329 research outputs found
Poincar\'e Husimi representation of eigenstates in quantum billiards
For the representation of eigenstates on a Poincar\'e section at the boundary
of a billiard different variants have been proposed. We compare these
Poincar\'e Husimi functions, discuss their properties and based on this select
one particularly suited definition. For the mean behaviour of these Poincar\'e
Husimi functions an asymptotic expression is derived, including a uniform
approximation. We establish the relation between the Poincar\'e Husimi
functions and the Husimi function in phase space from which a direct physical
interpretation follows. Using this, a quantum ergodicity theorem for the
Poincar\'e Husimi functions in the case of ergodic systems is shown.Comment: 17 pages, 5 figures. Figs. 1,2,5 are included in low resolution only.
For a version with better resolution see
http://www.physik.tu-dresden.de/~baecker
Universality in the flooding of regular islands by chaotic states
We investigate the structure of eigenstates in systems with a mixed phase
space in terms of their projection onto individual regular tori. Depending on
dynamical tunneling rates and the Heisenberg time, regular states disappear and
chaotic states flood the regular tori. For a quantitative understanding we
introduce a random matrix model. The resulting statistical properties of
eigenstates as a function of an effective coupling strength are in very good
agreement with numerical results for a kicked system. We discuss the
implications of these results for the applicability of the semiclassical
eigenfunction hypothesis.Comment: 11 pages, 12 figure
Dynamical tunneling in mushroom billiards
We study the fundamental question of dynamical tunneling in generic
two-dimensional Hamiltonian systems by considering regular-to-chaotic tunneling
rates. Experimentally, we use microwave spectra to investigate a mushroom
billiard with adjustable foot height. Numerically, we obtain tunneling rates
from high precision eigenvalues using the improved method of particular
solutions. Analytically, a prediction is given by extending an approach using a
fictitious integrable system to billiards. In contrast to previous approaches
for billiards, we find agreement with experimental and numerical data without
any free parameter.Comment: 4 pages, 4 figure
Autocorrelation function of eigenstates in chaotic and mixed systems
We study the autocorrelation function of different types of eigenfunctions in
quantum mechanical systems with either chaotic or mixed classical limits. We
obtain an expansion of the autocorrelation function in terms of the correlation
length. For localized states, like bouncing ball modes or states living on
tori, a simple model using only classical input gives good agreement with the
exact result. In particular, a prediction for irregular eigenfunctions in mixed
systems is derived and tested. For chaotic systems, the expansion of the
autocorrelation function can be used to test quantum ergodicity on different
length scales.Comment: 30 pages, 12 figures. Some of the pictures are included in low
resolution only. For a version with pictures in high resolution see
http://www.physik.uni-ulm.de/theo/qc/ or http://www.maths.bris.ac.uk/~maab
Chaotic eigenfunctions in momentum space
We study eigenstates of chaotic billiards in the momentum representation and
propose the radially integrated momentum distribution as useful measure to
detect localization effects. For the momentum distribution, the radially
integrated momentum distribution, and the angular integrated momentum
distribution explicit formulae in terms of the normal derivative along the
billiard boundary are derived. We present a detailed numerical study for the
stadium and the cardioid billiard, which shows in several cases that the
radially integrated momentum distribution is a good indicator of localized
eigenstates, such as scars, or bouncing ball modes. We also find examples,
where the localization is more strongly pronounced in position space than in
momentum space, which we discuss in detail. Finally applications and
generalizations are discussed.Comment: 30 pages. The figures are included in low resolution only. For a
version with figures in high resolution see
http://www.physik.uni-ulm.de/theo/qc/ulm-tp/tp99-2.htm
On the rate of quantum ergodicity in Euclidean billiards
For a large class of quantized ergodic flows the quantum ergodicity theorem
due to Shnirelman, Zelditch, Colin de Verdi\`ere and others states that almost
all eigenfunctions become equidistributed in the semiclassical limit. In this
work we first give a short introduction to the formulation of the quantum
ergodicity theorem for general observables in terms of pseudodifferential
