410 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
Fast Bit-Flipping based on a Stability Transition of Coupled Spins
A bipartite spin system is proposed for which a fast transfer from one
defined state into another exists. For sufficient coupling between the spins,
this implements a bit-flipping mechanism which is much faster than that induced
by tunneling. The states correspond in the semiclassical limit to equilibrium
points with a stability transition from elliptic-elliptic stability to complex
instability for increased coupling. The fast transfer is due to the spiraling
characteristics of the complex unstable dynamics. Based on the classical system
we find a universal scaling for the transfer time, which even applies in the
deep quantum regime. By investigating a simple model system, we show that the
classical stability transition is reflected in a fundamental change of the
structure of the eigenfunctions
Characterizing quantum chaoticity of kicked spin chains
Quantum many-body systems are commonly considered as quantum chaotic if their
spectral statistics, such as the level spacing distribution, agree with those
of random matrix theory. Using the example of the kicked Ising chain we
demonstrate that even if both level spacing distribution and eigenvector
statistics agree well with random matrix predictions, the entanglement entropy
deviates from the expected Page curve. To explain this observation we propose a
new measure of the effective spin interactions and obtain the corresponding
random matrix result. By this the deviations of the entanglement entropy can be
attributed to significantly different behavior of the -spin interactions
compared with RMT
Fractional-Power-Law Level-Statistics due to Dynamical Tunneling
For systems with a mixed phase space we demonstrate that dynamical tunneling
universally leads to a fractional power law of the level-spacing distribution
P(s) over a wide range of small spacings s. Going beyond Berry-Robnik
statistics, we take into account that dynamical tunneling rates between the
regular and the chaotic region vary over many orders of magnitude. This results
in a prediction of P(s) which excellently describes the spectral data of the
standard map. Moreover, we show that the power-law exponent is proportional to
the effective Planck constant h.Comment: 4 pages, 2 figure
Eigenfunction Statistics on Quantum Graphs
We investigate the spatial statistics of the energy eigenfunctions on large
quantum graphs. It has previously been conjectured that these should be
described by a Gaussian Random Wave Model, by analogy with quantum chaotic
systems, for which such a model was proposed by Berry in 1977. The
autocorrelation functions we calculate for an individual quantum graph exhibit
a universal component, which completely determines a Gaussian Random Wave
Model, and a system-dependent deviation. This deviation depends on the graph
only through its underlying classical dynamics. Classical criteria for quantum
universality to be met asymptotically in the large graph limit (i.e. for the
non-universal deviation to vanish) are then extracted. We use an exact field
theoretic expression in terms of a variant of a supersymmetric sigma model. A
saddle-point analysis of this expression leads to the estimates. In particular,
intensity correlations are used to discuss the possible equidistribution of the
energy eigenfunctions in the large graph limit. When equidistribution is
asymptotically realized, our theory predicts a rate of convergence that is a
significant refinement of previous estimates. The universal and
system-dependent components of intensity correlation functions are recovered by
means of an exact trace formula which we analyse in the diagonal approximation,
drawing in this way a parallel between the field theory and semiclassics. Our
results provide the first instance where an asymptotic Gaussian Random Wave
Model has been established microscopically for eigenfunctions in a system with
no disorder.Comment: 59 pages, 3 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
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
Expanded boundary integral method and chaotic time-reversal doublets in quantum billiards
We present the expanded boundary integral method for solving the planar
Helmholtz problem, which combines the ideas of the boundary integral method and
the scaling method and is applicable to arbitrary shapes. We apply the method
to a chaotic billiard with unidirectional transport, where we demonstrate
existence of doublets of chaotic eigenstates, which are quasi-degenerate due to
time-reversal symmetry, and a very particular level spacing distribution that
attains a chaotic Shnirelman peak at short energy ranges and exhibits GUE-like
statistics for large energy ranges. We show that, as a consequence of such
particular level statistics or algebraic tunneling between disjoint chaotic
components connected by time-reversal operation, the system exhibits quantum
current reversals.Comment: 18 pages, 8 figures, with 3 additional GIF animations available at
http://chaos.fiz.uni-lj.si/~veble/boundary
Correlations between spectra with different symmetry: any chance to be observed?
A standard assumption in quantum chaology is the absence of correlation
between spectra pertaining to different symmetries. Doubts were raised about
this statement for several reasons, in particular, because in semiclassics
spectra of different symmetry are expressed in terms of the same set of
periodic orbits. We reexamine this question and find absence of correlation in
the universal regime. In the case of continuous symmetry the problem is reduced
to parametric correlation, and we expect correlations to be present up to a
certain time which is essentially classical but larger than the ballistic time
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