Chaos-induced pulse trains in the ionization of hydrogen

Finding normally hyperbolic invariant manifolds in two and three degrees of freedom with Hénon-Heiles-type potential

We present a method based on a Lagrangian descriptor for revealing the high-dimensional phase space structures that are of interest in nonlinear Hamiltonian systems with index-1 saddle. These phase space structures include a normally hyperbolic invariant manifold and its stable and unstable manifolds, which act as codimension-1 barriers to phase space transport. In this article, finding the invariant manifolds in high-dimensional phase space will constitute identifying coordinates on these invariant manifolds. The method of Lagrangian descriptor is demonstrated by applying to classical two and three degrees of freedom Hamiltonian systems which have implications for myriad applications in chemistry, engineering, and physics.

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.

Geometric determination of classical actions of heteroclinic and unstable periodic orbits

Semiclassical sum rules, such as the Gutzwiller trace formula, depend on the properties of periodic, closed, or homoclinic (heteroclinic) orbits. The interferences embedded in such orbit sums are governed by classical action functions and Maslov indices. For chaotic systems, the relative actions of such orbits can be expressed in terms of phase-space areas bounded by segments of stable and unstable manifolds and Moser invariant curves. This also generates direct relations between periodic orbits and homoclinic (heteroclinic) orbit actions. Simpler, explicit approximate expressions following from the exact relations are given with error estimates. They arise from asymptotic scaling of certain bounded phase-space areas. The actions of infinite subsets of periodic orbits are determined by their periods and the locations of the limiting homoclinic points on which they accumulate.

Classical Fractals and Quantum Chaos in Ultracold Dipolar Collisions

We examine a dipolar-gas model to address fundamental issues regarding the correspondence between classical chaos and quantum observations in ultracold dipolar collisions. The theoretical model consists of a short-range Lennard-Jones potential well with an anisotropic, long-range dipole-dipole interaction between two atoms. Both the classical and quantum dynamics are explored for the same Hamiltonian of the system. The classical chaotic scattering is revealed by the fractals developed in the scattering function (defined as the final atom separation as a function of initial conditions), while the quantum chaotic features lead to the repulsion of the eigenphases from the corresponding quantum S matrix. The nearest-eigenphase-spacing statistics have an intermediate behavior between the Poisson and the Wigner-Dyson distributions. The character of the distribution can be controlled by changing an effective Planck constant or the dipole moment. The degree of quantum chaos shows a good correspondence with the overall average of the classical scattering function. The results presented here also provide helpful insights for understanding the role of the inherent dipole-dipole interaction in the currently ongoing experiments on ultracold collisions of highly magnetic atoms.

Matter, energy, and heat transfer in a classical ballistic atom pump

A ballistic atom pump is a system containing two reservoirs of neutral atoms or molecules and a junction connecting them containing a localized time-varying potential. Atoms move through the pump as independent particles. Under certain conditions, these pumps can create net transport of atoms from one reservoir to the other. While such systems are sometimes called "quantum pumps," they are also models of classical chaotic transport, and their quantum behavior cannot be understood without study of the corresponding classical behavior. Here we examine classically such a pump's effect on energy and temperature in the reservoirs, in addition to net particle transport. We show that the changes in particle number, of energy in each reservoir, and of temperature in each reservoir vary in unexpected ways as the incident particle energy is varied.

Topological analysis of chaotic transport through a ballistic atom pump

We examine a system consisting of two reservoirs of particles connected by a channel. In the channel are two oscillating repulsive potential-energy barriers. It is known that such a system can transport particles from one reservoir to the other, even when the chemical potentials in the reservoirs are equal. We use computations and the theory of chaotic transport to study this system. Chaotic transport is described by passage around or through a heteroclinic tangle. Topological properties of the tangle are described using a generalization of homotopic lobe dynamics, which is a theory that gives some properties of intermediate-time behavior from properties of short-time behavior. We compare these predicted properties with direct computation of trajectories.

New developments in classical chaotic scattering

Classical chaotic scattering is a topic of fundamental interest in nonlinear physics due to the numerous existing applications in fields such as celestial mechanics, atomic and nuclear physics and fluid mechanics, among others. Many new advances in chaotic scattering have been achieved in the last few decades. This work provides a current overview of the field, where our attention has been mainly focused on the most important contributions related to the theoretical framework of chaotic scattering, the fractal dimension, the basins boundaries and new applications, among others. Numerical techniques and algorithms, as well as analytical tools used for its analysis, are also included. We also show some of the experimental setups that have been implemented to study diverse manifestations of chaotic scattering. Furthermore, new theoretical aspects such as the study of this phenomenon in time-dependent systems, different transitions and bifurcations to chaotic scattering and a classification of boundaries in different types according to symbolic dynamics are also shown. Finally, some recent progress on chaotic scattering in higher dimensions is also described.

