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A technique to transform the waveform of a 14.4 keV photon (time dependence of the photon detection probability or, equivalently, the intensity of the single-photon wave packet) into a regular sequence of short, nearly bandwidth-limited pulses with a controlled number of pulses is proposed. It is based on coherent forward scattering of single X-ray photons from a synchrotron Mössbauer source (SMS) in an optically thick, vibrating, recoilless 57Fe resonant absorber. The possibility of compressing the waveform of an SMS photon into a single short bell-shaped pulse is predicted. The experiment is proposed for compressing a 100 ns duration 14.4 keV single-photon wave packet produced by SMS at the European Synchrotron Radiation Facility (ESRF) into a single bell-shaped pulse of less than 20 ns duration and more than twice the peak intensity. Such single-photon coherent pulses are promising for applications in the fast-developing field of X-ray quantum optics, including possible implementation of quantum memory. 

In a recent work (Antonov et al., Physical Review Letters 123, 243903 (2019)), it was shown that it is possible to amplify a train of attosecond pulses, which are produced from the radiation of high harmonics of the infrared field of the fundamental frequency, in the active medium of a plasma-based X-ray laser modulated by a replica of the infrared field of the same frequency. In this paper, we show that much higher amplification can be achieved using the second harmonic of the fundamental frequency for modulating of a hydrogen-like active medium. The physical reason for such enhanced amplification is the possibility to use all (even and odd) sidebands induced in the gain spectrum in the case of the modulating field of the doubled fundamental frequency, while only one set of sidebands (either even or odd) could participate in amplification in the case of the modulating field of the fundamental frequency due to the fact that the spectral components of the high-harmonic field are separated by twice the fundamental frequency. Using the plasma of hydrogen-like C5+ ions with an inverted transition wavelength of 3.38 nm in the water window as an example, it is shown that the use of a modulating field at a doubled fundamental frequency makes it possible to increase the intensity of amplified attosecond pulses by an order of magnitude in comparison with the previously studied case of a fundamental frequency modulating field.