Chapter 1 is the exposition of the principal model system used in the present study, the kicked harmonic oscillator. The Hamiltonian of the model system is introduced and some sample applications to physical systems are discussed. After scaling, the POINCARÉ map describing the classical dynamics over one period of the excitation is derived. In cases of resonance, this web map is then used to numerically generate classical stochastic webs in phase space and to explain some of their most important properties, such as the topology and the symmetries of the webs, and the diffusive dynamics in the channels. In cases of nonresonance, the discrete map does not give rise to web-like structures and the dynamics is typically diffusive. -- While this chapter does not contain much original material, it serves two important purposes. First, in comparison with the existing literature on the subject it gives a more readable account of -- some important aspects of -- the theory of stochastic webs. Second, it provides the basic classical results with which the quantum results of the following chapters are to be compared.
In chapter 2 I describe the quantum formulation of the problem. FLOQUET theory is employed to derive the quantum map which is the quantum analogue of the classical POINCARÉ map. The similarities and the fundamental differences of these two mappings are discussed.
Studying the quantum analogue of stochastic webs requires the iteration of the quantum map for a very large number of times. This cannot be done analytically; it can only be accomplished using numerical means. Since the numerical effort to be spent for a single iteration of the quantum map is much larger than for one iteration of the classical POINCARÉ map, it is important to select the most efficient algorithm that is available. In chapter 3 I present and compare three numerical methods that can be used to implement the quantum map on a computer. It turns out that representing the FLOQUET operator in the eigenbasis of the (unkicked) harmonic oscillator is better suited for the present study than using conventional finite differences methods.
Chapters 4 and 5 contain the core results of this study: for several parameter combinations, I iterate the quantum map very often, compare the resulting sequences of quantum states with the corresponding classical dynamics, and give analytical explanations for the observations. For this comparison, a technique is needed that is reviewed in appendix A: the theory of quantum phase space distribution functions can be used to define a quantum analogue of classical phase space; the quantum states are then described equivalently in terms of distribution functions in this quantum phase space that take the role of the classical LIOUVILLE distribution. In this way a direct comparison of the classical and quantum results becomes possible.
In chapter 4, the quantum dynamics in the resonance cases is studied numerically, with the result that there exist quantum stochastic webs if and only if there are classical stochastic webs. The quantum webs resemble their classical counterparts as closely as allowed by the value of , and generically the dynamics in the channels of the webs is diffusive, as in the classical case. In other words, in the quantum webs the dynamics is as classical as can be expected from a quantum wave packet. These numerical findings are then explained analytically using an argument that relies on exploiting the symmetries of the FLOQUET operator and on constructing groups of mutually commuting translation operators in the phase plane that also commute with the FLOQUET operator [BR95].
The complementary case of nonresonance is dealt with in chapter 5. It turns out that in this case the dynamics is similar -- in a well-defined way -- to the dynamics of the quantum kicked rotor, which is known for some time already to exhibit quantum suppression of diffusion, or quantum localization [CCIF79,FGP82]. In chapter 5 I show both numerically and analytically that the same is true for the model system considered here: the nonresonant quantum kicked harmonic oscillator is ANDERSON-localized.
Finally, appendix B contains some technical material needed for the proof of localization in chapter 5, and appendix C is a collection of sample quantum phase portraits of the dynamics of the kicked harmonic oscillator, both in cases of resonance and nonresonance.