Coherent x-ray-optical control of nuclear excitons with zeptosecond phase-stability

Coherent control of quantum dynamics is key to a multitude of fundamental studies and applications alike. In the visible or longer-wavelength domains, near-resonant light fields have become the primary tool to control electron dynamics. Recently, coherent control in the extreme-ultraviolet range was demonstrated, with timing stability of the applied light fields in the few-attosecond range. At hard x-ray energies, M\"ossbauer nuclei feature narrow nuclear resonances, and spectroscopy of these resonances is a widespread tool to study magnetic, structural and dynamical properties of matter. It has been shown that the power and scope of M\"ossbauer spectroscopy can be significantly advanced using various control techniques. However, the coherent control of atomic nuclei using near-resonant x-ray fields remains an open challenge, also because of the extreme stability requirements on the x-ray light in the few-zeptosecond range. Here, we demonstrate such control, and use the relative phase of two x-ray pulses to switch the nuclear dynamics between stimulated emission and enhanced coherent excitation. For this, we suggest and implement a method to shape single pulses delivered by modern x-ray facilities into tunable double-pulses, with the desired stability on the few-zeptosecond level. Our results unlock coherent optical control for nuclei, and pave the way for nuclear Ramsey spectroscopy and spin-echo-like techniques, which not only provide key concepts for advancing nuclear quantum optics, but also essential ingredients for possible x-ray clocks and frequency standards. As a long-term perspective, we envision time-resolved studies of nuclear out-of-equilibrium dynamics, which is a long-standing open challenge in M\"ossbauer science.