The concepts and ideas of coherent, nonlinear and quantum optics deeply penetrate into the range of 10-100 kiloelectronvolt (keV) photon energies, corresponding to soft gamma-ray (hard xray )) radiation. The recent experimental achievements in this frequency range include demonstration of the parametric down-conversion in the Langevin regime [1], cavity electromagnetically induced transparency [2], collective Lamb shift [3], and single-photon revival in the nuclear absorbing sandwiches [4]. Realization of a single photon coherent storage [5] and stimulated Raman adiabatic passage [6] were recently proposed. Still the number of tools for coherent manipulation of gamma-photon -nuclear ensemble interactions remains rather limited. In this work an efficient method to coherently control the waveforms of gamma-photons has been suggested and verified. In particular, the temporal compression of an individual gamma-photon into coherent ultrashort pulse train has been demonstrated. The method is based on the resonant interaction of gamma-photons with an ensemble of nuclei with modulated frequency of the resonant transition. The frequency modulation, achieved by uniform vibration of the resonant absorber due to the Doppler Effect, results in the time-dependence of the resonant absorption and dispersion, which allow shaping of the incident gamma-photons. The developed technique is expected to give a strong ) It is a historic tradition to call a radiation in this range x-ray radiation when it is produced by electron motion and to call it gamma-ray radiation if it is produced by nuclear transitions. 2impetus on emerging fields of coherent and quantum gamma-optics, providing a basis for realization of the gamma-photon -nuclear ensemble interfaces and quantum interference effects at the nuclear gamma-ray transitions.Quantum optics is the field of research dealing with interactions of quanta of electromagnetic radiation with quantum transitions of matter. It provides the basis for new fast growing fields of quantum cryptography, communication, and information. So far the experiments in these fields have been implemented either with microwave or optical photons, interacting with atomic electron transitions, and typically required cryogenic temperatures. The gamma-photons in the range of 10-100keV and the corresponding nuclear quantum transitions are the most suitable for realization of such experiments due to nearly 100% detector efficiency, extremely high Q-factor (~10 12 for 14.4keV transition in 57 Fe) of recoilless nuclear transitions even at room temperature, existence of radioactive materials (representing themselves the natural sources of single gamma-photons) and the cascade scheme of radiative decay of some radioactive sources (Fig.1a), allowing one to study the photon temporal shape via time-delayed coincidence measurement technique [7]. Moreover, the gamma-photons have important potential advantages over the microwave and optical photons for applications in cryptography, communication and information due to extremely...

We show here that taking into account the contribution of the nearest satellites of the resonant component removes misfit of our analytical approximation with the exact result for the probability amplitude of the photon, transmitted through the vibrating absorber. We analyze time evolution of the phase difference of the scattered field and the comb. We discuss the scheme how single and two-pulse bunches can be used to simulate spin 1/2 qubit and ququad.

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We suggest a technique to amplify a train of attosecond pulses, produced by high-harmonic generation (HHG) of an infrared (IR) laser field, in an active medium of a plasma-based X-ray laser. This technique is based on modulation of transition frequency of the X-ray laser by the same IR field, as used to generate the harmonics, via linear Stark effect, which results in redistribution of the resonant gain and simultaneous amplification of a wide set of harmonics in the incident field. We propose an experimental implementation of the suggested technique in active medium of C 5+ ions at wavelength 3.4 nm in the "water window" range and show the possibility to amplify by two orders of magnitude a train of attosecond pulses with pulse duration down to 100 as. We show also a possibility to isolate a single attosecond pulse from the incident attosecond pulse train during its amplification in optically deep modulated medium. Abstract:In this supplemental material we derive an analytical solution describing an amplification of the highharmonic field in an active medium of a plasma-based X-ray laser in the linear regime within the three-level medium model and constant population inversion approximation. We provide also more detailed numerical study of an amplification process in both linear and nonlinear regimes taking into account a saturation effect within the fivelevel medium model.

Coherent intense attosecond X-ray pulses could lead to a fast dynamical imaging of the biological macromolecules and other material nanostructures with a unique combination of a record high temporal and spatial resolution. Plasma based X-rays laser sources are capable to produce high energy X-ray pulses but with relatively long picosecond duration. The sources based on high-harmonic generation (HHG) of a laser field allow to produce much shorter pulses but of lower energy. We suggest two different paths towards intense sub-femtosecond X-ray sources, namely i) via efficient transformation of the picosecond radiation of the X-ray plasma lasers into the trains of sub-femtosecond pulses in a resonantly absorbing medium, and ii) via amplification of HHG radiation in the active medium of the X-ray plasma lasers. We show that essentially the same technique can be used for realization of both paths. This technique is a modulation of the parameters of the resonant transition (accordingly in absorbing or amplifying medium) produced under the action of sufficiently strong infrared or optical field. We propose experimental realization of the suggested technique in the passive/active media of i) Li III ions modulated by the mid-IR laser field and ii) C VI ions modulated by the optical laser radiation. I.

Recently, an exponentially decaying waveform (the time-dependence of detection probability) of a Mössbauer γ-ray photon was transformed into a regular sequence of short pulses in a sinusoidally vibrating recoilless resonant absorber [Nature, 508, 80-83 (2014)]. In the present paper, we show that the peak amplitude of the pulses can be considerably increased via joint adjustment of optical depth of the absorber and the initial phase of its vibration. This is due to reduction of the photoelectric absorption and maximizing the constructive temporal interference of spectral content of the single-photon wave packet in optically deep absorber. The ultimate capabilities for transforming a waveform of 14.4 keV photon from 57 Co radioactive source into a regular train of pulses in a harmonically vibrating 57 Fe recoilless resonant absorber are discussed. We show that the shortest pulse duration, produced by this technique, is limited by the highest available vibration frequency of a piezo-transducer and at present can be as short as 7.7 ps. The maximum achievable detection probability of the transformed photon at the experimentally feasible conditions is more than two times higher than peak detection probability of the photon emitted by the source and nearly 5.5 times higher than obtained in the above reference.

We propose a technique to form a single few-cycle attosecond pulse from vacuum ultraviolet or extreme ultraviolet radiation via resonant interaction with hydrogenlike atoms, irradiated by a high-intensity far-off-resonant laser field. The laser field strongly perturbs excited atomic energy levels via the Stark effect and ionizes atoms from the excited states. We show that an isolated attosecond pulse can be formed using either a short incident femtosecond pulse of the resonant radiation or a steep front edge of the laser field. We propose an experimental realization of a single subfemtosecond pulse formation at 121.6 nm in atomic hydrogen and a single sub-100 as pulse formation at 13.5 nm in Li(2+) plasma.