Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics B Atomic and Molecular Physics

Aligned atoms in a strong oscillating RF field. A simple treatment by a continued fraction of 2×2 matrices

E. Arimondo, G. Moruzzi

Published Jan 14, 1982• 3 citations
Publisher

Abstract

Continued fractions of matrices solve the spin J-RF-field interaction completely. However, the general method requires (2J+1) 2 *(2J+1) 2 matrices. In this work the authors develop a treatment for the interaction of an aligned system with arbitrary spin J and an oscillating RF field requiring 2*2 matrices only. A numerical evaluation of the Bloch-Siegert shift and a comparison with recent experimental results are presented.

DOI
10.1088/0022-3700/15/1/015

Related papers

Matrix continued-fraction solution for saturation effects in spin-½ radio-frequency spectroscopy

M. Allegrini, E. Arimondo, Arturo Bambini

We present a semiclassical solution for a Zeeman-split, optically pumped, spin-\textonehalf{} system interacting with a strong radio-frequency field. The method of solution presented here is based upon a matrix continued-fraction expansion, and can explain any resonance phenomena which occur for arbitrary geometrical configurations of the pumping beam or the rf field polarization axis. Theoretical predictions of line shapes for some cases are reported and discussed. The solution can also explain some deviations of the observed Bloch-Siegert shift at high rf power from the behavior predicted by a previous semiclassical theory, which can account for simple geometrical configurations only.

Physical Review A•Feb 1, 1977•18 citations

Bloch-Siegert shifts and multi-quantum transitions in spin-one systems

J.A. Hermann, S. Swain

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.

Journal of Physics B Atomic and Molecular Physics•Aug 1, 1977•17 citations

The resonance shifts of aligned atoms in intense oscillating radiofrequency fields

R I Jackson, S. Swain

For a spin-1 atomic system subjected to an oscillating radiofrequency field applied at right angles to a DC magnetic field, the Block-Siegert shift has been calculated to sixth order in V, the amplitude of the oscillating field, and the three-quantum resonance has been calculated to eight order. The results are obtained for the three different polarisations- pi , sigma and t (unpolarised)-and expressions for the maxima are also obtained. For simplicity the atomic levels were assumed to have zero natural lifetime. It is shown that the level-crossing criterion may be used to calculate the positions of the p-quantum resonance minimum to the order of 2(p-1) in the oscillating field amplitude. Expressions are given for this quantity to order V 6 , and the separation of the maxima is calculated to order V p . The author's analytic results are compared with the numerical and experimental work of Jacobson (1980) and reasonable agreement is found.

Journal of Physics B Atomic and Molecular Physics•Feb 28, 1982•3 citations

Resonance behavior of a two-level quantum system in a two-frequency field

Wilhelmus M. Ruyten

Based on the vanishing-damping continued-fraction method which I developed previously for the analysis of the resonance behavior of a two-level atom in a two-frequency field with equal intensities in the two field modes [W. M. Ruyten, Phys. Rev. A 40, 1447 (1989)], I analyze here the related problem in which the two field intensities are unequal. Several analytical results are derived, namely, expressions for the lowest-order Bloch-Siegert shifts of all resonances, and an expression for the positions of the resonances in the limit of one strong and one weak field. It is shown that, in this limit, the resonances exhibit level crossings of the field-atom system.

Physical Review A•Oct 1, 1992•12 citations

Aligned atoms in strong oscillating RF fields. I. The unmodulated resonance signals

E Jacobson

In radio frequency (RF) experiments on aligned atoms (e.g. optical double resonance) new phenomena can be found in the case of longitudinal excitation if a strong oscillating RF field is used: both the positions of the RF resonances (RF shift) and the widths (power broadening) depend on the direction of the light observed and on the state of its (linear) polarisation. This applies to the main (single-quantum) transitions as well as the (odd) multi-quanta transitions. These effects have been observed in an experiment on aligned mercury atoms in the metastable 6 3 P 2 state and have been confirmed by numerical calculations based on the semiclassical approach.

Journal of Physics B Atomic and Molecular Physics•Sep 28, 1980•2 citations

Saturation effects in RF spectroscopy. III. Mixing of nearly degenerate levels by strong off-resonant fields

Stig Stenholm

For pt. II see abstr. A31645 of 1972. The author solves the problem of a strong RF field interacting with a degenerate doublet in terms of complex order Bessel functions. The influence of phenomenological pumping and decay parameters is displayed. The solution is compared to numerical results obtained from the previously published continued fraction solution.

Journal of Physics B Atomic and Molecular Physics•Jun 1, 1973•13 citations
Continued fractions of matrices solve the spin J-RF-field interaction completely. However, the general method requires (2J+1) 2 (2J+1) 2 matrices. In this work the authors develop a treatment for the interaction of an aligned system with arbitrary spin J and an oscillating RF field requiring 22 matrices only. A numerical evaluation of the Bloch-Siegert shift and a comparison with recent experimental results are presented.
We present a semiclassical solution for a Zeeman-split, optically pumped, spin-\textonehalf{} system interacting with a strong radio-frequency field. The method of solution presented here is based upon a matrix continued-fraction expansion, and can explain any resonance phenomena which occur for arbitrary geometrical configurations of the pumping beam or the rf field polarization axis. Theoretical predictions of line shapes for some cases are reported and discussed. The solution can also explain some deviations of the observed Bloch-Siegert shift at high rf power from the behavior predicted by a previous semiclassical theory, which can account for simple geometrical configurations only.
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.
For a spin-1 atomic system subjected to an oscillating radiofrequency field applied at right angles to a DC magnetic field, the Block-Siegert shift has been calculated to sixth order in V, the amplitude of the oscillating field, and the three-quantum resonance has been calculated to eight order. The results are obtained for the three different polarisations- pi , sigma and t (unpolarised)-and expressions for the maxima are also obtained. For simplicity the atomic levels were assumed to have zero natural lifetime. It is shown that the level-crossing criterion may be used to calculate the positions of the p-quantum resonance minimum to the order of 2(p-1) in the oscillating field amplitude. Expressions are given for this quantity to order V 6 , and the separation of the maxima is calculated to order V p . The author's analytic results are compared with the numerical and experimental work of Jacobson (1980) and reasonable agreement is found.
Based on the vanishing-damping continued-fraction method which I developed previously for the analysis of the resonance behavior of a two-level atom in a two-frequency field with equal intensities in the two field modes [W. M. Ruyten, Phys. Rev. A 40, 1447 (1989)], I analyze here the related problem in which the two field intensities are unequal. Several analytical results are derived, namely, expressions for the lowest-order Bloch-Siegert shifts of all resonances, and an expression for the positions of the resonances in the limit of one strong and one weak field. It is shown that, in this limit, the resonances exhibit level crossings of the field-atom system.
In radio frequency (RF) experiments on aligned atoms (e.g. optical double resonance) new phenomena can be found in the case of longitudinal excitation if a strong oscillating RF field is used: both the positions of the RF resonances (RF shift) and the widths (power broadening) depend on the direction of the light observed and on the state of its (linear) polarisation. This applies to the main (single-quantum) transitions as well as the (odd) multi-quanta transitions. These effects have been observed in an experiment on aligned mercury atoms in the metastable 6 3 P 2 state and have been confirmed by numerical calculations based on the semiclassical approach.