Outline of this Study

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.