Numerical solution of time-dependent Schrödinger equation for multiphoton processes: A matrix iterative method

An implicit algorithm for integration of the three-dimensional (3D) time-dependent Schrödinger equation of an atomic system interacting with intense laser pulses is developed. It is based on a matrix iteration of the Crank-Nicholson approximant to the short-time propagator using the total Hamiltonian (unsplit) of the system directly. To test the method, 3D Schrödinger wave-packet propagation is carried out, and so-called above-threshold ionization and high-harmonic generation spectra for atomic hydrogen irradiated by intense laser pulses are obtained. They are also compared with that obtained using the popular split-operator method. The present algorithm is shown to provide an alternative to the the split-operator method, and proves to be more efficient in all the cases studied here. A procedure for optimizing the maximum grid size is also given, and its usefulness is illustrated.