155 research outputs found
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
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
Regular-to-Chaotic Tunneling Rates: From the Quantum to the Semiclassical Regime
We derive a prediction of dynamical tunneling rates from regular to chaotic
phase-space regions combining the direct regular-to-chaotic tunneling mechanism
in the quantum regime with an improved resonance-assisted tunneling theory in
the semiclassical regime. We give a qualitative recipe for identifying the
relevance of nonlinear resonances in a given -regime. For systems with
one or multiple dominant resonances we find excellent agreement to numerics.Comment: 4 pages, 3 figures, reference added, small text change
Dynamical Tunneling in Systems with a Mixed Phase Space
Tunneling is one of the most prominent features of quantum mechanics. While the tunneling process in one-dimensional integrable systems is well understood, its quantitative prediction for systems with mixed phase space is a long-standing open challenge. In such systems regions of regular and chaotic dynamics coexist in phase space, which are classically separated but quantum mechanically coupled by the process of dynamical tunneling. We derive a prediction of dynamical tunneling rates which describe the decay of states localized inside the regular region towards the so-called chaotic sea. This approach uses a fictitious integrable system which mimics the dynamics inside the regular domain and extends it into the chaotic region. Excellent agreement with numerical data is found for kicked systems, billiards, and optical microcavities, if nonlinear resonances are negligible.
Semiclassically, however, such nonlinear resonance chains dominate the tunneling process. Hence, we combine our approach with an improved resonance-assisted tunneling theory and derive a unified prediction which is valid from the quantum to the semiclassical regime. We obtain results which show a drastically improved accuracy of several orders of magnitude compared to previous studies.Der Tunnelprozess ist einer der bedeutensten Effekte in der Quantenmechanik. Während das Tunneln in eindimensionalen integrablen Systemen gut verstanden ist, gestaltet sich dessen Beschreibung für Systeme mit gemischtem Phasenraum weitaus schwieriger. Solche Systeme besitzen Gebiete regulärer und chaotischer Bewegung, die klassisch getrennt sind, aber quantenmechanisch durch den Prozess des dynamischen Tunnelns gekoppelt werden. In dieser Arbeit wird eine theoretische Vorhersage für dynamische Tunnelraten abgeleitet, die den Zerfall von Zuständen, die im regulären Gebiet lokalisiert sind, in die sogenannte chaotische See beschreibt. Dazu wird ein fiktives integrables System konstruiert, das im regulären Bereich eine nahezu gleiche Dynamik aufweist und diese Dynamik in das chaotische Gebiet fortsetzt. Die Theorie zeigt eine ausgezeichnete Übereinstimmung mit numerischen Daten für gekickte Systeme, Billards und optische Mikrokavitäten, falls nichtlineare Resonanzketten vernachlässigbar sind.
Semiklassisch jedoch bestimmen diese nichtlinearen Resonanzketten den Tunnelprozess. Daher kombinieren wir unseren Zugang mit einer verbesserten Theorie des Resonanz-unterstützten Tunnelns und erhalten eine Vorhersage,die vom Quanten- bis in den semiklassischen Bereich gültig ist. Ihre Resultate zeigen eine Genauigkeit, die verglichen mit früheren Theorien um mehrere Größenordnungen verbessert wurde
Experimental Observation of Resonance-Assisted Tunneling
We present the first experimental observation of resonance-assisted
tunneling, a wave phenomenon, where regular-to-chaotic tunneling is strongly
enhanced by the presence of a classical nonlinear resonance chain. For this we
use a microwave cavity made of oxygen free copper with the shape of a
desymmetrized cosine billiard designed with a large nonlinear resonance chain
in the regular region. It is opened in a region, where only chaotic dynamics
takes place, such that the tunneling rate of a regular mode to the chaotic
region increases the line width of the mode. Resonance-assisted tunneling is
demonstrated by (i) a parametric variation and (ii) the characteristic plateau
and peak structure towards the semiclassical limit.Comment: 5 pages, 2 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
Observing the emergence of chaos in a many-particle quantum system
Accessing the connection between classical chaos and quantum many-body
systems has been a long-standing experimental challenge. Here, we investigate
the onset of chaos in periodically driven two-component Bose-Einstein
condensates, whose small quantum uncertainties allow for exploring the phase
space with high resolution. By analyzing the uncertainties of time-evolved
many-body states, we find signatures of elliptic and hyperbolic periodic orbits
generated according to the Poincar\'e-Birkhoff theorem, and the formation of a
chaotic region at increasing driving strengths. The employed fluctuation
analysis allows for probing the phase-space structure by use of only short-time
quantum dynamics.Comment: 5+2 pages, 4 figure
Dynamical Tunneling in Systems with a Mixed Phase Space
Tunneling is one of the most prominent features of quantum mechanics. While the tunneling process in one-dimensional integrable systems is well understood, its quantitative prediction for systems with mixed phase space is a long-standing open challenge. In such systems regions of regular and chaotic dynamics coexist in phase space, which are classically separated but quantum mechanically coupled by the process of dynamical tunneling. We derive a prediction of dynamical tunneling rates which describe the decay of states localized inside the regular region towards the so-called chaotic sea. This approach uses a fictitious integrable system which mimics the dynamics inside the regular domain and extends it into the chaotic region. Excellent agreement with numerical data is found for kicked systems, billiards, and optical microcavities, if nonlinear resonances are negligible.
Semiclassically, however, such nonlinear resonance chains dominate the tunneling process. Hence, we combine our approach with an improved resonance-assisted tunneling theory and derive a unified prediction which is valid from the quantum to the semiclassical regime. We obtain results which show a drastically improved accuracy of several orders of magnitude compared to previous studies.Der Tunnelprozess ist einer der bedeutensten Effekte in der Quantenmechanik. Während das Tunneln in eindimensionalen integrablen Systemen gut verstanden ist, gestaltet sich dessen Beschreibung für Systeme mit gemischtem Phasenraum weitaus schwieriger. Solche Systeme besitzen Gebiete regulärer und chaotischer Bewegung, die klassisch getrennt sind, aber quantenmechanisch durch den Prozess des dynamischen Tunnelns gekoppelt werden. In dieser Arbeit wird eine theoretische Vorhersage für dynamische Tunnelraten abgeleitet, die den Zerfall von Zuständen, die im regulären Gebiet lokalisiert sind, in die sogenannte chaotische See beschreibt. Dazu wird ein fiktives integrables System konstruiert, das im regulären Bereich eine nahezu gleiche Dynamik aufweist und diese Dynamik in das chaotische Gebiet fortsetzt. Die Theorie zeigt eine ausgezeichnete Übereinstimmung mit numerischen Daten für gekickte Systeme, Billards und optische Mikrokavitäten, falls nichtlineare Resonanzketten vernachlässigbar sind.
Semiklassisch jedoch bestimmen diese nichtlinearen Resonanzketten den Tunnelprozess. Daher kombinieren wir unseren Zugang mit einer verbesserten Theorie des Resonanz-unterstützten Tunnelns und erhalten eine Vorhersage,die vom Quanten- bis in den semiklassischen Bereich gültig ist. Ihre Resultate zeigen eine Genauigkeit, die verglichen mit früheren Theorien um mehrere Größenordnungen verbessert wurde
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