and it is shown that there are several classes of resonances for which the behavior of the system is independent of the orientation of the pump beam. These are the geometries in which the static and oscillating magnetic fields are either orthogonal or parallel, the geometries leading to Haroche-like resonances (which, however, play a special role in modulated interaction components also), and a previously unknown class, labeled saddle-point resonances. However, other than for these special classes, the behavior of the system depends generally on the orientation of the optical pump beam, unlike as implied in some previous work on this problem. As in the recently studied case of a fully amplitude-modulated optical interaction [W. M. Ruyten, Phys. Rev. A 40, 1447 (1989)], it is found that the resonances in modulated interaction components are generally doubly branched, with one branch duplicating the resonances found in the time-averaged component. A new feature found for not-fully-amplitude-modulated and phase-modulated, off-resonance optical excitation (or magnetic resonance with nonparallel or nonorthogonal static and oscillating magnetic fields) is the occurrence of an extra type of resonance in modulated components of the interaction that cannot, like all other resonances of the system, be classified as power-shifted multiple quantum transitions.
Using a novel semiclassical approach, the authors analyse the resonance behaviour of spin-1/2 atoms subjected to static and oscillating magnetic fields. Unlike a number of previous calculations, the orientation of the pump beam is taken into account explicitly. Thus it is shown that much previous work is valid only if the pump beam is along the static magnetic field. They also show how one such previous calculation can be generalised, namely that based on diagonalisation of the Floquet Hamiltonian.
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.
We derive algebraic expressions for the atomic multipole moments mk,q(F){m}_{k,q}(F)mk,q​(F) describing the steady-state polarization of an atomic ensemble with angular momentum FFF in a static magnetic field of arbitrary direction and general multipole relaxation rates ${\ensuremath{\Gamma}}_{q}^{(k)}$. The longitudinal moments mk,0{m}_{k,0}mk,0​ are given in terms of truncated continued fractions, while the transverse moments ${m}_{k,q\ensuremath{\ne}0}$, representing the ensemble coherence, are given by products of truncated continued fractions. The special case of isotropic relaxation leads to particularly simple and elegant expressions. Our results are relevant for all domains of physics that consider the evolution of a spin system interacting with vector-type perturbations.
A general perturbation procedure is described for writing down the contributions to any order for the probability of a quantum system described by a time-independent Hamiltonian making a transition from an arbitrary initial state to an arbitrary final state. The results are expressed in terms of certain propagators which involve continued fractions, so that some processes are summed to all orders. The general theory is then applied to a single multilevel atom interacting with any number of modes of the electromagnetic field in the electric dipole approximation. The fundamental theory is considered primarily but a detailed discussion is given of spontaneous and stimulated emission. Expressions for the lineshape are derived incorporating the generalized Lamb shifts and damping constants.