Chaos, photonic localization and Kepler map

The classical chaos border ( $\epsilon \approx 1/50{\omega^{1/3}n^5}$) for the electron motion in the atom under the action of microwave monochromatic field was obtained in [13] for arbitrary field frequency $\omega$ basing on the Chirikov criterion of overlapping resonances. This approach also allowed to obtain expression for the diffusion rate and ionization times at different $\omega$. Before that there were only estimates of Meerson, Oks and Sasorov for $\omega \approx 1/n^3$ while the estimates given independently by Jensen (1982) for arbitrary $\omega$ were not correct. These results [13] allowed to make comparison of diffusive ionization rate with one-photon ionization rate. This comparison showed that here we have matter with unusual photoelectric effect when ionization rate by direct one-photon transition is much smaller then diffusive ionization at lower frequencies when it is necessary to absorb about 100 photons to ionize the atom [22].

The understanding of the properties of quantum chaos and localization of chaos obtained from the analysis of simple models like kicked rotator allowed to understand the quantum process of microwave ionization. According to the relation $l \sim D$ the localization length is determined by the classical diffusion rate. It is convenient to express the localization length in photonic basis where the probability distribution is exponentially localized with the localization length $l \approx 3.3 \epsilon^2/{\omega^{10/3}}$ [17,26,28,33]. If this length is much less than the number of photons required for ionization $N_{I}=1/{2n^2\omega}$ then the quantum ionization probability is exponentially small in comparison with the classical one. The condition $l \approx N_{I}$ determines the quantum delocalization border $\epsilon_{q}$ above which the quantum ionization process is close to the classical diffusive ionization. It is interesting to note that for fixed $\omega$ the boder $\epsilon_{q}$ grows with the initial value of the principal quantum number $n$ and becomes higher than the classical chaos border [33].

The ionization process is also well described by the simple Kepler map [28,33] which gives the change of photon number $N$ and field phase $\omega t$ after one orbital period of the electron. This map is quite close to the Chirikov standard map and it gives very simple picture of ionization. In the quantum case its close connection with the kicked rotator allows to understand the peculiarity of the quantum ionization. The numerical simulations of the quantum Kepler map give very good agreement with the ionization border obtained in the laboratory experiments [40].

The existence of the quantum delocalization border explains why in the first experiments of Bayfield and Koch (1974-1987) the quantum suppression of chaos was not observed: the experimental conditions were above delocalization border [26,33]. After theoretical explanation of this fact and intensive numerical simulations started in [15] the experiments had been done in the localization regime and the quantum suppression of chaos had been clearly observed for hydrogen atom by Koch (1988), Bayfield (1989) and also for alcali atoms by Walther (1991).