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.
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.
The so-called gamma-echo effect has been observed experimentally and analyzed using the semiclassical optical theory. Here the effect is reinterpreted using a new 1D quantum mechanical model. This leads to a different interpretation of the effect as a π phase-shift induced transparency. In the basic time-differential Mössbauer spectroscopic technique the forward-scattered recoil-free radiation is observed, in delayed coincidence, after passing through a nuclear-resonant absorber. The effect in question is produced most efficiently when the source of recoil-free radiation is moved abruptly causing a π phase shift of the source radiation during its radiative lifetime. Using the 1D model the effect is seen to arise from the constructive interference between the source radiation at a later time, and the radiation coming from the absorber excited at an earlier time. The exact form of the source modulation and the nuclear-resonant thickness of the resonant absorber determines the shape of the time-differential resonant gamma ray transmission spectrum. Numerical results are given using the familiar 57Fe recoil-free resonant transition. The π phase-shift-induced transparency allows the resonant gamma radiation, incident on the resonant absorber, to be transmitted through the absorber without appreciable attenuation.
Mössbauer radiation from a single-line source was converted into a set of harmonics by transmission through an ultrasound-vibrated SS foil. Energy and time analysis of the radiation revealed characteristic sidebands and quantum beats, respectively. When the radiation was further transmitted through a sample (a second SS foil) at rest, the original quantum beat pattern was shifted in time proportional to the sample thickness. This time beat shift originates from interference in energy space. It is analogous to the spatial fringe shifts in classical interferometry.
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.
Abstract Recently, an observation of acoustically induced transparency (AIT) of a stainless-steel foil for resonant 14.4-keV photons from a radioactive 57Co Mössbauer source due to collective uniform oscillations of atomic nuclei was reported [Phys Rev Lett 124,163602, 2020]. In this paper, we propose to use the steep resonant dispersion of the absorber within the AIT spectral window to dramatically reduce a propagation velocity of γ-ray and x-ray photons. In particular, we show that a significant fraction (more than 40%) of a 97-ns γ-ray single-photon wave packet from a 57Co radioactive source can be slowed down up to 3 m/s and delayed by 144 ns in a 57Fe-enriched stainless-steel foil at room temperature. We also show that a similarly significant slowing down up to 24 m/s and a delay by 42 ns can be achieved for more than 70% of the 100-ns 14.4-keV x-ray single-photon pulse from a synchrotron Mössbauer source available at European Synchrotron Radiation Facility (ESRF) and Spring-8 facility. The propagation velocity can be widely controlled by changing the absorber vibration frequency. Achieving the propagation velocity on the order of 1–50 m/s would set a record in the hard x-ray range, comparable to what was obtained in the optical range.