This thesis describes a study into the eigenvalues and eigenstates of twodimensional
(2D) quantum systems. The research is summarized in four scientific
publications by the author. The underlying motivation for this work is the grand
question of quantum chaos: how does chaos, as known in classical mechanics,
manifest in quantum mechanics? The search and analysis of these quantum fingerprints
of chaos requires efficient numerical tools and methods, the development
of which is given a special emphasis in this thesis.
The first publication in this thesis concerns the eigenspectrum analysis of a
nanoscale device. It is shown that a measured addition energy spectrum can
be explained by a simple confinement of interacting electrons in a potential
well. This result is derived using density-functional theory (DFT) and numerical
optimization, guided by the asymmetric geometry of the device. The calculations
also show that an observed decrease in the conductance can be explained by
a
change in the shape of the quantum wave function.
The study of quantum chaos by statistical properties of eigenvalues requires
a way to solve the eigenvalue spectrum of a quantum system up to highly
excited states. The second publication describes a numerical program, itp2d,
that uses modern advances in the imaginary time propagation (ITP) method
to solve the eigenvalue spectrum of generic 2D systems up to thousands of
eigenstates. The program provides means to sophisticated eigenvalue analysis
involving long-range correlations, and a unique view to highly excited eigenstates
of complicated 2D systems, such as those involving magnetic fields and strong
disorder.
After the spectrum is solved, the next step in the eigenvalue analysis is to
remove the trivial part of the spectrum in a process known as unfolding. Unless
the system belongs to a special class for which the trivial part is known, unfolding
is an ambiguous process that can cause substantial artifacts to the eigenvalue
statistics. Recently it has been proposed that these artifacts can be mitigated by
employing the empirical mode decomposition (EMD) algorithm in the unfolding.
The third publication describes an efficient implementation of this algorithm,
which is also highly useful in other kinds of data analysis.
Quantum scarring refers to the condensation of quantum eigenstate probability
density around unstable classical periodic orbits in chaotic systems. It
represents a useful and visually striking quantum suppression of chaos. The
final publication describes the discovery of a new kind of quantum scarring
in symmetric 2D systems perturbed by local disorder. These unusually strong
quantum scars are not explained by ordinary scar theory. Instead, they are
caused by classical resonances and resulting quantum near-degeneracy in the
unperturbed system. Wave-packet analysis shows that the scars greatly influence
the transport properties of these systems, even to the extent that wave packets
launched along the scar path travel with higher fidelity than in the corresponding
unperturbed system. This discovery raises interesting possibilities of selectively
enhancing the conductance of quantum systems by adding local perturbations.
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