On Quantum Chaos

*Quantum chaos* is a controversial subject
[SZ94].
On the basis of the correspondence principle one might be tempted to take
quantum chaos for granted, because classical mechanics is supposed to be
a special limiting case of quantum mechanics, and clearly there are
classical dynamical systems that are known to exhibit chaos
-- cf. for example [AFH95,KH95a]
and the comprehensive lists of references therein.

But this approach is problematic:
while stating that in the limit the quantum
dynamics transforms into the classical dynamics of the system,
the correspondence principle does
not say *how* this quantum-classical transition comes about.
There is no
well-defined algorithm that maps the two
dynamical theories onto each other;
the semiclassical limit of quantum mechanics is mathematically
singular [Ber01].
With such an algorithm, given a classically
chaotic system, it would be possible to uniquely identify those properties
of the corresponding quantum system that mark its quantum chaoticity,
based on the well-established concepts of classical chaos theory:
LIAPUNOV exponents, (fractal) dimensions, entropies, etc.
Conversely, since there is no such algorithm and the concepts of
chaotic dynamics originate in the theory of classical mechanics,
until today nobody can define in a straightforward way what makes a
quantum system chaotic and thus what quantum chaos really is.
With such a definition it would be possible to identify a quantum system
as chaotic without having to discuss its classical mechanics beforehand.

Another aspect of the same problem is the fact that quantum mechanical phase space is quantized into cells of finite size, given by . As a consequence, the concept of sensitive dependence on initial conditions -- an essential ingredient of the theory of classical chaos -- cannot be applied to quantum mechanics in a straightforward way. What is more, if the (region of) phase space under consideration is of finite volume then the corresponding quantum dynamics effectively evolves on different phase space cells only, which makes the HILBERT space effectively finite-dimensional, brings about quasiperiodic dynamics and thus excludes chaos in the classical sense [KW96]. This is to be contrasted with the continuum of phase space points in any nontrivial classical phase space and illustrates again the fundamental differences between the two dynamical theories.

Note that the above reasoning leading to the absence of classical chaos in
quantum mechanics is also based on the *linearity* of the
SCHRÖDINGER equation;
the canonical equations of motion of classical mechanics, on the other
hand, are in general *nonlinear*.
BERRY considers this distinction to be ``the ultimate reason for the
absence of quantum chaos'' [Ber89].

To give an example, since the days of POINCARÉ the three-body problem [Bru94] is a paradigm of classically chaotic nonlinear dynamics -- see [Klu97,Hip97] for some recent studies of classical molecular three-body systems and [Con02] for applications in astronomy. However, its quantum counterparts, for example the ion and the He atom, can be analyzed with relative ease using conventional quantum techniques [Nol02], their eigenvalues and eigenfunctions are obtained accurately, and from this point of view it is difficult to see why these apparently well-behaved quantum three-body problems should be examples of quantum chaos.

Nevertheless, motivated by random matrix theory, one also has the
BOHIGAS-GIANNONI-SCHMIT conjecture
[BGS84]
which
claims
that *there are* manifestations of classical
chaos in quantum energy level spacing distributions: classical
integrability is associated with POISSON distributions
and level clustering, whereas for
(hard) classical chaos WIGNER distributions and level repulsion are
expected.
The conjecture is yet unproven, but there is
much evidence
for its validity [Stö99].
--
Similarly, there is the general conjecture that classically chaotic
systems with diffusive dynamics are characterized
by quantum suppression of diffusion
[CCIF79,Ber89]:
another quantum fingerprint of classical chaos.

A pragmatic way out of this dilemma is to adopt a working definition of
quantum chaos that avoids the correspondence principle altogether, and to
define *quantum chaos* as the study of quantum systems whose
classical counterparts exhibit classical chaos
[CGS93].
This definition
-- often also termed *quantum chaology* [Ber89],
trying to avoid the
impression that the quantum dynamics might be chaotic per se --
appears to be accepted by more and more researchers in the field
(see for example
[GVZJ91,CC95,Stö99,Haa01]).
Note that this definition makes no ad hoc statement about the actual
properties of the quantum systems, other than their classical chaoticity.

