Spectral properties of Bunimovich mushroom billiards

Properties of a quantum mushroom billiard in the form of a superconducting microwave resonator have been investigated. They reveal unexpected nonuniversal features such as, e.g., a supershell effect in the level density and a dip in the nearest-neighbor spacing distribution. Theoretical predictions for the quantum properties of mixed systems rely on the sharp separability of phase space—an unusual property met by mushroom billiards. We however find deviations which are ascribed to the presence of dynamic tunneling.

Figure 1

(Color online) (a) Typical shape of a mushroom billiard with the critical caustic shown as a dotted line, (b) desymmetrized version (quarter-circle as hat) with triangular stem, and (c) photograph of the corresponding experimental microwave resonator. The positions of the six antennae in the experiment are indicated in (b).

Figure 2

Part of five transmission spectra between 9 and $10\mathrm{GHz}$, measured with the microwave resonator shown in Fig. 1c. The curves labeled with $\mid S_{1j}\mid$, $j=2,\dots ,6$, are obtained with the antennae 1 and $j$ whose positions are indicated in Fig. 1b. The spectra have been displaced from each other along the $y$ axis for illustration and are plotted in a logarithmic scale. One notices a bunching behavior—i.e., large gaps marked by arrows—between stretches of resonances.

Figure 3

Fluctuating part of the number of resonances below a given frequency $f$. The curve oscillates and shows a beating behavior with some noise at the nodes marked by arrows.

Figure 4

(a) Experimental length spectrum of periodic orbits of the mushroom billiard. The two peaks of lengths of about $0.7\mathrm{m}$ are due to two regular orbits in the hat of the mushroom, and that at length $1.12\mathrm{m}$ is due to a chaotic one. (b) The computed length spectrum of a quarter-circle billiard is shown flipped upside down, where only eigenvalues were considered which correspond to angular momenta larger than the critical one.

Figure 5

(a) Nearest-neighbor spacing distribution of the mushroom billiard together with those for Poisson statistics and the GOE (solid lines) expected for systems with regular and chaotic dynamics, respectively. Note the dip at $s=0.7$. It vanishes (b) when the contribution of the dominant periodic orbits in Fig. 4 to the resonance density is subtracted. The dashed line in both (a) and (b) shows the Berry-Robnik prediction for a mixed system with $q_c=82.9\%$ chaos.

Figure 6

(Color online) Measured intensity distributions. The eigenfunctions are usually either regular (a) or chaotic (b) and correspond to eigenfunctions of a quarter-circle billiard in the former case. We also found some mixed eigenfunctions like the one in (c). In (d) the averaged distribution of 239 chaotic eigenfunctions is shown. The radial intensity profile in the hat is plotted in (e) between the quarter-circle center and the circular boundary (points), and compared to the classical probability of finding chaotic orbits (solid line). The decrease beyond the critical caustic at $r∕R=2∕3$ is revealed in the measurements, and deviations at the boundaries and at the knee are due to quantum effects.

Figure 7

Deviations of the experimental eigenfrequencies of the mushroom billiard from those of a quarter-circle billiard versus frequency. The symbols (points, triangles, squares) refer to regular states. Eigenvalues with equal radial quantum number $n$ are connected by lines. For a given $n$ there are almost no deviations for high frequencies where the eigenfunctions are localized close to the circular boundary. For smaller frequencies, the influence of the stem leads to increasing deviations pointing to the importance of dynamic tunneling.