Resonance Transitions Induced by Perturbations at Two or More Different Frequencies

Resonance transition probabilities for a system in the presence of oscillatory perturbations at two or more different frequencies are discussed. It is shown that if resonance transitions are induced by a perturbation at one frequency, then the presence of the other perturbations at nonresonant frequencies alters the resonance frequency for the first perturbation. Theoretical expressions for the alteration of the resonance frequency are derived. Various applications of the derived formulas are discussed, including resonance experiments with more than one oscillatory field, molecular beam experiments in a nonuniform constant field, evaluation of "collision-narrowing" effects in nuclear paramagnetic resonance, etc. The effects of the extraneous oscillatory perturbations are shown to be especially important if their frequencies are close to Bohr frequencies for the perturbation.