The ionization rate of a hydrogen atom in an alternating electric field, like that of a laser, can be treated, in the appropriate limit, as the DC ionization rate averaged over a single period of the electric field's oscillation. Multiphoton and tunnel ionization of an atom or a molecule describes the same process by which a bounded electron, through the absorption of more than one photon from the laser field, is ionized. The difference between them is a matter of definition under different conditions. They can henceforth be called multiphoton ionization (MPI) whenever the distinction is not necessary. The dynamics of the MPI can be described by finding the time evolution of the state of the atom which is described by the Schrödinger equation.
When the intensity of the laser is strong, the lowest-order perturbation theory is not sufficient to describe the MPI process. In this case, the laser field on larger distances from the nucleus is more important than the Coulomb potential and the dynamic of the electron in the field should be properly taken into account. The first work in this category was published by Leonid Keldysh. He modeled the MPI process as a transition of the electron from the ground state of the atom to the Volkov states (the state of a free electron in the electromagnetic field). In this model, the perturbation of the ground state by the laser field is neglected and the details of atomic structure in determining the ionization probability are not taken into account. The major difficulty with Keldyshs model was its neglect of the effects of Coulomb interaction on the final state of the electron. As is observed from the figure, the Coulomb field is not very small in magnitude compared to the potential of the laser at larger distances from the nucleus. This is in contrast to the approximation made by neglecting the potential of the laser at regions near the nucleus. A. M. Perelomov, V. S. Popov and M. V. Terentev  included the Coulomb interaction at larger internuclear distances. Their model (which is called the PPT model after their initials) was derived for short-range potential and includes the effect of the long-range Coulomb interaction through the first-order correction in the quasi-classical action. In the quasi-static limit, the PPT model approaches the ADK model by M. V. Ammosov, N. B. Delone, and V. P. Krainov.
Many experiments have been carried out on the MPI of rare gas atoms using strong laser pulses, through measuring both the total ion yield and the kinetic energy of the electrons. Here, one only considers the experiments designed to measure the total ion yield. Among these experiments are those by S. L. Chin et al., S. Augst et al. and T. Auguste et al. Chin et al. used a 10.6 μm CO laser in their experiment. Due to the very small frequency of the laser, the tunneling is strictly quasi-static, a characteristic that is not easily attainable using pulses in the near infrared or visible region of frequencies. These findings weakened the suspicion on the applicability of models basically founded on the assumption of a structureless atom. S. Larochelle et al. have compared the theoretically predicted ion versus intensity curves of rare gas atoms interacting with a :Ti:sapphire laser with experimental measurement. They have shown that the total ionization rate predicted by the PPT model fits very well the experimental ion yields for all rare gases in the intermediate regime of Keldysh parameter.