  To shorten the duration of x-ray pulses, we present a nonlinear optical
technique using atoms with core-hole vacancies (core-hole atoms) generated by
inner-shell photoionization. The weak Coulomb screening in the core-hole atoms
results in decreased absorption at photon energies immediately above the
absorption edge. By employing this phenomenon, referred to as saturable
absorption, we successfully reduce the duration of x-ray free-electron laser
pulses (photon energy: 9.000 keV, duration: 6-7 fs, fluence:
2.0-3.5$\times$10$^5$ J/cm$^2$) by $\sim$35%. This finding that core-hole atoms
are applicable to nonlinear x-ray optics is an essential stepping stone for
extending nonlinear technologies commonplace at optical wavelengths to the hard
x-ray region.
