21 research outputs found
Periodic Orbits and Spectral Statistics of Pseudointegrable Billiards
We demonstrate for a generic pseudointegrable billiard that the number of
periodic orbit families with length less than increases as , where is a constant and is the average area occupied by these families. We also find that
increases with before saturating. Finally, we show
that periodic orbits provide a good estimate of spectral correlations in the
corresponding quantum spectrum and thus conclude that diffraction effects are
not as significant in such studies.Comment: 13 pages in RevTex including 5 figure
Semiclassical Inequivalence of Polygonalized Billiards
Polygonalization of any smooth billiard boundary can be carried out in
several ways. We show here that the semiclassical description depends on the
polygonalization process and the results can be inequivalent. We also establish
that generalized tangent-polygons are closest to the corresponding smooth
billiard and for de Broglie wavelengths larger than the average length of the
edges, the two are semiclassically equivalent.Comment: revtex, 4 ps figure
Low rank perturbations and the spectral statistics of pseudointegrable billiards
We present an efficient method to solve Schr\"odinger's equation for
perturbations of low rank. In particular, the method allows to calculate the
level counting function with very little numerical effort. To illustrate the
power of the method, we calculate the number variance for two pseudointegrable
quantum billiards: the barrier billiard and the right triangle billiard
(smallest angle ). In this way, we obtain precise estimates for the
level compressibility in the semiclassical (high energy) limit. In both cases,
our results confirm recent theoretical predictions, based on periodic orbit
summation.Comment: 4 page
Classical and quantum decay of one dimensional finite wells with oscillating walls
To study the time decay laws (tdl) of quasibounded hamiltonian systems we
have considered two finite potential wells with oscillating walls filled by non
interacting particles. We show that the tdl can be qualitatively different for
different movement of the oscillating wall at classical level according to the
characteristic of trapped periodic orbits. However, the quantum dynamics do not
show such differences.Comment: RevTeX, 15 pages, 14 PostScript figures, submitted to Phys. Rev.
Hexagonal dielectric resonators and microcrystal lasers
We study long-lived resonances (lowest-loss modes) in hexagonally shaped
dielectric resonators in order to gain insight into the physics of a class of
microcrystal lasers. Numerical results on resonance positions and lifetimes,
near-field intensity patterns, far-field emission patterns, and effects of
rounding of corners are presented. Most features are explained by a
semiclassical approximation based on pseudointegrable ray dynamics and boundary
waves. The semiclassical model is also relevant for other microlasers of
polygonal geometry.Comment: 12 pages, 17 figures (3 with reduced quality
A pseudointegrable Andreev billiard
A circular Andreev billiard in a uniform magnetic field is studied. It is
demonstrated that the classical dynamics is pseudointegrable in the same sense
as for rational polygonal billiards. The relation to a specific polygon, the
asymmetric barrier billiard, is discussed. Numerical evidence is presented
indicating that the Poincare map is typically weak mixing on the invariant
sets. This link between these different classes of dynamical systems throws
some light on the proximity effect in chaotic Andreev billiards.Comment: 5 pages, 5 figures, to appear in PR
Mode structure and ray dynamics of a parabolic dome microcavity
We consider the wave and ray dynamics of the electromagnetic field in a
parabolic dome microcavity. The structure of the fundamental s-wave involves a
main lobe in which the electromagnetic field is confined around the focal point
in an effective volume of the order of a cubic wavelength, while the modes with
finite angular momentum have a structure that avoids the focal area and have
correspondingly larger effective volume. The ray dynamics indicates that the
fundamental s-wave is robust with respect to small geometrical deformations of
the cavity, while the higher order modes are associated with ray chaos and
short-lived. We discuss the incidence of these results on the modification of
the spontaneous emission dynamics of an emitter placed in such a parabolic dome
microcavity.Comment: 50 pages, 17 figure
Evanescent wave approach to diffractive phenomena in convex billiards with corners
What we are going to call in this paper "diffractive phenomena" in billiards
is far from being deeply understood. These are sorts of singularities that, for
example, some kind of corners introduce in the energy eigenfunctions. In this
paper we use the well-known scaling quantization procedure to study them. We
show how the scaling method can be applied to convex billiards with corners,
taking into account the strong diffraction at them and the techniques needed to
solve their Helmholtz equation. As an example we study a classically
pseudointegrable billiard, the truncated triangle. Then we focus our attention
on the spectral behavior. A numerical study of the statistical properties of
high-lying energy levels is carried out. It is found that all computed
statistical quantities are roughly described by the so-called semi-Poisson
statistics, but it is not clear whether the semi-Poisson statistics is the
correct one in the semiclassical limit.Comment: 7 pages, 8 figure
Level statistics and eigenfunctions of pseudointegrable systems: dependence on energy and genus number
We study the level statistics (second half moment and rigidity
) and the eigenfunctions of pseudointegrable systems with rough
boundaries of different genus numbers . We find that the levels form energy
intervals with a characteristic behavior of the level statistics and the
eigenfunctions in each interval. At low enough energies, the boundary roughness
is not resolved and accordingly, the eigenfunctions are quite regular functions
and the level statistics shows Poisson-like behavior. At higher energies, the
level statistics of most systems moves from Poisson-like towards Wigner-like
behavior with increasing . Investigating the wavefunctions, we find many
chaotic functions that can be described as a random superposition of regular
wavefunctions. The amplitude distribution of these chaotic functions
was found to be Gaussian with the typical value of the localization volume
. For systems with periodic boundaries we find
several additional energy regimes, where is relatively close to the
Poisson-limit. In these regimes, the eigenfunctions are either regular or
localized functions, where is close to the distribution of a sine or
cosine function in the first case and strongly peaked in the second case. Also
an interesting intermediate case between chaotic and localized eigenfunctions
appears
Spectral properties of quantized barrier billiards
The properties of energy levels in a family of classically pseudointegrable
systems, the barrier billiards, are investigated. An extensive numerical study
of nearest-neighbor spacing distributions, next-to-nearest spacing
distributions, number variances, spectral form factors, and the level dynamics
is carried out. For a special member of the billiard family, the form factor is
calculated analytically for small arguments in the diagonal approximation. All
results together are consistent with the so-called semi-Poisson statistics.Comment: 8 pages, 9 figure