Nonperiodic echoes from mushroom billiard hats

Mushroom billiards have the remarkable property to show one or more clear cut integrable islands in one or several chaotic seas, without any fractal boundaries. The islands correspond to orbits confined to the hats of the mushrooms, which they share with the chaotic orbits. It is thus interesting to ask how long a chaotic orbit will remain in the hat before returning to the stem. This question is equivalent to the inquiry about delay times for scattering from the hat of the mushroom into an opening where the stem should be. For fixed angular momentum we find that no more than three different delay times are possible. This induces striking nonperiodic structures in the delay times that may be of importance for mesoscopic devices and should be accessible to microwave experiments.

Figure 1

Different shapes of mushroom hats. Circular mushroom billiards consist of a hat with a circular boundary and a stem, which has been chosen in triangular form for all mushroom billiards here. Stable orbits in a circular mushroom hat never enter the stem. They form a caustic in the hat, and the radius of the smallest possible caustic is one-half the width of the stem, as indicated in (a). Other shapes of mushroom billiards may display an elliptic hat or a shifted stem (b), very wide or narrow stems (c) or hats which are a circle sector of an angle smaller than 180° (d), in this case 90° and $180°\times (\sqrt{5}-1)∕2$.

Figure 2

The $\Theta$ billiard is a circle billiard with a straight line of length $2r$ along a diameter of the circle of length $2R$, which defines the starting and ending points of the particle orbits taken into consideration. In a semicircle mushroom hat with the opening located at the position of the straight line these orbits correspond to those of particles entering and leaving it. In this sketch, a particle starts from the straight line at $P_S$, undergoes four reflections with the boundary at $P_1$, $P_2$, $P_3$, and $P_4$ before it finally ends on the straight line at $P_E$.

Figure 3

A $\Theta$ billiard with the straight line chosen symmetric with respect to the center of the circle. Initial conditions for a particle orbit in the $\Theta$ billiard are the angular momentum $M$, i.e., the radius of its caustic (dotted line circle), and the orientation ${\theta }_0$ of its first line segment. The orientation of each line segment of a particle orbit is defined in terms of the total angle covered by the vector pointing from the center of the circle to the point of contact of the line segment with the caustic during its rotation while the particle propagates. In this example the orientation ${\theta }_0$ of the initial line segment is negative and determined by extending the latter back beyond its point of contact with the caustic as indicated in the figure by the dotted line. The angle $\psi$ between the reflected particle orbit and the tangent to the circle boundary at the point of reflection is a constant. At each reflection the orientation angle increases by $2\psi$.

Figure 4

Illustrative representation of particle orbits starting from the straight line in the $\Theta$ billiard in terms of the orientations of their line segments. The orientations of the ensemble of all particle orbits starting from the straight line are restricted to the interval $[-\chi ,\chi ]$ [see Eq. (3) in the main text]. Their evolution is illustrated by shifting this interval by $2\psi$ at each reflection of the particle with the billiard boundary, thereby obtaining the range of values of the orientations of successive line segments of the particle orbits [see Eq. (8)]. Thus, a string of “orbit intervals” is obtained. The straight line is represented by a string of “line intervals” of size $2\chi$ around each multiple of $\pi$ on the angle axis. When an overlap of a line and an orbit interval occurs, the overlapping part of the orbit interval corresponds to angles ${\theta }_n$, which fulfill the inequality Eq. (7), that is, to initial angles ${\theta }_0$, for which the particle orbit ends after $n$ reflections. Accordingly, the overlapping part is cut off the orbit interval in the subsequent iterations.

Figure 5

Magic numbers plotted as a function of angular momentum for a semicircle mushroom hat with radius $R$ and a half-opening $r=R∕3$ chosen symmetrically with respect to the circle center. The angular momentum is given in units of $R$ as the absolute momentum is set to unity. One observes lines where a magic number remains constant over a certain range of angular momentum values, and sudden discontinuities (arrows), where one member of the triplet changes its value. Note further a singularity, where two of the three members of the triplet tend to infinity. Singularities of this kind are observed at angular momenta of unstable periodic orbits, which do not intersect the straight line in the $\Theta$ billiard in some orientations and do it in others. Between singularities of this kind, every magic number except the first and second, is the sum of two smaller ones, as proven in the text. Note that the chosen interval of angular momentum is small and corresponds to particles which reach the straight line close to one of its ends.

Figure 6

Counts of the numbers of bounce particles starting from the straight line of half-length $r=R∕3$ in the $\Theta$-billiard experience with the circle boundary until they reach the straight line again; they were obtained using the linear inequality Eq. (7) for the orientation of line segments between two reflections. Note the persisting selectivity. The organized and regular long time behavior (large number of bounces) is dominated by unstable periodic orbits, while the short time behavior (small number of bounces) cannot be explained by those.

Figure 7

Counts of magic numbers for mushroom hats with half-opening $r=0.3R$ and different shapes: 180°-mushroom hat (open triangles), $180°\times (\sqrt{5}-1)∕2$-mushroom hat (open circles), and 90°-mushroom hat (open squares) [see inset and Figs. 1a, 1d]. Note that the selectivity is observed in all examples, as the density of points remains almost constant in the logarithmic plot. For wider openings there is no correspondence between the number of reflections and the physical time any more.

Figure 8

Magic numbers of an elliptic $\Theta$ billiard, corresponding to a mushroom billiard with an elliptic hat [see Fig. 1b], with the length of the semimajor axis chosen equal to $R$, that of the semiminor axis equal to $\sqrt{3}∕2R$ while the focal points have a distance of $0.5R$ from the center. Circles refer to a straight line chosen symmetrically between the focal points (length $2r=0.99R$), where no selectivity is observed. For a straight line of length $2r=1.1R$ selectivity is evident even for long times, because here only deformed starlike periodic stable orbits of the ellipse survive. Those particles, which intersect the straight line between the focal points, will intersect it again after just one bounce with the elliptic boundary. The diamonds refer to the case of two straight lines of length $0.1R$, whose outer endpoints are located in the focal points. They cut the caustic of the stable orbits of particles crossing the horizontal line between the two focal points from outside. Again a selectivity is observed.