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

The resonance shifts of aligned atoms in intense oscillating radiofrequency fields

R I Jackson, S. Swain

Published Feb 28, 1982• 3 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3700/15/4/010

Related papers

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

Resonance lineshapes in multi-level atomic systems

J.A. Hermann, S. Swain

Exact expressions for the transition probabilities of a three-level atom interacting with a rotating magnetic field are obtained, and used to obtain the double-quantum resonance shift of sixth order and the linewidths to fourth order of perturbation theory. The treatment is extended to deal with overlap effects, oscillating fields and atomic systems with more energy levels. Numerical calculations are performed to check the validity of the analytic expressions and to obtain results applicable at higher field strengths. Although the majority of the detailed work is concerned with the double-quantum resonance, the formalism is general, and the methods may be applied to single-quantum or higher multiple-quantum transitions.

Journal of Physics B Atomic and Molecular Physics•Jul 11, 1977•11 citations

Interaction of a decaying atom with a linearly polarized oscillating field

Mark P. Silverman, F. M. Pipkin

The Schrodinger equation for the interaction of a decaying two-level atom with an external oscillating field is separated via a series of matrix transformations into a time independent and a time dependent part; the former yields the radiofrequency Stark shift for decaying states, the latter is normally negligible for atomic beam resonance experiments. In the limit of stable states, or for the case of two levels of equal lifetime, the shift reduces to that of the Bloch-Siegert effect. Exact analytical expressions for the state amplitudes are obtained for the case when the time dependent matrix can be neglected; the resulting transition probabilities are investigated as functions of frequency, time, and interaction strength and are compared with the rotating field theory and the exact numerical solutions.

Journal of Physics B Atomic and Molecular Physics•Oct 1, 1972•43 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

Bloch-Siegert shifts in resonances between excited states

David A. Andrews, G Newton

Expressions are presented for the Bloch-Siegert shift appropriate to electric dipole resonance experiments between excited states, when the maximum change in lifetime due to the applied field is used as the criterion for determining the resonance centre. The results differ from those reported by Silverman and Pipkin (1972) in the magnitude of the power shift, and in that the present authors predict an additional power-independent shift. The effect of neighbouring levels on the power shift is calculated, and shown to be substantial in certain cases.

Journal of Physics B Atomic and Molecular Physics•Jun 21, 1976•10 citations

Bloch-Siegert shift in optically oriented199Hg vapour

E. Arimondo, G. Moruzzi

The Bloch-Siegert shift displayed by a spin 1/2 particle system is measured in a double resonance experiment on the ground state of an optically oriented 199 Hg vapour. The displacement of the nuclear magnetic resonance peaks corresponding to the one and three quantum transitions are studied as a function of the radiofrequency field intensity. Similarly, the displacement of the component of the vapour polarization which is modulated at twice the RF field frequency is observed. The experimental results are compared with the existing theoretical approaches to the Bloch-Siegert effect, and a good agreement is found with the semiclassical theory leading to the continued fraction solution.

Journal of Physics B Atomic and Molecular Physics•Nov 1, 1973•34 citations
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.
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.
Exact expressions for the transition probabilities of a three-level atom interacting with a rotating magnetic field are obtained, and used to obtain the double-quantum resonance shift of sixth order and the linewidths to fourth order of perturbation theory. The treatment is extended to deal with overlap effects, oscillating fields and atomic systems with more energy levels. Numerical calculations are performed to check the validity of the analytic expressions and to obtain results applicable at higher field strengths. Although the majority of the detailed work is concerned with the double-quantum resonance, the formalism is general, and the methods may be applied to single-quantum or higher multiple-quantum transitions.
The Schrodinger equation for the interaction of a decaying two-level atom with an external oscillating field is separated via a series of matrix transformations into a time independent and a time dependent part; the former yields the radiofrequency Stark shift for decaying states, the latter is normally negligible for atomic beam resonance experiments. In the limit of stable states, or for the case of two levels of equal lifetime, the shift reduces to that of the Bloch-Siegert effect. Exact analytical expressions for the state amplitudes are obtained for the case when the time dependent matrix can be neglected; the resulting transition probabilities are investigated as functions of frequency, time, and interaction strength and are compared with the rotating field theory and the exact numerical solutions.
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.
Expressions are presented for the Bloch-Siegert shift appropriate to electric dipole resonance experiments between excited states, when the maximum change in lifetime due to the applied field is used as the criterion for determining the resonance centre. The results differ from those reported by Silverman and Pipkin (1972) in the magnitude of the power shift, and in that the present authors predict an additional power-independent shift. The effect of neighbouring levels on the power shift is calculated, and shown to be substantial in certain cases.
The Bloch-Siegert shift displayed by a spin 1/2 particle system is measured in a double resonance experiment on the ground state of an optically oriented 199 Hg vapour. The displacement of the nuclear magnetic resonance peaks corresponding to the one and three quantum transitions are studied as a function of the radiofrequency field intensity. Similarly, the displacement of the component of the vapour polarization which is modulated at twice the RF field frequency is observed. The experimental results are compared with the existing theoretical approaches to the Bloch-Siegert effect, and a good agreement is found with the semiclassical theory leading to the continued fraction solution.