The kicked hydrogen atom

In this chapter we restrict ourselves to a classical analysis of the ionization process. Within the classical description, we ask the central question  

How does an atom ionize Obviously, subsets of phase space are leaving for the continuum. While this is undisputed, another, more interesting, question arises What is the nature of the sets leaving for the continuum, and what is the nature of the sets that stay bounded  

In Section 2.3 we studied the tent map, a schematic model for ionization that was able to produce fractal structures as a result of ionization. An important question is therefore whether the results presented in Section 2.3 are only of academic interest, or whether fractal structures can appear as a result of ionization in physical systems. In order to answer this question we return to the microwave-driven one-dimensional hydrogen atom. As we know from the previous chapter, this model is ionizing and realistic enough to qualitatively reproduce measured ionization data. Therefore this model is expected to be a fair representative for a large class of chaotic ionization processes. 

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The electron is restricted to move in the half-space x > 0. There is a totally reflecting wall at x = 0. Since the Hamiltonian (8.1.1) of the kicked hydrogen atom and the Hamiltonian of microwave-driven surface state electrons are so similar, we can use many of the results that were derived in Chapter 6. The most important result is the transformation to action and angle variables I and 6, respectively, defined in (6.1.18). The... 

Fig. 8.3. Powerlaw decay of the phase-space probability for the kicked hydrogen atom.

This estimate is independent of a particular location in phase space. This implies that all points of phase space, in particular the period-1 points, are linearly unstable. Thus we have proved that the positively kicked hydrogen atom does indeed not possess any first order eUiptic islands in phase space. It is possible to extend this proof to period-iV points and to show that all period-N points, N integer, are Unearly unstable (Bliimel (1993c)). This implies that the positively kicked hydrogen atom is completely chaotic. We emphasize that, as far as we know, the kicked hydrogen atom is the only model for a physically realizable system where a numerically motivated chaos conjecture was followed up by an analytical proof. In this sense the kicked hydrogen atom is a most remarkable system. 

Prom the physical point of view the absence of stable islands means that all the phase-space probability eventually ionizes. Since all the atomic physics systems investigated to date possess a mixed phase space that shows regular islands embedded in a chaotic sea, the absence of stable islands in the kicked hydrogen atom is a very unique property. 

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