Theoretical description of quantum beats of recoil-freeγradiation

A classical optical theory is given for the phenomenon of $\gamma$-ray quantum beats observed by Perlow in a study of the Mössbauer effect of a frequency-modulated source. The intensity $I({\omega }_0-{\omega }_0^{\prime },t)$ of the radiation from such a source transmitted through a resonant absorber is obtained as a function of ${\omega }_0-{\omega }_0^{\prime }$ and $t$, where ${\omega }_0^{\prime }$ is the frequency of an absorber resonance, ${\omega }_0$ 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 ${\omega }_0-{\omega }_0^{\prime }$, the calculated intensity displays beats in the time spectrum, and when viewed at fixed $t$, the intensity shows dispersion at appropriate values of ${\omega }_0-{\omega }_0^{\prime }$, 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 ${\omega }_0-{\omega }_0^{\prime }=0\mathrm{}$ of the ratio of Fourier components $\frac{D_1}{D_2}$ is explained. The use of $\frac{D_1}{D_2}$ 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 $I({\omega }_0-{\omega }_0^{\prime },t)$ 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 $I({\omega }_0-{\omega }_0^{\prime },t)$ accordingly. Finally, the effect of a distribution of phase and amplitude of the motion of the vibrating source is discussed.