Bound electron in a strong light field

A study is made of a Green's function of an atomic electron in a light field that takes into account different many-photon transitions. It is shown that in weak fields these transitions lead to damping and a level shift whereas in strong light fields an electron may execute periodic damped oscillations between levels. A two-stage transition through an intermediate resonance state is considered and it is shown that the probability of the total process is determined by the smaller probability of the stages of which it is composed. In a two-level model, a study is made of scattering of light with a transition from the Is state of the hydrogen atom to the $2p$ state and vice versa, this being accompanied by the emission of a photon with a multiple frequency.