Until about 1990 only a very small number of experiments on the quantum mechanics of chaotic systems existed, apart from the early studies of nuclear spectra [Por65]. In this context the experiments on hydrogen atoms in strong microwave fields by Bayfield and Koch [Bay74], and in strong magnetic fields by Welge and his group [Hol86, Mai86] have to be mentioned in particular. The studies of irregularly shaped microwave cavities by Stöckmann and Stein [Stö90] have introduced a new type of quantum chaos research. The microwave billiards and a number of variants to be discussed in this chapter are analogue systems, as they use the equivalence of the Helmholtz equation and the time independent Schrödinger equation. Whether there is a complete correspondence with quantum mechanics or not depends on the respective boundary conditions. As most of the phenomena discussed in the following are common to all types of waves, this does not reduce the conclusiveness of the analogue experiments.

Starting with a historical review, the state of the art in billiard experiments is presented with emphasis on a general survey and the technical background. The results and their quantum mechanical implications will be presented later. The hydrogen experiments, too, will be described in the proper context. The discussion of mesoscopic systems is restricted to billiard-like structures such as antidot lattices [Wei91], quantum dots [Mar92], and tunnelling devices [Fro94].