Ionisation dynamics through a Fano resonance: Time domain interpretation of spectral amplitudes

We investigate a conjecture used in recent experiments to reconstruct the complete dynamics of Fano autoionisation processes out of measured spectral amplitudes [Gruson et al, Science 354 734 (2016); Beaulieu el al, Science 358 1288 (2017); Busto et al, J. Phys. B: At. Mol. Opt. Phys. 51 044002 (2018)]. The validity of the conjecture is established analytically within the formalism of Fano, and tested numerically on model atoms displaying adjustable autoionising states. A general condition for which the conjecture is valid, beyond the Fano case, is then derived.