operators and show that it is equivalent to the semiclassical eigenfunction
hypothesis for the Wigner function in the case of ergodic systems. Of great
importance is the rate by which the quantum mechanical expectation values of an
observable tend to their mean value. This is studied numerically for three
Euclidean billiards (stadium, cosine and cardioid billiard) using up to 6000
eigenfunctions. We find that in configuration space the rate of quantum
ergodicity is strongly influenced by localized eigenfunctions like bouncing
ball modes or scarred eigenfunctions. We give a detailed discussion and
explanation of these effects using a simple but powerful model. For the rate of
quantum ergodicity in momentum space we observe a slower decay. We also study
the suitably normalized fluctuations of the expectation values around their
mean, and find good agreement with a Gaussian distribution.Comment: 40 pages, LaTeX2e. This version does not contain any figures. A
version with all figures can be obtained from
http://www.physik.uni-ulm.de/theo/qc/ (File:
http://www.physik.uni-ulm.de/theo/qc/ulm-tp/tp97-8.ps.gz) In case of any
problems contact Arnd B\"acker (e-mail: [email protected]) or Roman
Schubert (e-mail: [email protected]
Geometrical theory of diffraction and spectral statistics
We investigate the influence of diffraction on the statistics of energy
levels in quantum systems with a chaotic classical limit. By applying the
geometrical theory of diffraction we show that diffraction on singularities of
the potential can lead to modifications in semiclassical approximations for
spectral statistics that persist in the semiclassical limit . This
result is obtained by deriving a classical sum rule for trajectories that
connect two points in coordinate space.Comment: 14 pages, no figure, to appear in J. Phys.
Stochastic stabilization of cosmological photons
The stability of photon trajectories in models of the Universe that have
constant spatial curvature is determined by the sign of the curvature: they are
exponentially unstable if the curvature is negative and stable if it is
positive or zero. We demonstrate that random fluctuations in the curvature
provide an additional stabilizing mechanism. This mechanism is analogous to the
one responsible for stabilizing the stochastic Kapitsa pendulum. When the mean
curvature is negative it is capable of stabilizing the photon trajectories;
when the mean curvature is zero or positive it determines the characteristic
frequency with which neighbouring trajectories oscillate about each other. In
constant negative curvature models of the Universe that have compact topology,
exponential instability implies chaos (e.g. mixing) in the photon dynamics. We
discuss some consequences of stochastic stabilization in this context.Comment: 4 pages, 3 postscript figures in color which are also appropriate for
black and white printers; v2 emphasizes relevance to flat as well as
negatively curved cosmologies; to appear in J. Phys.
Localization of Eigenfunctions in the Stadium Billiard
We present a systematic survey of scarring and symmetry effects in the
stadium billiard. The localization of individual eigenfunctions in Husimi phase
space is studied first, and it is demonstrated that on average there is more
localization than can be accounted for on the basis of random-matrix theory,
even after removal of bouncing-ball states and visible scars. A major point of
the paper is that symmetry considerations, including parity and time-reversal
symmetries, enter to influence the total amount of localization. The properties
of the local density of states spectrum are also investigated, as a function of
phase space location. Aside from the bouncing-ball region of phase space,
excess localization of the spectrum is found on short periodic orbits and along
certain symmetry-related lines; the origin of all these sources of localization
is discussed quantitatively and comparison is made with analytical predictions.
Scarring is observed to be present in all the energy ranges considered. In
light of these results the excess localization in individual eigenstates is
interpreted as being primarily due to symmetry effects; another source of
excess localization, scarring by multiple unstable periodic orbits, is smaller
by a factor of .Comment: 31 pages, including 10 figure
Intermediate statistics in quantum maps
We present a one-parameter family of quantum maps whose spectral statistics
are of the same intermediate type as observed in polygonal quantum billiards.
Our central result is the evaluation of the spectral two-point correlation form
factor at small argument, which in turn yields the asymptotic level
compressibility for macroscopic correlation lengths
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