Dimension and escape rate of chaotic scattering from classical and semiclassical cross section data

Open quantum maps from complex scaling of kicked scattering systems

We derive open quantum maps from periodically kicked scattering systems and discuss the computation of their resonance spectra in terms of theoretically grounded methods, such as complex scaling and sufficiently weak absorbing potentials. In contrast, we also show that current implementations of open quantum maps, based on strong absorptive or even projective openings, fail to produce the resonance spectra of kicked scattering systems. This comparison pinpoints flaws in current implementations of open quantum maps, namely, the inability to separate resonance eigenvalues from the continuum as well as the presence of diffraction effects due to strong absorption. The reported deviations from the true resonance spectra appear, even if the openings do not affect the classical trapped set, and become appreciable for shorter-lived resonances, e.g., those associated with chaotic orbits. This makes the open quantum maps, which we derive in this paper, a valuable alternative for future explorations of quantum-chaotic scattering systems, for example, in the context of the fractal Weyl law. The results are illustrated for a quantum map model whose classical dynamics exhibits key features of ionization and a trapped set which is organized by a topological horseshoe.

Quantum signatures of classical multifractal measures

A clear signature of classical chaoticity in the quantum regime is the fractal Weyl law, which connects the density of eigenstates to the dimension D(0) of the classical invariant set of open systems. Quantum systems of interest are often partially open (e.g., cavities in which trajectories are partially reflected or absorbed). In the corresponding classical systems D(0) is trivial (equal to the phase-space dimension), and the fractality is manifested in the (multifractal) spectrum of Rényi dimensions D(q). In this paper we investigate the effect of such multifractality on the Weyl law. Our numerical simulations in area-preserving maps show for a wide range of configurations and system sizes M that (i) the Weyl law is governed by a dimension different from D(0)=2, and (ii) the observed dimension oscillates as a function of M and other relevant parameters. We propose a classical model that considers an undersampled measure of the chaotic invariant set, explains our two observations, and predicts that the Weyl law is governed by a nontrivial dimension D(asymptotic)<D(0) in the semiclassical limit M→∞.

Asymptotic observability of low-dimensional powder chaos in a three-degrees-of-freedom scattering system

We treat a chaotic Hamiltonian scattering system with three degrees of freedom where the chaotic invariant set is of low dimension. Then the chaos and its structure are not visible in scattering functions plotted along one-dimensional lines in the set of asymptotic initial conditions. We show that an asymptotic observer can nevertheless see the structure of the chaotic set in an appropriate scattering function on the two-dimensional impact parameter plane and in the doubly differential cross section. Rainbow singularities in the cross section carry over the symbolic dynamics of the chaotic set into the cross section. A smooth image of the fractal structure of the chaotic set can be reconstructed on the domain of the cross section.

Transition to chaotic scattering: signatures in the differential cross section

We show that bifurcations in chaotic scattering manifest themselves through the appearance of an infinitely fine-scale structure of singularities in the cross section. These "rainbow singularities" are created in a cascade, which is closely related to the bifurcation cascade undergone by the set of trapped orbits (the chaotic saddle). This cascade provides a signature in the differential cross section of the complex pattern of bifurcations of orbits underlying the transition to chaotic scattering. We show that there is a power law with a universal coefficient governing the sequence of births of rainbow singularities and we verify this prediction by numerical simulations.

Classical versus quantum structure of the scattering probability matrix: chaotic waveguides

The purely classical counterpart of the scattering probability matrix (SPM)/S(n,m)/(2) of the quantum scattering matrix S is defined for two-dimensional quantum waveguides for an arbitrary number of propagating modes M. We compare the quantum and classical structures of /S(n,m)/(2) for a waveguide with generic Hamiltonian chaos. It is shown that even for a moderate number of channels, knowledge of the classical structure of the SPM allows us to predict the global structure of the quantum one and, hence, understand important quantum transport properties of waveguides in terms of purely classical dynamics. It is also shown that the SPM, being an intensity measure, can give additional dynamical information to that obtained by the Poincaré maps.

Rainbow transition in chaotic scattering

We study the effects of classical chaotic scattering on the differential cross section, which is the measurable quantity in most scattering experiments. We show that the fractal set of singularities in the deflection function is not, in general, reflected on the differential cross section. We show that there are systems in which, as the energy (or some other parameter) crosses a critical value, the system's differential cross-section changes from a singular function having an infinite set of rainbow singularities with structure in all scales to a smooth function with no singularities, the scattering being chaotic on both sides of the transition. We call this metamorphosis the rainbow transition. We exemplify this transition with a physically relevant class of systems. These results have important consequences for the problem of inverse scattering in chaotic systems and for the experimental observation of chaotic scattering.

Output functions and fractal dimensions in dynamical systems

We present a novel method for the calculation of the fractal dimension of boundaries in dynamical systems, which is in many cases many orders of magnitude more efficient than the uncertainty method. We call it the output function evaluation (OFE) method. We show analytically that the OFE method is much more efficient than the uncertainty method for boundaries with D<0.5, where D is the dimension of the intersection of the boundary with a one-dimensional manifold. We apply the OFE method to a scattering system, and compare it to the uncertainty method. We use the OFE method to study the behavior of the fractal dimension as the system's dynamics undergoes a topological transition.

Chaotic scattering: An introduction

In recent years chaotic behavior in scattering problems has been found to be important in a host of physical situations. Concurrently, a fundamental understanding of the dynamics in these situations has been developed, and such issues as symbolic dynamics, fractal dimension, entropy, and bifurcations have been studied. The quantum manifestations of classical chaotic scattering is also an extremely active field, with new analytical techniques being developed and with experiments being carried out. This issue of Chaos provides an up-to-date survey of the range of work in this important field of study.

Quantum irregular scattering induced by tunneling

Irregular behavior in a simple two-dimensional scattering model is investigated in the quantum domain. The scattering potential is composed from Dirac deltas on a stadium shaped curve. The unusual feature of the model is that the irregular patterns disappear in the classical limit because the main mechanism leading to resonances in the cross section data is the quantum tunneling. Calculations for the standard characteristics such as nearest-neighbor distribution of eigenphases of the S-matrix, the distribution of the S-matrix elements and the correlation function of the total cross section are performed. Deviations from the usual predictions for irregular scattering have been found in certain regions, which can be traced back to the fact that the model does not have such a characteristic time like the classical escape rate, which survives the classical limit.