Microwave experiments in chaotic and disordered systems

Hans-Jürgen Stöckmann

Fachbereich Physik
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.

1. The random-superposition-of-plane-waves approach

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,

$$\psi(r)=\sum\limits_n a_n e^{\imath {\bf k}_n {\bf r}_n}\,,$$

where the modulus $k=\vert{\bf k}_n\vert$ of the incoming waves is fixed, but directions ${\bf k}_n/k$ and amplitudes $a_n$ 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

2. Open microwave billiards as scattering systems

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,

$$S_{ij}=\delta_{ij}-2\imath\gamma \sum\limits_n\frac{\psi_n(r_i)\psi_n(r_j)} {k^2-k_n^2+\frac{\imath}{2}\Gamma_n}\,,$$

which is a direct equivalent of the Breit-Wigner formula known from nuclear physics for many years (Stein et al 1995, see chapter 6 of Stöckmann 1990 for details). In the equation $\psi_n(r_i)$ is the value of the wave function of the billiard (with Dirichlet boundary conditions at the wall, and Neumann ones at the opening) at the coupling position $r_i$. $\gamma$ is a parameter describing the coupling to the wave guide or the antenna.

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