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Physical Review A

Further Calculations on Multiple-Quantum Transitions

C. S. Chang, P. Stehle

Published Mar 1, 1972• 21 citations
Publisher

Abstract

We develop a detailed theory to calculate multiple-quantum transition probabilities. It is shown that the quantum electrodynamical (QED) approach must be used in order to explain quantitatively the experimental observations of Kusch. The anomalous behavior of shifts in resonance frequency and saturation of widths of transition probabilities can be explained on the basis of a QED renormalization principle. In this paper, we work out the detailed calculations for the case of a double-quantum transition quantitatively. The agreement between QED calculation and Kusch's observation is remarkable. We conclude that the renormalization principle of QED is important even for low-frequency and finite-intensity fields.

DOI
10.1103/physreva.5.1087

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Journal of Physics B Atomic and Molecular Physics•Jul 11, 1977•11 citations
We develop a detailed theory to calculate multiple-quantum transition probabilities. It is shown that the quantum electrodynamical (QED) approach must be used in order to explain quantitatively the experimental observations of Kusch. The anomalous behavior of shifts in resonance frequency and saturation of widths of transition probabilities can be explained on the basis of a QED renormalization principle. In this paper, we work out the detailed calculations for the case of a double-quantum transition quantitatively. The agreement between QED calculation and Kusch's observation is remarkable. We conclude that the renormalization principle of QED is important even for low-frequency and finite-intensity fields.
It is shown that a recent quantum-electrodynamic derivation by Chang and Stehle of double-quantum resonance-frequency shifts is equivalent to a previously known semiclassical approximation, and that neither of these methods is adequate to account for the results of Kusch's experiments. A more-exact semiclassical computation is given, which is in reasonable agreement with Kusch's results in the region where these results are meaningful. It is concluded that the assertion of Chang and Stehle that quantum electrodynamics is necessary for such situations is unfounded.
Multiple-quantum as well as the single-quantum transitions, $\ensuremath{\Delta}F=0$, $\ensuremath{\Delta}m=\ifmmode\pm\else\textpm\fi{}1, 2, 3, 4$, have been observed in K39{\mathrm{K}}^{39}K39. In the neighborhood of the rf amplitude, required to give a maximum transition probability, the lines have the behavior predicted for transitions induced by a rectangular rf pulse. For higher rf amplitudes, the line width becomes constant or else decreases after having reached a maximum width, the line frequency is modified less rapidly than predicted and anomalous transition probabilities occur. All these effects are markedly dependent on the details of construction of the rf circuit in which transitions are induced. It is suggested that these effects may arise from the end regions of the circuit through an adiabatic traverse of the circuit by the atom.
The present paper applies the method of high-intensity quantum electrodynamics developed earlier to several problems of current experimental interest. In this paper the intensitydependent shifts and widths of atomic levels are calculated and applied to a given transition near resonance. The technique is demonstrated with three examples: the multiphoton ionization of atomic cesium and hydrogen in atomic beams and third-harmonic production in an atomic vapor. It is found that the multiphoton-ionization rate in a relatively weak laser beam (of order 108{10}^{8}108 W/cm2{\mathrm{cm}}^{2}cm2) is determined by the intensity-dependent level width (6f6f6f configuration) as well as the interference of the fine-structure states, and that the rates for hydrogen, in a much more intense beam (of order 1012{10}^{12}1012 W/cm2{\mathrm{cm}}^{2}cm2) are determined by both the intensity-dependent widths and shifts as‘\stackrel{\mathrm{`}}{\mathrm{s}}s‘ well as the wave-function renormalization constants. The agreement of these two examples of calculations with the experiments done recently is satisfactory. For third-harmonic production the role of two-photon resonance is emphasized, and the dependence of the generated third-harmonic intensity on the intensity and frequency of the fundamental is discussed. Near resonance, the ratio of third-harmonic intensity to fundamental intensity saturates.
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.