X-ray free-electron lasers (XFELs) provide ultra-short x-ray pulses of unprecedented peak brightness, opening a pathway to transfer quantum electronics and quantum optical processes to the x-ray domain. We present results of the first experimental demonstration of an x-ray laser based on amplification of x-ray radiation on an atomic inner-shell transition in Neon . The population inversion is achieved by rapid K-shell photo-ionization [2,3] or resonant excitation  with an XFEL pulse. Gain saturation of the K-atransition in Neon could be demonstrated in a recent experiment at the Linac Coherent Light Source. In addition to experimental results, we present numerical solutions of the generalized Maxwell-Bloch equations. The emitted x-ray pulses have ultra short time durations, ranging from sub femtoseconds to a few femtoseconds and are nearly transform limited, despite pumping with a stochastic XFEL source of only limited temporal coherence. We will discuss the extension of the x-ray lasing scheme to amplified spontaneous emission and self-stimulated x-ray Raman scattering in diatomic molecules, thereby creating a tunable x-ray laser based on inner-shell transitions in molecules.
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