History of the problem

The researches in this field were initiated by the pioneer experiment of Bayfield and Koch (1974) in which they had observed a strong ionization of an atom with principal quantum number $n \approx 66$ in a linearly polarized microwave field with a field $\epsilon \sim 10 V/cm$ and a frequency $\omega/2\pi = 9.9 GHz$. In this case the value of $\epsilon$ was much less than the value at which ionization takes place in the static electric field, $\epsilon_{st} =0.13/n^4$. Also for ionization it was necessary to absorb approximately 100 photons (here and below we use atomic units).

It happened that this system with quite simple equations of motion lies on the intersection of few modern lines of development in physics being the following: classical and quantum chaos, Anderson localization, multiphoton ionization, Rydberg atoms. Only the knowledge of the physics of these fields allowed to understand the origin of the fast ionization observed in the experiment.

For an explanation of the results of these experiments Delone, Zon and Krainov (1978) put forward a hypothesis about a diffusive mechanism of ionization. On those grounds the diffusion rate in energy and the estimate for a ionization time were obtained. However, the suggested border for the appearance of diffusive ionization $\epsilon \approx 1/n^5$ was based on the quantum perturbation theory and was not correct. Leopold and Percival (1978), on the basis that the principal quantum number $n>>1$, applied for the description of ionization the method of numerical simulation of the classical electron dynamics. As the result they obtained a satisfactory agreement between the ionization probability of the classical atom and its experimental value. The explanation of the physical reason for the appearance of the diffusion and ionization in the classical system was given by Meerson, Oks and Sasorov (1979). They showed that for a field strength above some critical value the overlapping of the resonances took place and the motion of the electron became chaotic leading to its ionization. It is necessary to stress here that the field is strictly monochromatic and there are no random forces acting on the atom. Further laboratory and numerical experiments were made for different values of the initially excited level $n$ and for differents values of the field strength by Jensen, Koch, Leopold and Richards (1985). They showed that the ionization probability obtained in the experiment was close to its classical value and in such a way confirmed the classical picture of the ionization process. These results led these authors to the conclusion that the ionization process, except some fine details, is excellently described by classical mechanics and that the quantum effects had unimportant influence.

The first quantum investigations of microwave ionization of hydrogen atom were done in ref. [15]. The numerical simulations carried out there showed that quantum interference can suppress under certain conditions the chaotic diffusion giving ionization probability much smaller than in the classical case. Further detailed investigations allowed to understand the properties of the quantum system [13,15,17,26,33]. These researches indicated the optimal parameter region in which the effect of the quantum localization of chaos was finally observed in the laboratory experiments of Koch (1988) and Bayfield (1989).