Chaos in classic and quantum physics

A special session in 

Fourth International Conference on Dynamical Systems and Differential Equations

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


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

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
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
Title: Classical and Quantum Chaos in Fundamental Field Theories
H. Markum, R. Pullirsch, W. Sakuker

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.