The sign of matrix elements in free-electron-laser multiphoton ionization

A low-lying atomic state can be ionized by photons smaller than the binding energy, by absorbing as many photons as needed. This Multiphoton Ionization (MPI) can be understood in terms of off-resonant intermediate states. We study short-pulse two-photon ionization from state i to a continuum state f via a series of high-lying intermediate states n. In second order perturbation theory, two-photon processes are the sum of products of matrix elements over all the possible intermediate states, weighed by the detuning from the ground state for the laser with frequency w.