Abstract: Narrow optical resonances corresponding to quantum transitions in atoms, molecules, quantum dots, rare-earth ions and color centers constitute the basis of quantum optics with numerous applications in sensing, imaging, computation, communication, etc. Very high-quality atomic resonances with Qfactor ~10¹⁵-10²⁰ are on demand for atomic clocks, chronometric geodesy and gravimetry, search for time variation of the fundamental constants and dark matter. Their realization requires low atomic density, vacuum environment, laser cooling below 100 nK temperature, and magnetic traps or optical lattices. Nuclear resonances with similar high-quality factor can be achieved at solid density and much higher temperature, as the nuclei are naturally trapped in a crystal lattice. The major advantage of nuclear vs atomic transitions is a smaller sensitivity to frequency shifts caused by electric and magnetic fields perturbations. Besides, the Mössbauer effect makes it possible to effectively eliminate a thermal-motion broadening. Thus, nuclear transitions offer an appealing platform for a new precision metrology capable of, for example, detecting a gravitational red shift with a sub-mm displacement. They could provide the basis for nuclear clocks and super-dense quantum nuclear memory. However, their resonant excitation, coherent control and interfacing with the resonant x-ray photons is challenging due to absence of the bright coherent sources and high-quality cavities in the hard x-ray range.

In this talk we will review recent progress in the emerging field of quantum x-ray optics including experimental demonstrations of the coherent waveform shaping of the x-ray photons, acoustically induced transparency, slow light, and very recent realizations of quantum nuclear memory and resonant excitation of the 12.4 keV long-lived (0.46 s) nuclear transition in 45Sc, the most promising Mössbauer nuclear clock candidate.