Invariant manifolds and the geometry of front propagation in fluid flows

Recent theoretical and experimental work has demonstrated the existence of one-sided, invariant barriers to the propagation of reaction-diffusion fronts in quasi-two-dimensional periodically driven fluid flows. These barriers were called burning invariant manifolds (BIMs). We provide a detailed theoretical analysis of BIMs, providing criteria for their existence, a classification of their stability, a formalization of their barrier property, and mechanisms by which the barriers can be circumvented. This analysis assumes the sharp front limit and negligible feedback of the front on the fluid velocity. A low-dimensional dynamical systems analysis provides the core of our results.

Chaotic escape from an open vase-shaped cavity. I. Numerical and experimental results

We present part I in a two-part study of an open chaotic cavity shaped as a vase. The vase possesses an unstable periodic orbit in its neck. Trajectories passing through this orbit escape without return. For our analysis, we consider a family of trajectories launched from a point on the vase boundary. We imagine a vertical array of detectors past the unstable periodic orbit and, for each escaping trajectory, record the propagation time and the vertical detector position. We find that the escape time exhibits a complicated recursive structure. This recursive structure is explored in part I of our study. We present an approximation to the Helmholtz equation for waves escaping the vase. By choosing a set of detector points, we interpolate trajectories connecting the source to the different detector points. We use these interpolated classical trajectories to construct the solution to the wave equation at a detector point. Finally, we construct a plot of the detector position versus the escape time and compare this graph to the results of an experiment using classical ultrasound waves. We find that generally the classical trajectories organize the escaping ultrasound waves.

Chaotic escape from an open vase-shaped cavity. II. Topological theory

We present part II of a study of chaotic escape from an open two-dimensional vase-shaped cavity. A surface of section reveals that the chaotic dynamics is controlled by a homoclinic tangle, the union of stable and unstable manifolds attached to a hyperbolic fixed point. Furthermore, the surface of section rectifies escape-time graphs into sequences of escape segments; each sequence is called an epistrophe. Some of the escape segments (and therefore some of the epistrophes) are forced by the topology of the dynamics of the homoclinic tangle. These topologically forced structures can be predicted using the method called homotopic lobe dynamics (HLD). HLD takes a finite length of the unstable manifold and a judiciously altered topology and returns a set of symbolic dynamical equations that encode the folding and stretching of the unstable manifold. We present three applications of this method to three different lengths of the unstable manifold. Using each set of dynamical equations, we compute minimal sets of escape segments associated with the unstable manifold, and minimal sets associated with a burst of trajectories emanating from a point on the vase's boundary. The topological theory predicts most of the early escape segments that are found in numerical computations.

Demonstration of turnstiles as a chaotic ionization mechanism in Rydberg atoms

We present an experimental and theoretical study of the chaotic ionization of quasi-one-dimensional potassium Rydberg wave packets via a classical phase-space turnstile mechanism. Turnstiles form a general transport mechanism for numerous chaotic systems, and this study explicitly illuminates their relevance to atomic ionization. We create time-dependent Rydberg wave packets, subject them to alternating applied electric-field pulses, and measure the electron survival probability. Ionization depends not only on the initial electron energy, but also on the classical phase-space position of the electron with respect to the turnstile--that part of the electron packet inside the turnstile ionizes after the applied ionization sequence, while that part outside the turnstile does not. The survival data thus encode information on the shape and location of the turnstile, in good agreement with theoretical predictions.

Escape of quantum particles from an open cavity

A negative ion irradiated by a laser provides a coherent source of electrons propagating out from the location of the negative ion. The total escape rates of the electrons when the negative ion is placed inside an open cavity in the shape of a wedge are studied. It is shown that the wedge induces significant oscillations in the total escape rates because of quantum interference effects. In particular, it is shown that, for a wedge with an opening angle of π/N, where N is an arbitrary positive integer, there are (2N-1) induced oscillations in the rates. As a demonstration, the case for a wedge with an opening angle π/5 is calculated and analyzed in detail.

Escape of trajectories from a vase-shaped cavity

We consider the escape of ballistic trajectories from an open, vase-shaped cavity. Such a system serves as a model for microwaves escaping from a cavity or electrons escaping from a microjunction. Fixing the initial position of a particle and recording its escape time as a function of the initial launch direction, the resulting escape-time plot shows "epistrophic fractal" structure--repeated structure within structure at all levels of resolution with new features appearing in the fractal at longer time scales. By launching trajectories simultaneously in all directions (modeling an outgoing wave), a detector placed outside the cavity would measure a train of escaping pulses. We connect the structure of this chaotic pulse train with the fractal structure of the escape-time plot.

Partitioning the phase space in a natural way for scattering systems

In this paper, we demonstrate a recent procedure for the construction of a symbolic dynamics for open systems by applying it to a model potential, the driven inverted Gaussian, which has proven very useful in describing laser-atom interaction. The symbolic dynamics and the corresponding partition of the Poincaré map are natural from the point of view of an asymptotic observer since the resulting branching tree coincides with the one extracted from the scattering functions. In general, the whole procedure is approximate because it only describes the globally unstable part of the chaotic invariant set, that is, the part that can be seen by an asymptotic observer in scattering data. It ignores Kolmogorov-Arnold-Moser islands and their fractal surroundings.