43 research outputs found
Hamilton-Jacobi Formulation of KS Entropy for Classical and Quantum Dynamics
A Hamilton-Jacobi formulation of the Lyapunov spectrum and KS entropy is
developed. It is numerically efficient and reveals a close relation between the
KS invariant and the classical action. This formulation is extended to the
quantum domain using the Madelung-Bohm orbits associated with the Schroedinger
equation. The resulting quantum KS invariant for a given orbit equals the mean
decay rate of the probability density along the orbit, while its ensemble
average measures the mean growth rate of configuration-space information for
the quantum system.Comment: preprint, 8 pages (revtex
Scarring on invariant manifolds for perturbed quantized hyperbolic toral automorphisms
We exhibit scarring for certain nonlinear ergodic toral automorphisms. There
are perturbed quantized hyperbolic toral automorphisms preserving certain
co-isotropic submanifolds. The classical dynamics is ergodic, hence in the
semiclassical limit almost all eigenstates converge to the volume measure of
the torus. Nevertheless, we show that for each of the invariant submanifolds,
there are also eigenstates which localize and converge to the volume measure of
the corresponding submanifold.Comment: 17 page
Probabilistic Weyl laws for quantized tori
For the Toeplitz quantization of complex-valued functions on a
-dimensional torus we prove that the expected number of eigenvalues of
small random perturbations of a quantized observable satisfies a natural Weyl
law. In numerical experiments the same Weyl law also holds for ``false''
eigenvalues created by pseudospectral effects.Comment: 33 pages, 3 figures, v2 corrected listed titl
Spectral statistics for quantized skew translations on the torus
We study the spectral statistics for quantized skew translations on the
torus, which are ergodic but not mixing for irrational parameters. It is shown
explicitly that in this case the level--spacing distribution and other common
spectral statistics, like the number variance, do not exist in the
semiclassical limit.Comment: 7 pages. One figure, include
Quantization of multidimensional cat maps
In this work we study cat maps with many degrees of freedom. Classical cat
maps are classified using the Cayley parametrization of symplectic matrices and
the closely associated center and chord generating functions. Particular
attention is dedicated to loxodromic behavior, which is a new feature of
two-dimensional maps. The maps are then quantized using a recently developed
Weyl representation on the torus and the general condition on the Floquet
angles is derived for a particular map to be quantizable. The semiclassical
approximation is exact, regardless of the dimensionality or of the nature of
the fixed points.Comment: 33 pages, latex, 6 figures, Submitted to Nonlinearit
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
Quantum ergodicity for Pauli Hamiltonians with spin 1/2
Quantum ergodicity, which expresses the semiclassical convergence of almost
all expectation values of observables in eigenstates of the quantum Hamiltonian
to the corresponding classical microcanonical average, is proven for
non-relativistic quantum particles with spin 1/2. It is shown that quantum
ergodicity holds, if a suitable combination of the classical translational
dynamics and the spin dynamics along the trajectories of the translational
motion is ergodic.Comment: 20 pages, no figure
Delocalization of slowly damped eigenmodes on Anosov manifolds
We look at the properties of high frequency eigenmodes for the damped wave
equation on a compact manifold with an Anosov geodesic flow. We study
eigenmodes with spectral parameters which are asymptotically close enough to
the real axis. We prove that such modes cannot be completely localized on
subsets satisfying a condition of negative topological pressure. As an
application, one can deduce the existence of a "strip" of logarithmic size
without eigenvalues below the real axis under this dynamical assumption on the
set of undamped trajectories.Comment: 28 pages; compared with version 1, minor modifications, add two
reference
Discrete Wigner functions and the phase space representation of quantum teleportation
We present a phase space description of the process of quantum teleportation
for a system with an dimensional space of states. For this purpose we
define a discrete Wigner function which is a minor variation of previously
existing ones. This function is useful to represent composite quantum system in
phase space and to analyze situations where entanglement between subsystems is
relevant (dimensionality of the space of states of each subsystem is
arbitrary). We also describe how a direct tomographic measurement of this
Wigner function can be performed.Comment: 8 pages, 1 figure, to appear in Phys Rev
Semi-classical study of the Quantum Hall conductivity
The semi-classical study of the integer Quantum Hall conductivity is
investigated for electrons in a bi-periodic potential .
The Hall conductivity is due to the tunnelling effect and we concentrate our
study to potentials having three wells in a periodic cell. A non-zero
topological conductivity requires special conditions for the positions, and
shapes of the wells. The results are derived analytically and well confirmed by
numerical calculations.Comment: 23 pages, 13 figure