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Physical Review A

Semiclassical Calculations on Multiple-Quantum Transitions

David Thomas Pegg

Published Nov 1, 1973• 10 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.8.2214

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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.
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.
The author attempts to show that, in contradiction with work by Chang and Stehle, the semiclassical description of RF resonance experiments is valid. The density matrix equation of motion is set up for a two-level atom interacting with a one-mode quantized field. The terms giving the semiclassical description are separated and the contributions arising from the rest are carefully evaluated. Corrections to the semiclassical treatment are found to be of order n 0 -2 , where n 0 is the average number of RF photons. In practice such corrections are negligible.
The quantum-electrodynamic frequency shifts of a two-level system in spontaneous emission and in an intense quantized radiation field are given up to eighth order in the coupling parameter, using the recursive perturbation method developed previously by one of the authors. The first six orders of the latter agree with the semiclassical result of Shirley and thus disagree with Chang and Stehle's calculation. The eighth-order term agrees with a recent semiclassical calculation by Ahmad and Bullough. The transition from the quantum mechanical case (the number of photons small) to the classical case (the number of photons very large) is also exhibited in our approach.
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.