  We present a comprehensive experimental and theoretical study on
superfluorescence in the extreme ultraviolet wavelength regime. Focusing a
high-intensity free-electron laser pulse in a cell filled with Xe or Kr gas,
the medium is quasi instantaneously population-inverted by inner-shell
ionization on the giant resonance followed by Auger decay. On the timescale of
100 ps a macroscopic polarization builds up in the medium, resulting in
superfluorescent emission of several Xe and Kr lines in the forward direction.
As the number of emitters in the system is increased by either raising the
pressure or the pump-pulse energy, the emission shows an exponential growth of
over 4 orders of magnitude and reaches saturation. With increasing yield, we
observe line broadening, a manifestation of superfluorescence in the spectral
domain. Our novel theoretical approach, based on a full quantum treatment of
the atomic system and the irradiated field, shows quantitative agreement with
the experiment and supports our interpretation.
