Polygonal billiards


JAN WIERSIG


Billiards are important models in mesoscopic physics. They are simpler than general systems, both classically and quantum mechanically, which makes them suitable objects to study how the quantum mechanical properties are influenced by the dynamics of the underlying classical system.

Classical mechanics

Polygonal billiards are interesting examples whose classical dynamics is neither integrable nor chaotic. The motion in a typical polygon is conjectured to be ergodic on the three-dimensional constant-energy surfaces. But this is not rigorously proven so far. There is numerical evidence that motion on these energy surfaces may exhibit even stronger ergodic properties, e.g. mixing.

The motion in a rational polygon (all angles are rationally related to Pi) is restricted to two-dimensional invariant surfaces. That is similar to integrable systems, but the genus of these surfaces is larger than 1, so they do not have the topology of tori. Rational polygonal billiards are therefore also characterized as pseudointegrable. It is proven that the flow on such a surface is ergodic and not mixing. It is an open question whether this flow is typically weak mixing. Weak mixing as maximal ergodic property implies interesting (classical) spectral properties. I have studied the spectra of the barrier billiard, see Fig. 1, in [1]. Recently, I have discovered an interesting relation to Andreev billiards [3].

Figure 1: Trajectory in the barrier billiard.

Quantum mechanics

While the classical dynamics in rational polygons is close to integrability, it has been found that the energy eigenstates are similar to those in fully chaotic cavities. They look typically "irregular" as can be seen in Fig. 2. I have resolved the paradoxon by showing that appropriate superpositions of energy eigenstates share properties of eigenstates in integrable systems [2].

In collaboration with T. Gorin, G. Carlo and A. Bäcker, I study the structure of the energy eigenstates in more detail. Preliminary numerical results indicate that the eigenstates have multifractal properties in momentum space.

Figure 2: Wave function in the barrier billiard.

The statistical properties of high-lying energy levels in rational polygons are conjectured to be close to a third universality class beside the Poisson statistics for integrable systems and the random-matrix statistics GOE for chaotic systems with time-reversal symmetry: the semi-Poisson statistics. The black curve in Fig. 3 is numerical data obtained from the barrier billiard [4]. An explanation for these ``critical statistics'' is still lacking.

Figure 3: Nearest-neighbor distribution P(s) of energy levels of the barrier billiard.

In order to describe polygonal billiards in the framework of semiclassical periodic-orbit theory, so-called diffractive orbits, which start and end at (critical) corners of the polygon, have to be included.

Polygonal billiards have interesting applications in mesoscopic optics. Presently, I investigate the emission properties of coupled dielectric resonators of hexagonal shape.


Bibliography

1
J. W. Singular continuous spectra in a pseudointegrable billiard. Phys. Rev. E, 62:R21-24,2000.

2
J. W. The quantum-classical correspondence in polygonal billiards. Phys. Rev. E, 64:026212, 2001.

3
J. W. Pseudointegrable Andreev billiard. Phys. Rev. E, 65:036221, 2002.

4
J. W. Spectral properties of quantized barrier billiards. Phys. Rev. E, 65:04627, 2002.