With the above definition, quantum chaos integrates smoothly into the
development of the theory of nonlinear dynamics, or chaos theory,
as it has taken place since the middle of the last century.
Using both the increasing theoretical understanding of the dynamics and
computers which constantly became more powerful,
a lot of knowledge about classical dynamical systems has been accumulated.
The natural next step
is
to look for the quantum manifestations of classical chaos.
With the present study I want to make a contribution to this
development by investigating the quantum mechanics of a system the
chaotic classical mechanics of which has been studied
in some detail
in the past: the *kicked harmonic oscillator.*

Among the classically chaotic systems,
those that exhibit *hard chaos* take an exceptional role.
Hard chaotic systems are characterized by complete absence of
regular regions in classical phase space:
all orbits, including the periodic orbits, are unstable.
Once all periodic orbits of such a system are known, the quantum mechanics
of the system can be studied in the semiclassical approximation
using, for example, GUTZWILLER's
trace formulas
[Gut90,Gut91].
This theory is especially successful for bound dynamics
[BB97],
but has been extended to scattering dynamics as well
[Wir99,CAM^{+}03].

Billiards represent a frequently studied class of model systems for semiclassical quantization. Inside a billiard, the dynamics is free, and therefore easy to solve classically and quantum mechanically. The billiard boundaries then account for the nontrivial part of the dynamics; in particular, defocusing boundaries can be chosen such that hard chaotic classical dynamics with a complete symbolic dynamics is obtained, and the methods of semiclassical quantization can be applied [Hor93,Jun97,Hau00]. Two-dimensional quantum billiard systems have the striking additional advantage of being accessible to direct experimental verification of the numerical and analytical results: these systems can be modelled by microwave resonators, also known as microwave billiards, which allow to measure the wave function of the system using a macroscopic experimental setup; this method is based on the fact that for two-dimensional systems, both the SCHRÖDINGER equation for the quantum billiard and the MAXWELL equations for the microwave resonator give rise to the same HELMHOLTZ equation with essentially identical boundary conditions [Stö99].

The generic
situation
in nonlinear Hamiltonian dynamical systems,
on the other hand, is
*soft* or *weak chaos*.
Here, the phase portrait is mixed and consists of regions of
regular and irregular or chaotic dynamics at the same time.
Often such systems can be analyzed
from a perturbation point of view,
starting from a completely integrable system to which an increasingly
strong perturbation is added, rendering the system nonintegrable and
leading to growing phase space regions of chaotic dynamics.

The kicked harmonic oscillator is a
typical
representative of this class of systems, the integrable component being
the well-known harmonic oscillator to which an impulsive time-periodic
forcing is added.
The kicked harmonic oscillator is *un*typical, however,
in that respect
that it can generate a very special kind of classical phase portraits,
namely *stochastic webs*. These extended, periodic structures in
phase space contain both bounded cells of regular dynamics and infinitely
long channels of chaotic, or stochastic, motion.
Due to the degeneracy of the harmonic oscillator, its classical dynamics
cannot be analyzed in terms of the elegant standard theory of perturbed
integrable systems, KAM theory
[LL92,JS98]. Rather, a theory
of its own had to be developed in order to understand the stochastic webs
that are generated by this system [ZSUC91].

In contrast to classically hard chaotic systems, the quantum mechanics of systems exhibiting weak chaos is much less understood from a general point of view. The methods of semiclassical quantization via periodic orbit theory, as mentioned above, cannot be applied in a straightforward way. Therefore, a significant part of the quantum investigations of weakly chaotic systems have been performed using quantum mechanical methods without -related approximations, allowing to consider both the full quantum regime and the semiclassical case [Zas85,Hei92,CC95,BR97] (and references therein). The same approach is followed in the present study.

Beyond the lack of a suitable semiclassical theory, studying the quantum dynamics of the kicked harmonic oscillator is further complicated by the quantum fingerprints of its diffusive dynamics in the channels of the classical webs: in cases of resonance between the harmonic oscillator's eigenfrequency and the frequency of the perturbation, the evolving quantum states can extend quite rapidly into large areas in phase space, which greatly increases the numerical effort needed to study the system. It is interesting to note how this numerical complexity of the system has been assessed by some prominent researchers in this field:

``The numerical investigation of this model [the quantum kicked harmonic oscillator] is much more difficult than in the kicked HARPER case, since the dynamics leads to spreading in the whole -plane... This seems to be the reason why there were practically no attempts to investigate the quantum dynamics of [this] model'' [SS92].While this remark is slightly outdated by now, it is still true that the kicked harmonic oscillator is a seldom studied object with respect to quantum chaos. To put that into perspective, the quantum dynamics of the paradigmatic example of nonlinear Hamiltonian kick dynamics, the