Superfluorescence, free-induction decay, and four-wave mixing: propagation of free-electron laser pulses through a dense sample of helium ions

We report an experimental and numerical study of the propagation of free-electron laser pulses (wavelength \\SI{24.3}{ano\\metre}) through helium gas. Ionisation and excitation populates the He$^{+}$~4$p$ state. Strong, directional emission was observed at wavelengths of \\SIlist{469;164;30.4;25.6}{ano\\metre}. We interpret the emissions at \\SIlist{469;164}{ano\\metre} as 4$p$-3$s$-2$p$ cascade s…