65 research outputs found
Eigenfunctions in chaotic quantum systems
The structure of wavefunctions of quantum systems strongly depends on the underlying classical dynamics. In this text a selection of articles on eigenfunctions in systems with fully chaotic dynamics and systems with a mixed phase space is summarized. Of particular interest are statistical properties like amplitude distribution and spatial autocorrelation function and the implication of eigenfunction structures on transport properties. For systems with a mixed phase space the separation into regular and chaotic states does not always hold away from the semiclassical limit, such that chaotic states may completely penetrate into the region of the regular island. The consequences of this flooding are discussed and universal aspects highlighted
Homoclinic points of 2-D and 4-D maps via the Parametrization Method
An interesting problem in solid state physics is to compute discrete breather
solutions in coupled 1--dimensional Hamiltonian particle chains
and investigate the richness of their interactions. One way to do this is to
compute the homoclinic intersections of invariant manifolds of a saddle point
located at the origin of a class of --dimensional invertible
maps. In this paper we apply the parametrization method to express these
manifolds analytically as series expansions and compute their intersections
numerically to high precision. We first carry out this procedure for a
2--dimensional (2--D) family of generalized Henon maps (=1), prove
the existence of a hyperbolic set in the non-dissipative case and show that it
is directly connected to the existence of a homoclinic orbit at the origin.
Introducing dissipation we demonstrate that a homoclinic tangency occurs beyond
which the homoclinic intersection disappears. Proceeding to , we
use the same approach to determine the homoclinic intersections of the
invariant manifolds of a saddle point at the origin of a 4--D map consisting of
two coupled 2--D cubic H\'enon maps. In dependence of the coupling the
homoclinic intersection is determined, which ceases to exist once a certain
amount of dissipation is present. We discuss an application of our results to
the study of discrete breathers in two linearly coupled 1--dimensional particle
chains with nearest--neighbor interactions and a Klein--Gordon on site
potential.Comment: 24 pages, 10 figures, videos can be found at
https://comp-phys.tu-dresden.de/supp
Coupling of bouncing-ball modes to the chaotic sea and their counting function
We study the coupling of bouncing-ball modes to chaotic modes in
two-dimensional billiards with two parallel boundary segments. Analytically, we
predict the corresponding decay rates using the fictitious integrable system
approach. Agreement with numerically determined rates is found for the stadium
and the cosine billiard. We use this result to predict the asymptotic behavior
of the counting function N_bb(E) ~ E^\delta. For the stadium billiard we find
agreement with the previous result \delta = 3/4. For the cosine billiard we
derive \delta = 5/8, which is confirmed numerically and is well below the
previously predicted upper bound \delta=9/10.Comment: 10 pages, 6 figure
Temporal flooding of regular islands by chaotic wave packets
We investigate the time evolution of wave packets in systems with a mixed
phase space where regular islands and chaotic motion coexist. For wave packets
started in the chaotic sea on average the weight on a quantized torus of the
regular island increases due to dynamical tunneling. This flooding weight
initially increases linearly and saturates to a value which varies from torus
to torus. We demonstrate for the asymptotic flooding weight universal scaling
with an effective tunneling coupling for quantum maps and the mushroom
billiard. This universality is reproduced by a suitable random matrix model
Localization of Chaotic Resonance States due to a Partial Transport Barrier
Chaotic eigenstates of quantum systems are known to localize on either side
of a classical partial transport barrier if the flux connecting the two sides
is quantum mechanically not resolved due to Heisenberg's uncertainty.
Surprisingly, in open systems with escape chaotic resonance states can localize
even if the flux is quantum mechanically resolved. We explain this using the
concept of conditionally invariant measures from classical dynamical systems by
introducing a new quantum mechanically relevant class of such fractal measures.
We numerically find quantum-to-classical correspondence for localization
transitions depending on the openness of the system and on the decay rate of
resonance states.Comment: 5+1 pages, 4 figure
Partial Weyl Law for Billiards
For two-dimensional quantum billiards we derive the partial Weyl law, i.e.
the average density of states, for a subset of eigenstates concentrating on an
invariant region of phase space. The leading term is proportional to
the area of the billiard times the phase-space fraction of . The
boundary term is proportional to the fraction of the boundary where parallel
trajectories belong to . Our result is numerically confirmed for the
mushroom billiard and the generic cosine billiard, where we count the number of
chaotic and regular states, and for the elliptical billiard, where we consider
rotating and oscillating states.Comment: 5 pages, 3 figures, derivation extended, cosine billiard adde
Linear and logarithmic entanglement production in an interacting chaotic system
We investigate entanglement growth for a pair of coupled kicked rotors. For
weak coupling, the growth of the entanglement entropy is found to be initially
linear followed by a logarithmic growth. We calculate analytically the time
after which the entanglement entropy changes its profile, and a good agreement
with the numerical result is found. We further show that the different regimes
of entanglement growth are associated with different rates of energy growth
displayed by a rotor. At a large time, energy grows diffusively, which is
preceded by an intermediate dynamical localization. The time-span of
intermediate dynamical localization decreases with increasing coupling
strength. We argue that the observed diffusive energy growth is the result of
one rotor acting as an environment to the other which destroys the coherence.
We show that the decay of the coherence is initially exponential followed by a
power-law.Comment: 6 pages, 3 figure
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