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Journal of Physics B Atomic and Molecular Physics

Quantum theory of RF resonances: the semiclassical limit

Stig Stenholm

Published Aug 1, 1973• 20 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3700/6/8/042

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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.
A simple model of a maser or laser consisting of a single-mode field coupled to NNN identical two-level atoms is considered. The semiclassical approximation is equivalent to neglecting the statistical correlations between the atoms and the field. The accuracy of this approximation is investigated by writing equations of motion for the correlations. To obtain a complete, self-consistent set of equations, it is necessary to include correlations of the field with itself (related to the coherence of the field) and correlations between different atoms. Relaxations and an energy source are introduced phenomenologically into the equations of motion. They are then solved for the case of steady-state oscillation. It is found that the correlations are smaller than the terms kept in the semiclassical theory by the order of 1 over the number of photons present in the field, larger if thermal photons abound, but smaller if the field relaxation is dominant. Expressions are also found for the amplitude and phase fluctuations of the field. The latter yield a linewidth for the maser oscillator in agreement with earlier calculations, but obtained by a different method.
The semiclassical description of an RF resonance measurement on a doublet excited state is formulated. The solution of the density matrix equation of motion is obtained as a continued fraction. The single quantum and multiple quantum transitions are seen as resonances in the measured quantities. An analytic solution is obtained for the case of degenerate doublet (the Hanle phenomena) and the RF modification of the g-value is obtained. It is shown that the continued fraction solution gives simple computations of strong field saturation effects.
The author presents a quantum mechanical method capable of providing exact non-perturbative solutions in problems involving the interaction of atoms and molecules with the electromagnetic field. To illustrate the method two examples of double-resonance phenomena are considered. In the first example the author studies the interaction of a near-resonance rf field with the Zeeman sublevels of atomic hydrogen. In the second the interaction of a three-level molecule with two applied electromagnetic fields is studied. Both the molecular levels and the electromagnetic field are quantized. The method eliminates the deficiencies of a semiclassical treatment, yet does not require the sophisticated computations of the resolvant formalism of Cohen-Tannoudji.
One of the simplest nontrivial problems involving the atom–field interaction is the coupling of a two-level atom with a single mode of the electromagnetic field. A two-level atom description is valid if the two atomic levels involved are resonant or nearly resonant with the driving field, while all other levels are highly detuned. Under certain realistic approximations, it is possible to reduce this problem to a form which can be solved exactly; allowing essential features of the atom-field interaction to be extracted. In this chapter we present a semiclassical theory of the interaction of a single two-level atom with a single mode of the field in which the atom is treated as a quantum two-level system and the field is treated classically. A fully quantum mechanical theory will be presented in Chapter 6. A two-level atom is formally analogous to a spin-1/2 system with two possible states. In the dipole approximation, when the field wavelength is larger than the atomic size, the atom–field interaction problem is mathematically equivalent to a spin-1/2 particle interacting with a time-dependent magnetic field. Just as the spin-1/2 system undergoes the so-called Rabi oscillations between the spin-up and spin-down states under the action of an oscillating magnetic field, the two-level atom also undergoes optical Rabi oscillations under the action of the driving electromagnetic field. These oscillations are damped if the atomic levels decay. An understanding of this simple model of the atom–field interaction enables us to consider more complicated problems involving an ensemble of atoms interacting with the field.