Theoretical description of quantum beats of recoil-freeγradiation

A classical optical theory is given for the phenomenon of γ-ray quantum beats observed by Perlow in a study of the M\"ossbauer effect of a frequency-modulated source. The intensity I(ω₀-ω₀^',t) of the radiation from such a source transmitted through a resonant absorber is obtained as a function of ω₀-ω₀^' and t, where ω₀^' is the frequency of an absorber resonance, ω₀ is the central frequency of the frequency-modulated source, and t is the laboratory time. An average is taken over the unobserved initial formation time of the excited nuclear state in the source. When viewed at fixed ω₀-ω₀^', the calculated intensity displays beats in the time spectrum, and when viewed at fixed t, the intensity shows dispersion at appropriate values of ω₀-ω₀^', responsible for the observed enhancement of intensity above background. The harmonic content of the quantum beats is calculated explicitly in the thin-absorber limit, and the observed linear variation about ω₀-ω₀^'=0 of the ratio of Fourier components D₁D₂ is explained. The use of D₁D₂ to measure small frequency shifts is analyzed by a statistical comparison with the method of switching between steepest points of the line, as used in gravitational-red-shift measurements. The variances are comparable. The effect on I(ω₀-ω₀^',t) of line broadening due to sample inhomogeneities is calculated for a Lorentz distribution of center frequencies in source and absorber, and a prescription is given for modifying the various terms in I(ω₀-ω₀^',t) accordingly. Finally, the effect of a distribution of phase and amplitude of the motion of the vibrating source is discussed.