Photoelectron spectrum from laser-induced autoionization in the presence of spontaneous emission and off-diagonal relaxation

The exact solution of the Fano model, including both radiative recombination to the initial state (recycling) and off-diagonal phase-jitter relaxation, is obtained by two different methods. We derive a simple substitution rule stating that, by replacing $\pm i{\omega }_L\to \pm i{\omega }_L+{\gamma }_T$, the phase-jitter effects are included. The time dependence of the solution has a convolution form. The results for the electron spectrum exhibit an increasing elastic-peak line narrowing the faster the spontaneous decay becomes. For very fast spontaneous decay, the elastic linewidth saturates to ${\gamma }_T$. The relevant thresholds are discussed and demonstrated.