Nonsequential double ionization: a comparison of classical simulations and $S$-matrix results

In a fully classical simulation, we investigate the recollision mechanism of non-sequential double ionization (NSDI) and its manifestation in the end-of-pulse electron momentum distributions. We compare two different electron-electron potentials: a soft-core Coulombic potential and a Yukawa potential. We also implement the strong-field approximation (SFA), which is commonly made in quantum-mechanical $S$-matrix calculations, in our classical simulations and study its consequences. We find that, regardless of the form of the e-e potential, the SFA modifies the momentum distributions and recollision dynamics significantly, but more so for the long-range than for the short-range e-e interaction. Surprisingly, our classical results, especially those obtained with the Yukawa potential under the SFA, agree well with the results from the $S$-matrix calculations with a contact e-e interaction. This implies that the recollision dynamics initiated by a quantum tunneling process and a purely classical process do not deviate much from each other. Furthermore, our classical predictions of travel times are consistent with the results from the simple-man model, and the most probable thermalization time is found to be within 0.15 and 0.25 laser periods.