Photoionization is one of the most fundamental processes in laser-matter interaction. It plays a crucial role also from the practical point of view since the electron-density dynamics in a gaseous medium is a process that affects the field propagation. Photoionization has been addressed in different regimes: linear, multiphoton, and tunneling. The ideal tool that allows for the description of ionization, in all aforementioned regimes, is the numerical evaluation of the time-dependent Schrödinger equation. The determination of the electron density needs the computation of the time-dependent ionization probability which unfortunately is an ambiguous quantity due to the gauge dependence of the latter. In this paper, we show how to overcome this difficulty by properly defining the time-dependent ionization probability in the context of the resolvent operator method. We show in particular that the velocity gauge allows for a definition of adiabatic states that is suitable to define an ionization threshold at all times during the interaction to compute ionization probability. Applications to linear, multiphoton, and tunneling regimes are presented for the one-dimensional problem. The extension to the nondipole case is discussed and we show that time-dependent ionization probability cannot be defined unambiguously due to the introduction of the magnetic-field component. We also discuss the case of gauge invariance in a subspace of the eigenbasis defined by the Hamiltonian.

• Received 19 May 2022
• Accepted 4 November 2022

DOI:https://doi.org/10.1103/PhysRevA.106.053115