© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Proceedings of the Royal Society of London A Mathematical and Physical Sciences

On the reduction of a problem in magnetic resonance

David Thomas Pegg, George William Series

Published Mar 27, 1973• 28 citations

Abstract

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.

DOI: 10.1098/rspa.1973.0026

Related papers

An improved coordinate transformation for the treatment of a paramagnetic atom in a linear oscillatory and a constant transverse magnetic field

F Ansbacher

The introduction of a suitably moving coordinate system results in approximate equations of motion which lead in a direct manner to the Block-Siegert resonance shift, to 2nd or higher order, and to a simple visualization of multiple quantum transitions, including their correct quantitative treatment. The relation between the present method and that due to Shirley (1965), whose results are confirmed, is pointed out.

Journal of Physics B Atomic and Molecular Physics•Aug 1, 1973•3 citations

Semi-classical model of magnetic resonance in intense RF fields

David Thomas Pegg

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.

Journal of Physics B Atomic and Molecular Physics•Feb 1, 1973•60 citations

Theory of the Bloch-Siegert shift

F Ahmad, R. Bullough

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.

Journal of Physics B Atomic and Molecular Physics•Jun 1, 1974•37 citations

A unitary perturbation method for a paramagnetic atom in a magnetic field with constant and oscillatory components

F Ansbacher

The method is based on replacing the actual field B by an 'integrable' field, i.e. one for which the Schrodinger equation can be integrated. A simple criterion allows the best choice of field to be made for a given degree of approximation. Two examples are discussed: (1) the classic case of a small oscillating field transverse to a large steady field to an approximation beyond the rotating wave approximation, which reproduces, with certain modifications, the results of Bloch and Siegert valid for any j; (2) the field configuration of Pegg and Series (1970).

Journal of Physics B Atomic and Molecular Physics•Aug 1, 1973•8 citations

The exact solution of magnetic resonance for arbitrary spin using an operator-based formalism

Sanya Bharti, James K. Freericks

The time evolution of a spin precessing in a large static field with a perpendicular rotating field is a classic problem in magnetic resonance that appears in nearly all quantum textbooks. It can be solved exactly by going to the rotating frame. Often, however, the final transition probabilities are determined only for the spin 1/2 case. In this work, we show how an operator-based approach naturally produces a time-ordered product via a Trotter product formula and then by employing simple operator identities such as the Hadamard lemma, the Baker–Campbell–Hausdorff formula, and the exponential disentangling identity, the problem can be directly solved for arbitrary spin.

American Journal of Physics•Jan 22, 2026•0 citations

Magnetic and optical resonance of two-level quantum systems in modulated fields. I. Bloch equation approach

Wilhelmus M. Ruyten

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.

Physical Review A•Oct 1, 1990•17 citations

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.

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 method is based on replacing the actual field B by an 'integrable' field, i.e. one for which the Schrodinger equation can be integrated. A simple criterion allows the best choice of field to be made for a given degree of approximation. Two examples are discussed: (1) the classic case of a small oscillating field transverse to a large steady field to an approximation beyond the rotating wave approximation, which reproduces, with certain modifications, the results of Bloch and Siegert valid for any j; (2) the field configuration of Pegg and Series (1970).

The time evolution of a spin precessing in a large static field with a perpendicular rotating field is a classic problem in magnetic resonance that appears in nearly all quantum textbooks. It can be solved exactly by going to the rotating frame. Often, however, the final transition probabilities are determined only for the spin 1/2 case. In this work, we show how an operator-based approach naturally produces a time-ordered product via a Trotter product formula and then by employing simple operator identities such as the Hadamard lemma, the Baker–Campbell–Hausdorff formula, and the exponential disentangling identity, the problem can be directly solved for arbitrary spin.

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.