Dynamical Tunneling in the Open Annular Billiard

The ray picture is not always adequate to describe resonance lifetimes in asymmetric resonant cavities (ARCs). Important corrections may arise due to classically forbidden tunneling phenomena. This can be studied in the annular billiard, a circular resonator with a non-concentric inner circle that cannot be penetrated. The outer circle could be a glass-air interface, and the inner circle a metallic inclusion. In addition, we coated the outer surface with a thin but penetrable metallic film to increase all resonance lifetimes. The annular billiard with impenetrable outer walls has been studied (among others) by Doron and Frischat, as an example for the effects of chaos on tunneling processes. The work I did with Gregor Hackenbroich shows that the leaky annular billiard shows resonance lifetimes that fluctuate strongly with deformation, due to dynamical tunneling. That is a wave correction to the geometric-optics picture which arises only in systems for which the wave equation is non-separable, i.e. whose short-wavelength limit can exhibit chaos. The paper we wrote on this subject only scratches the surface of this poorly understood phenomenon. The reason why this is a hard problem lies in the fact that the classical phase space structure in itself is very complex because chaos and regularity coexist in these resonator billiards.

Shown on the left are the phase-space portraits of the annular billiard of the shape indicated on the right. The false-color plot represents a metastable wave function in the annular region between inner and outer circle, projected onto the phase space. This representation is called a Husimi plot. The highest weight in the first plot is concentrated on a region where the classical dynamics is regular (not chaotic). For these trajectories, it is difficult to leak out of the resonator, so that the corresponding lifetime is very long.
At the same deformation, but at a slightly different frequency, one finds a broad resonance associated with chaotic rays that can classically diffuse to regions in phase space where escape is easiest, namely the bottom of the plot. In this region, trajectories impinge on the outer boundary with large momentum (χ is the angle of incidence with respect to the normal).
When the deformation is changed by a minute amount, the two states above are coupled due to their energetic overlap, and the wave function has weight on the regular and chaotic regions alike. This is called dynamical tunneling because it represents a phase space distribution that cannot be generated by a classical trajectory.

This page © Copyright Jens Uwe Nöckel, 05/2001