Nonperiodic echoes from quantum mushroom-billiard hats

Nonperiodic tunable quantum echoes have been observed in experiments with an open microwave billiard whose geometry under certain conditions provides Fibonacci-like sequences of classical delay times. These sequences combined with the reflection at the opening induced by the wave character of the experiment and the size of the opening allow to shape quantum pulses. The pulses are obtained by response of an integrable scattering system.

Figure 1

Sketch of the experimental setup (top view). The inner white part of the light gray area indicates the quarter circle shape of the flat microwave resonator. The radius is $R$ and the size of the adjustable opening is $r$. An antenna $(\times )$ in front of the opening near point C (edge of the bar) couples the microwave signal in and out and is attached to a VNA. The dark gray bar A indicates microwave absorber material inserted into a part of the opening of the cavity.

Figure 2

Upper panel: reflection spectrum at the antenna shown in Fig. 1 (linear scale). Lower panel: Fourier transform of the spectrum in semilog-scale. The black arrows mark times at which classical echoes occur, whereas gray arrows mark times of trajectories facilitated by quantum effects. Bounce numbers corresponding to these times are also shown. Exponentially decaying peak sequences of equal spacings are connected by straight lines. Some peaks correspond to a combination of bounce numbers (e.g., $4+4$) or more complex systematics $(\ast )$.

Figure 3

Same as the lower panel of Fig. 2 for opening sizes $r/R=1/4$ (upper panel) and $r/R=1$ (lower panel), where we find whispering gallery dynamics. The time between two peaks is $\pi R/c$.