Eigenstate expansion method for simulations of non-perturbative multiphoton processes Original Research Article

A method to solve the time-dependent Schrödinger equation of the hydrogen atom and hydrogen-like ions in intense laser pulses, based on an expansion of the total wavefunction in bound and continuum eigenstates of the unperturbed system, is presented. The problem arising from the well-known singular continuum–continuum dipole matrix elements is fully analyzed and resolved by a combination of analytical integration of the singular contribution and a numerical integration of the singularity-free difference contribution. Unlike in usual finite-difference methods using a spatial grid, in the present approach, probability amplitudes of bound–bound, bound–free and free–free transitions are obtained without additional calculations. The efficacy and accuracy of the present method is demonstrated by calculating above-threshold ionization spectra (ATI) and high harmonic generation spectra (HHG) for different values of laser parameters. Finally, the results obtained by using the present method are compared with that of a finite-difference method involving a spatial grid. The present method proves to be an efficient alternative to the usual spatial grid methods for the solution of the time-dependent Schrödinger equation and the estimation of ATI and HHG spectra in case of intense laser pulses.