The radiation from a Mössbauer source of 57Co in a copper matrix is frequency-modulated by vibrating the source with a quartz piezo-crystal driven by an oscillator. If the radiation passes through a resonant absorber (a "filter"), a time structure containing the oscillator frequency and its harmonics appears in the resulting counting rate. It may be seen by sorting the time of each count with respect to the cross-over time of a subharmonic of the oscillator. These quantum beats are caused by interferences between the frequency components of the photon amplitude, and vanish unless there is some alteration of relative phase or amplitude of the original radiation by the filter. The harmonic constitution of the beats is a sensitive function of the relative shift of source and filter and may be used as a sensitive probe of small shifts. This is demonstrated by measuring the temperature shift in 57Fe-Be. If a velocity spectrum is made with counts collected during only a part of the vibration cycle, prominent dispersion effects are seen, with counting rates higher than background in some portions of the spectrum. A classical optical theory of the phenomenon is presented in summary. It explains the beats, the dispersion effects, and the sensitivity of the harmonic ratios to relative shifts.
Interference experiments were carried out in which the source was so faint that effects could be attributed to separate quanta of radiation. The energy radiated in the helium line 4471A from a glow discharge was measured by comparison with a black body, and from the known decay constant of the atomic radiation process, the separation of atomic radiations in time was deduced. With an echelon diffraction grating the characteristic double order pattern was photographed when each quantum of light passed separately through the instrument. Under these conditions it gave rise to a pattern due to simultaneous passage through several steps. With an air film between two parallel plates, interference fringes were photographed in 4471, when the quanta emitted from the volume of the source used were completely separated in time, showing that a single quantum obeys the classical laws of partial transmission and reflection at a half silvered mirror and of subsequent combination with the phase difference required by the wave theory of light.
A classical optical theory is given for the phenomenon of $\ensuremath{\gamma}$-ray quantum beats observed by Perlow in a study of the M\"ossbauer effect of a frequency-modulated source. The intensity $I({\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}},t)$ of the radiation from such a source transmitted through a resonant absorber is obtained as a function of ${\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}}$ and
t, where ${\ensuremath{\omega}}_{0}^{\ensuremath{'}}$ is the frequency of an absorber resonance, ${\ensuremath{\omega}}_{0}$ 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 ${\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}}$, the calculated intensity displays beats in the time spectrum, and when viewed at fixed
t, the intensity shows dispersion at appropriate values of ${\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}}$, 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 ${\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}}=0$ of the ratio of Fourier components
D2D1 is explained. The use of
D2D1 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 $I({\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}},t)$ 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 $I({\ensuremath{\omega}}_{0}\ensuremath{-}{\ensuremath{\omega}}_{0}^{\ensuremath{'}},t)$ accordingly. Finally, the effect of a distribution of phase and amplitude of the motion of the vibrating source is discussed.
Quantum-beat and sideband spectra of recoilless $\ensuremath{\gamma}$ radiation of
57Fe were measured as a function of frequency and the voltage on the quartz crystal on which a
57Co source was deposited. The sideband spectra were analyzed using a Rice distribution of the modulation index. The width of this distribution was shown to depend on the frequency; the narrowest distribution was found at 9.795 MHz. The results of the quantum-beat measurements are in agreement with the optical theory of Monahan and Perlow.