Philipps-Universität Marburg, Germany

In a sequence of two talks microwave experiments on spectra, line widths, and field distributions in various closed and open microwave resonators are presented with special emphasis on universal features common to all chaotic systems.

According to a conjecture of Berry (1977) at any point in a chaotic billiard
the wave function may be described by a random superposition of plane waves,

where the modulus of the incoming waves is fixed, but directions and amplitudes are considered as random. As an consequence of the central-limit theorem the approach predicts Gaussian distributions for the wave function amplitudes, or, equivalently, Porter-Thomas distributions for their squares. Such distributions have been observed for the first time for wave functions of chaotic billiards (McDonald and Kaufman 1988), and subsequently in numerous simulations and experiments on chaotic and disordered systems.

In this lecture microwave experiments are presented, exploiting further
consequences of the Berry conjecture. Results for field distributions and
spatial correlation functions in three-dimensional Sinai resonators (Dörr
*et al* 1998) are presented. For spectral level dynamics in a
disordered system with the position of one impurity as the parameter the
approach allows to calculate velocity distributions and velocity
autocorrelation functions which are in complete agreement with the
experiment (Barth *et al* 1999). In open billiards and billiards with
broken time-reversal symmetry the distributions of currents and vortices, as
well as the vortex distance distribution are measured and compared with the
prediction from the Berry conjecture (Barth and Stöckmann, Vranicar
*et al*).

**References**

Berry M 1977 *J. Phys. A* **10** 2083

McDonald S and Kaufman A 1988 *Phys. Rev. A* **37** 3067

Dörr U, Stöckmann H J, Barth M and Kuhl U 1998 *Phys. Rev. Lett.*
**80** 1030

Barth M, Kuhl U and Stöckmann H J 1999 *Phys. Rev. Lett.* **82**
2026

Barth M and Stöckmann H J *Current and vortex statistics in microwave
billiards, to be published*

Vranicar M *et al.* *`Persistent currents' and eigenfunctions
in microwave resonators with broken time reversal symmetry, to be
published*

Whenever a microwave experiment is performed, the system has to be opened either by attaching wave guides or introducing antennas. This has the unavoidable consequence that the system is perturbed, and the measurement always yields an unwanted combination of properties of the system and the apparatus. A tailor-made approach to cope with this situation is provided by scattering theory. For the case of isolated resonances an expression for the matrix elements of the scattering matrix is obtained,

which is a direct equivalent of the Breit-Wigner formula known from nuclear physics for many years (Stein

This correspondence of microwave billiards with atomic nuclei can be used to check
predictions from theory which are unaccessible in nuclear physics. As an example the
first unambiguous demonstration of resonance trapping is presented, namely the phenomenon
that with increasing coupling strength the widths of the resonances do not increase
unlimited but finally decrease again (Persson *et al* 2000). If the transmission
through a cavity with a number of incoming and outgoing channels is measured as a
function of frequency, irregular fluctuations are observed, an equivalent to the Ericson
fluctuations observed in nuclear scattering processes. The distribution of these
fluctuations was studied in an open microwave billiard in dependence of the number of
channels, both for systems with and without time-reversal symmetry, and the results were
compared with random matrix predictions (Schanze *et al* 2001). A new parameter comes
into play if absorption is involved, which is unavoidable in experiments anyway, but has
been considered by theory only recently (Beenakker and Brouwer 2001). Again the
experiment is able to verify the theoretical predictions perfectly (Méndez *et
al*).

**References**

Stein J, Stöckmann H J and Stoffregen U *Phys. Rev. Lett.* **75** 53

Stöckmann H J 1990 *Quantum Chaos - An Introduction* (Cambridge:
Cambridge University press)

Persson E, Rotter I, Stöckmann H J and Barth M 2000 *Phys. Rev.
Lett.* **85** 2478

Schanze H, Alves E, Lewenkopf C and Stöckmann H J 2001 *Phys. Rev. E*
**64** 065201(R)

Beenakker C and Brouwer P 2001 *Physica E* **9** 463

Méndez-Sánchez R *et al.* *Distribution of reflection
eigenvalues in absorbing chaotic microwave cavities, to be published*