441 research outputs found
Resonance-assisted tunneling in deformed optical microdisks with a mixed phase space
The lifetimes of optical modes in whispering-gallery cavities depend crucially on the underlying classical ray dynamics, and they may be spoiled by the presence of classical nonlinear resonances due to resonance-assisted tunneling. Here we present an intuitive semiclassical picture that allows for an accurate prediction of decay rates of optical modes in systems with a mixed phase space. We also extend the perturbative description from near-integrable systems to systems with a mixed phase space, and we find equally good agreement. Both approaches are based on the approximation of the actual ray dynamics by an integrable Hamiltonian, which enables us to perform a semiclassical quantization of the system and to introduce a ray-based description of the decay of optical modes. The coupling between them is determined either perturbatively or semiclassically in terms of complex paths
Mode fluctuations as fingerprint of chaotic and non-chaotic systems
The mode-fluctuation distribution is studied for chaotic as well as
for non-chaotic quantum billiards. This statistic is discussed in the broader
framework of the functions being the probability of finding energy
levels in a randomly chosen interval of length , and the distribution of
, where is the number of levels in such an interval, and their
cumulants . It is demonstrated that the cumulants provide a possible
measure for the distinction between chaotic and non-chaotic systems. The
vanishing of the normalized cumulants , , implies a Gaussian
behaviour of , which is realized in the case of chaotic systems, whereas
non-chaotic systems display non-vanishing values for these cumulants leading to
a non-Gaussian behaviour of . For some integrable systems there exist
rigorous proofs of the non-Gaussian behaviour which are also discussed. Our
numerical results and the rigorous results for integrable systems suggest that
a clear fingerprint of chaotic systems is provided by a Gaussian distribution
of the mode-fluctuation distribution .Comment: 44 pages, Postscript. The figures are included in low resolution
only. A full version is available at
http://www.physik.uni-ulm.de/theo/qc/baecker.htm
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
Structure of resonance eigenfunctions for chaotic systems with partial escape
Physical systems are often neither completely closed nor completely open, but instead are best described by dynamical systems with partial escape or absorption. In this paper we introduce classical measures that explain the main properties of resonance eigenfunctions of chaotic quantum systems with partial escape. We construct a family of conditionally invariant measures with varying decay rates by interpolating between the natural measures of the forward and backward dynamics. Numerical simulations in a representative system show that our classical measures correctly describe the main features of the quantum eigenfunctions: their multifractal phase-space distribution, their product structure along stable and unstable directions, and their dependence on the decay rate. The (Jensen-Shannon) distance between classical and quantum measures goes to zero in the semiclassical limit for long- and short-lived eigenfunctions, while it remains finite for intermediate cases
Isolated resonances in conductance fluctuations in ballistic billiards
We study numerically quantum transport through a billiard with a classically
mixed phase space. In particular, we calculate the conductance and Wigner delay
time by employing a recursive Green's function method. We find sharp, isolated
resonances with a broad distribution of resonance widths in both the
conductance and the Wigner time, in contrast to the well-known smooth
conductance fluctuations of completely chaotic billiards. In order to elucidate
the origin of the isolated resonances, we calculate the associated scattering
states as well as the eigenstates of the corresponding closed system. As a
result, we find a one-to-one correspondence between the resonant scattering
states and eigenstates of the closed system. The broad distribution of
resonance widths is traced to the structure of the classical phase space.
Husimi representations of the resonant scattering states show a strong overlap
either with the regular regions in phase space or with the hierarchical parts
surrounding the regular regions. We are thus lead to a classification of the
resonant states into regular and hierarchical, depending on their phase space
portrait.Comment: 2 pages, 5 figures, to be published in J. Phys. Soc. Jpn.,
proceedings Localisation 2002 (Tokyo, Japan
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
About ergodicity in the family of limacon billiards
By continuation from the hyperbolic limit of the cardioid billiard we show
that there is an abundance of bifurcations in the family of limacon billiards.
The statistics of these bifurcation shows that the size of the stable intervals
decreases with approximately the same rate as their number increases with the
period. In particular, we give numerical evidence that arbitrarily close to the
cardioid there are elliptic islands due to orbits created in saddle node
bifurcations. This shows explicitly that if in this one parameter family of
maps ergodicity occurs for more than one parameter the set of these parameter
values has a complicated structure.Comment: 17 pages, 9 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
Regular-to-chaotic tunneling rates using a fictitious integrable system
We derive a formula predicting dynamical tunneling rates from regular states
to the chaotic sea in systems with a mixed phase space. Our approach is based
on the introduction of a fictitious integrable system that resembles the
regular dynamics within the island. For the standard map and other kicked
systems we find agreement with numerical results for all regular states in a
regime where resonance-assisted tunneling is not relevant.Comment: 4 pages, 4 figure
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