Microwave transport experiments have been performed in a quasi-two-dimensional resonator with randomly distributed conical scatterers. At high frequencies, the flow shows branching structures similar to those observed in stationary imaging of electron flow. Semiclassical simulations confirm that caustics in the ray dynamics are responsible for these structures. At lower frequencies, large deviations from Rayleigh's law for the wave height distribution are observed, which can only partially be described by existing multiple-scattering theories. In particular, there are "hot spots" with intensities far beyond those expected in a random wave field. The results are analogous to flow patterns observed in the ocean in the presence of spatially varying currents or depth variations in the sea floor, where branches and hot spots lead to an enhanced frequency of freak or rogue wave formation.

We study the fundamental question of dynamical tunneling in generic two-dimensional Hamiltonian systems by considering regular-to-chaotic tunneling rates. Experimentally, we use microwave spectra to investigate a mushroom billiard with adjustable foot height. Numerically, we obtain tunneling rates from high precision eigenvalues using the improved method of particular solutions. Analytically, a prediction is given by extending an approach using a fictitious integrable system to billiards. In contrast to previous approaches for billiards, we find agreement with experimental and numerical data without any free parameter. Typical Hamiltonian systems have a mixed phase space in which regular and chaotic motion coexist. While classically these regions are separated, quantum mechanically they are coupled by tunneling. This process has been called "dynamical tunneling" [1] as it occurs across a dynamically generated barrier in phase space. Tunneling has been studied between symmetry related regular regions (chaos-assisted tunneling) [2,3,4,5,6,7] and from a single regular region to the chaotic sea [8,9,10,11,12]. In contrast to the well understood 1D tunneling through a barrier, the quantitative prediction of dynamical tunneling is a major challenge. Results have been found for specific systems or system classes only, e.g. recently for 2D quantum maps with an approach using a fictitious integrable system [12]. However, a precise knowledge of tunneling rates is of great importance. Recent examples are spectral statistics in systems with a mixed phase space [13], eigenstates affected by flooding of regular islands [14], and emission properties of optical micro-cavities [15].Billiards are an important class of Hamiltonian systems. Classically, a point particle moves along straight lines inside a domain with elastic reflections at its boundary. Quantum-mechanical approaches for dynamical tunneling rates have so far escaped a full quantitative prediction as they required fitting by a factor of 6 for the annular billiard [3] and by a factor of 100 (see below) for the mushroom billiard [16].In this paper we present a combined experimental, theoretical, and numerical investigation of dynamical tunneling rates in mushroom billiards [17], which are of great current interest [13,16,18,19,20] due to their sharply divided phase space. Experiments are performed using a microwave cavity. Extending the approach using a fictitious integrable system [12] to billiards, we find quantitative agreement in the experimentally accessible regime, see Fig. 1, without a free parameter. In addition, numerical computations verify the predictions over 18 orders of magnitude with errors typically smaller than a factor of 2, see Fig. 4. The theoretical approach thus provides unprecedented agreement for tunneling rates in billiards.We consider the desymmetrized mushroom billiard, i.e. the 2D autonomous system H(p, q) = p 2 /2M + V (q) shown in Fig. 2b, characterized by the radius of the quarter circle R, the foot width a and the foot height l. The pote...

From the measurement of a reflection spectrum of an open microwave cavity, the poles of the scattering matrix in the complex plane have been determined. The resonances have been extracted by means of the harmonic inversion method. By this, it became possible to resolve the resonances in a regime where the linewidths exceed the mean level spacing up to a factor of 10, a value inaccessible in experiments up to now. The obtained experimental distributions of linewidths were found to be in perfect agreement with predictions from random matrix theory when wall absorption and fluctuations caused by couplings to additional channels are considered.

We consider the energy spectra of quantum Hamilton systems whose classical dynamics is of mixed type, i.e. regular for some initial conditions in the classical phase space and chaotic for the complementary initial conditions. In the strict semiclassical limit when the effective Planck constanth eff is sufficiently small the Berry-Robnik (BR) statistics applies, while at larger values ofh eff (or smaller energies) one sees deviations from BR due to localization and tunneling effects. We derive a two-level random matrix model, which describes these effects and can be treated analytically in a closed form. The coupling between the regular and chaotic levels due to tunneling is assumed to be Gaussian distributed. This two-level model describes most of the features of matrices of large dimensions (here N = 1000), which we treat numerically, and is predicted to apply in mixed type systems at low energies. The proposed analytical level spacing distribution function has two parameters, the BR parameter ρ, characterizing the classical phase space, and the coupling (antenna distortion or tunneling) parameter σ between states. Localization effects so far are not included in our analysis except in the special case where ρ can be replaced by some effective ρ eff .

From a reflection measurement in a rectangular microwave billiard with randomly distributed scatterers the scattering and the ordinary fidelity was studied. The position of one of the scatterers is the perturbation parameter. Such perturbations can be considered as local since wave functions are influenced only locally, in contrast to, e.g., the situation where the fidelity decay is caused by the shift of one billiard wall. Using the random-plane-wave conjecture, an analytic expression for the fidelity decay due to the shift of one scatterer has been obtained, yielding an algebraic 1/t decay for long times. A perfect agreement between experiment and theory has been found, including a predicted scaling behavior concerning the dependence of the fidelity decay on the shift distance. The only free parameter has been determined independently from the variance of the level velocities.

Abstract. The theoretical interpretation of measurements of "wavefunctions" and spectra in electromagnetic cavities excited by antennas is considered. Assuming that the characteristic wavelength of the field inside the cavity is much larger than the radius of the antenna, we describe antennas as "pointlike perturbations". This approach strongly simplifies the problem reducing the whole information on the antenna to four effective constants. In the framework of this approach we overcame the divergency of series of the phenomenological scattering theory and justify assumptions lying at the heart of "wavefunction measurements". This selfconsistent approach allowed us to go beyond the one-pole approximation, in particular, to treat the experiments with degenerated states. The central idea of the approach is to introduce "renormalized" Green function, which contains the information on boundary reflections and has no singularity inside the cavity. PACS numbers:On the theory of cavities with point-like perturbations. Part I: General theory 2

We present microwave measurements for the density and spatial correlation of current critical points in an open billiard system, and compare them with the predictions of the Random Wave Model (RWM). In particular, due to a novel improvement of the experimental set-up, we determine experimentally the spatial correlation of saddle points of the current field. An asymptotic expression for the vortex-saddle and saddle-saddle correlation functions based on the RWM is derived, with experiment and theory agreeing well. We also derive an expression for the density of saddle points in the presence of a straight boundary with general mixed boundary conditions in the RWM, and compare with experimental measurements of the vortex and saddle density in the vicinity of a straight wall satisfying Dirichlet conditions.

Nodal domains are studied both for real ψR and imaginary part ψI of the wavefunctions of an open microwave cavity and found to show the same behavior as wavefunctions in closed billiards. In addition we investigate the variation of the number of nodal domains and the signed area correlation by changing the global phase ϕg according to ψR +iψI = e iϕg (ψ). This variation can be qualitatively, and the correlation quantitatively explained in terms of the phase rigidity characterising the openness of the billiard.