Self-pulsing effect in chaotic scattering

Abstract

We study the quantum and classical scattering of Hamiltonian systems whose chaotic saddle is described by binary or ternary horseshoes. We are interested in parameters of the system for which a stable island, associated with the inner fundamental periodic orbit of the system exists and is large, but chaos around this island is well developed. In this situation, in classical systems, decay from the interaction region is algebraic, while in quantum systems it is exponential due to tunneling. In both cases, the most surprising effect is a periodic response to an incoming wave packet. The period of this self-pulsing effect or scattering echoes coincides with the mean period, by which the scattering trajectories rotate around the stable orbit. This period of rotation is directly related to the development stage of the underlying horseshoe. Therefore the predicted echoes will provide experimental access to topological information. We numerically test these results in kicked one dimensional models and in open billiards.

A Study of Ray-Chaotic Cylindrical Scatterers

“…For the profile in (2), the ray equation in (26) admits the following closed-form solution: (27) where is determined by the ray-launch position. The ray trajectory in (3) is finally obtained by inverting (27) with respect to , and enforcing the remaining condition on the ray departure angle …”

The vibrational dynamics of 3D HOCl above dissociation

“…[25][26][27], and has been observed experimentally in microwave cavities [28] The period of the echoes (about 180 d.u.) gives a measure of the time needed to complete one revolution, along the chaotic strip, around the large KAM island in Fig.…”

“…5.2.a (about τ f x = 180 d.u.) [25,27]. The period of such echoes has also been related to the development parameter [25] of the stable and unstable manifolds of the InM saddle point.…”

Phase-space volume of regions of trapped motion: multiple ring components and arcs

“…In turn, the phase-space volume of the regions of trapped motion are ultimately related to the invariant manifolds of the unstable periodic orbit and their homoclinic connections, i.e., the underlying horseshoe structure. The connection among these aspects, for low developed horseshoes, can be established by the relation between the formal development parameter which characterizes the horseshoe development (Jung et al 1999) and the characteristic period of an outermost stable periodic orbits (Jung et al 2004). In particular, notice that certain segments of the manifolds of the unstable periodic orbit (at a locally turning point of the manifolds or tips) are quite close to the unstable periodic orbits that are the companions of the outermost secondary islands, thus mimicking the periodicity of such islands.…”

“…5, we obtain that the formal development parameter is β = 1/4. This value of β is related with the period T β = 3/2 − log 2 β = 3.5 given in units of the average return time to the surface of section (see Jung et al 2004). As noted in Jung et al 2004, T β is an average period with an associated error of ±0.5.…”