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.