Fourth International Conference on Dynamical Systems and Differential Equations

Wilmington, NC, USA, May 24-27, 2002

- Helmut Kroger, Laval University, Quebec, Canada (hkroger@phy.ulaval.ca)

**List of invited speakers:**

(1) J. Bird, Arizona State Univ.

(2) J. Delos, College of William and Mary, Williamsburg (tentative)

(3) H. Friedrich, Tech. Univ. Munich

(4) F. Haake, Essen University

(5) R. Jensen, Weslyan University

(6) H. Kr\"oger, Laval University.

(7) H. Markum, Tech. Univ. Vienna (undecided)

(8) L. Reichl, Austin, Univ. of Texas

(9) S. Sridhar, Northeastern Univ, Boston

(10) H. Taylor, Univ. of Southern California, Los Angeles (tentative)

**Titles Abstracts:**

**Dr. J. Bird**,

Title: Open Quantum Dots as a Probe of Quantum Chaos: Going Beyond
Convenient

Mathematical Approaches

Authors: J. P. Bird, R. Akis, D. K. Ferry, A. P. de Moura & Y.-C. Lai

Address: Department of Electrical Engineering, Arizona State University,
Tempe, AZ

85287-5706

Abstract: Semiconductor quantum dots are sub-micron sized structures that
consist of a

mesoscopic scattering region, fabricated on length scales much smaller than

the electron mean-free path. The relevant current-flow process through such

structures is therefore one in which electrons are injected into the cavity,

and undergo multiple scattering from the walls of the dot before finally

escaping to the external reservoirs. For this reason, these ballistic

structures may be viewed as the solid-state analog of classical scattering

billiards, and their electrical properties have attracted much interest as

an experimental probe of quantum chaos (for a recent review, see Ref. [1]).

In the development of a theoretical description of these structures, there

has unfortunately been a tendency to introduce a number of limiting

assumptions, which provide for mathematical simplicity at the expense of

physical reality. In this presentation, we will therefore discuss the

results of experimental and theoretical studies of open quantum dots, which

go beyond these artificial approaches to develop a better understanding of

electron dynamics in these structures. From quantum-mechanical simulations

of our experimental results, we find strong evidence for the role of scarred

wavefunction states in transport through the dots [2]. This behavior is in

turn shown to be related to an effect analogous to resonance trapping, in

which transport through the open system can be understood in terms of

selected eigenstates of the closed structure. We will also discuss the

results of a more resent semiclassical analysis, which has pointed to a

consistent picture in which the scars are found to be associated with

classically-inaccessible orbits, which end up playing an important role in

transport due to the process of dynamical (or phase-space) tunneling [3].

[1] J. P. Bird, "Recent experimental studies of electron transport in open

quantum dots", J. Phys.: Condens. Matter 11, R413 (1999).

[2] J. P. Bird, R. Akis, D. K. Ferry, D. Vasileska, J. Cooper, Y. Aoyagi,

and T. Sugano, "Lead orientation dependent wavefunction scarring in open

quantum dots", Phys. Rev. Lett. 82, 4691 (1999).

[3] A. P. S. de Moura, Y.-C. Lai, R. Akis, J. P. Bird, and D. K. Ferry,

"Tunneling and nonhyperbolicity in quantum dots", submitted for
publication.

**Dr. Harald Friedrich**, Technische Universit\"at M\"unchen

Title: Semiclassical and anticlassical limits of the Schr"odinger equation

Abstract: The natural place to study how the chaotic nature of a classical
system

manifests itself in the corresponding quantum system is in or near the

semiclassical limit of the quantum system. For billard systems this is,

quite generally, the high-energy limit, but for more realistic one- or

many-body systems described by the appropriate Schr"odinger equation,

the semiclassical limit may not be so easy to identify. In this talk I

identify the semiclassical limit of the Schr"odinger equation - and the

anticlassical or extreme quantum limit - for several examples including

one-electron atoms in external fields and two- or more-electron atoms.

As a curiosity I mention a class of one-dimensional bound systems where

the quantum number grows to infinity without approaching the

semiclassical limit.

**Dr. Helmut Kr\"oger, **Laval University

Title: Quantum Chaos at Finite Temperature in the Paul Trap

Abstract: In quantum chaos, we have no "local" information of
the degree of chaoticity, being available in classical chaos via Lyapunov exponents
and Poincar\'e sections from phase space.
Also little is known about the role of temperature in quantum chaos.
For example, an analysis of level densities is insensitive to temperature.
Here we present a new method aiming to overcome those problems.

Recently, my co-workers and I have proposed the concept of a quantum

action. This action has a mathematical structure like the classical

action, but takes into account quantum effects (quantum fluctuations)

via renormalized action parameters. This bridges the gap between

classical physics and quantum physics. Lyapunov exponents,

Poincar\'e sections etc. can be computed to study quantum chaos via

the quantum action. The quantum action has been applied and proven

useful to precisely define and quantitatively compute quantum instantons.

Here we use the quantum action to study quantum chaos at finite

temperature. We present a numerical study of 2-D Hamiltonian systems

which are classically chaotic. First we consider harmonic oscillators

with weak anharmonic coupling. We construct the quantum action

non-perturbatively and find temperature dependent quantum corrections

in the action parameters. We compare Poincar\'e sections of the quantum

action at finite temperature with those of the classical action. We observe

chaotic behavior for both. Secondly, we consider the Hamiltonian of

the Paul trap, which is quite important for atomic clocks, in Bose-Einstein

condensation etc. We present Poincar\'e sections as function of temperature

and compare its chaotic behavior with its classical counterpart.

**F. Haake
**Title: Long period orbits from symbolic dynamics

Abstract: Working with a sequence of local sections of the symbol sequence one can construct periodic orbits of extremely long periods. The accuracy attained grows exponentially with the length of the local sections. By properly including correlations between symbols one obtains orbits behaving ergodically.

**John Delos**

Title: Chaotic Ionization of Hydrogen in Parallel Fields

J. B. Delos and K. A. Mitchell

Abstract:

We examine the classical ionization of photoexcited states of hydrogen in
parallel electric and magnetic fields. This system reduces to an area-preserving
chaotic map of the plane and is a useful and experimentally accessible model of
chaotic decay and scattering. Decay is studied by examining segments of a line
of initial conditions that escape at various iterates of the map. These segments
exhibit what we call "epistrophic self-similarity": the segments
organize themselves into self-similar geometric sequences, but the beginnings of
these sequences are only partially predictable. Points that remain form a
Cantor set which we call an "epistrophic fractal". These studies
should be important for the analysis of classical decay rates, which numerical
studies have shown typically exhibit algebraic rather than exponential decay.
They should also be important for the analysis of semiclassical decay and
scattering, including the construction of the semi-classical propagator and
S-matrix.

**Harald Markum**

Institut f"ur Kernphysik, Technische Universit"at Wien, Wiedner
Hauptstrasse 8-10

A-1040 Wien, Austria, Tel: 43-1-58801-5579, FAX: 43-1-564203

E-mail: markum@kph.tuwien.ac.at

Title: Classical and Quantum Chaos in Fundamental Field Theories

H. Markum, R. Pullirsch, W. Sakuker

Abstract:

The role of chaotic field dynamics for the confinement of quarks is a
longstanding question. Concerning classical chaos, we analyze the leading
Lyapunov exponents of Yang-Mills field configurations on the lattice. Concerning
the quantum case, we investigate the eigenvalue spectrum of the staggered Dirac
operator in QCD at nonzero temperature. The quasi-zero modes and their role for
chiral symmetry breaking and the deconfinement transition are examined. The bulk
of the spectrum and its relation to quantum chaos is considered. Our results
demonstrate that chaos is present when particles are confined, but it persists
also into the quark-gluon-plasma phase. Further, we decompose U(1) gauge fields
into a monopole and photon part across the phase transition from the confinement
to the Coulomb phase. We analyze the leading Lyapunov exponents of such gauge
field configurations on the lattice which are initialized by quantum Monte Carlo
simulations. It turns out that there is a strong relation between the
sizes of the monopole density and the Lyapunov exponent.