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arXiv:nlin/0612021 · Physical Review Letters

First Experimental Observation of Superscars in a Pseudointegrable Barrier Billiard

E. Bogomolny, Barbara Dietz, Thomas Friedrich, M. Miski-Oglu, A. Richter, F. Schäfer, C. Schmit

Published Dec 20, 2006• 59 citations
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Abstract

With a perturbation body technique intensity distributions of the electric field strength in a flat microwave billiard with a barrier inside up to mode numbers as large as about 700 were measured. A method for the reconstruction of the amplitudes and phases of the electric field strength from those intensity distributions has been developed. Recently predicted superscars have been identified experimentally and--using the well-known analogy between the electric field strength and the quantum mechanical wave function in a two-dimensional microwave billiard--their properties determined.

DOI
10.1103/physrevlett.97.254102

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With a perturbation body technique intensity distributions of the electric field strength in a flat microwave billiard with a barrier inside up to mode numbers as large as about 700 were measured. A method for the reconstruction of the amplitudes and phases of the electric field strength from those intensity distributions has been developed. Recently predicted superscars have been identified experimentally and--using the well-known analogy between the electric field strength and the quantum mechanical wave function in a two-dimensional microwave billiard--their properties determined.
In a unifying way, the doorway mechanism explains spectral properties in a rich variety of open mesoscopic quantum systems, ranging from atoms to nuclei. A distinct state and a background of other states couple to each other which sensitively affects the strength function. The recently measured superscars in the barrier billiard provide an ideal model for an in-depth investigation of this mechanism. We introduce two new statistical observables: the full distribution of the maximum coupling coefficient to the doorway and directed spatial correlators. Using random matrix theory and random plane waves, we obtain a consistent understanding of the experimental data.
The recently measured superscars in a pseudointegrable barrier billiard [Phys. Rev. Lett. 97, 254102 (2006)] serve as a paradigm for strength function phenomena known in a variety of open mesoscopic quantum systems ranging from atoms to nuclei. The superscarring state is thereby of a simple structure whose strength is spread over the many other nonscarring chaotic states in the billiard, i.e. it acts as a so called doorway state. In this talk the observed spreading of four different superscars is briefly discussed in terms of a model based on Random Matrix Theory and also by investigating the spatial correlations of the measured wave functions. It is shown that the results derived are fully consistent with each other. Furthermore, some properties of the nodal domains of the experimental wave functions are analyzed. It is found that the mean value of the rescaled nodal domains shows a deviation from what is expected from a percolation model. This deviation can, however, be qualitatively understood in terms of the doorway state picture.
Two experiments with super- and normal conducting microwave cavities which are used as an analog to two-dimensional, infinitely deep quantum potentials are presented. The eigenvalues of such a potential can be measured directly through the resonance frequencies of the cavity, while the eigenvectors can be determined by measuring the field distributions inside the cavity. In the case of open systems another information – the imaginary part of the eigenvalues – is observable by measuring the widths of the resonances. As examples for this experimental approach to quantum chaos the semiclassical reconstruction of length spectra of billiards with varying chaoticity based of various trace formulas proposed by Gutzwiller, by Ullmo, Grinberg and Tomsovic and by Berry and Tabor for systems with chaotic, mixed and regular dynamics, respectively, and the first observation of square root singularities – so called exceptional points recently proposed by Heiss – in the energy spectrum of coupled cavities are presented.
This paper reports on a joint theoretical and experimental study of the Pauli quantum-mechanical stress tensor T_{alphabeta}(x,y) for open two-dimensional chaotic billiards. In the case of a finite current flow through the system the interior wave function is expressed as psi=u+iv . With the assumption that u and v are Gaussian random fields we derive analytic expressions for the statistical distributions for the quantum stress tensor components T_{alphabeta} . The Gaussian random field model is tested for a Sinai billiard with two opposite leads by analyzing the scattering wave functions obtained numerically from the corresponding Schrödinger equation. Two-dimensional quantum billiards may be emulated from planar microwave analogs. Hence we report on microwave measurements for an open two-dimensional cavity and how the quantum stress tensor analog is extracted from the recorded electric field. The agreement with the theoretical predictions for the distributions for T_{alphabeta}(x,y) is quite satisfactory for small net currents. However, a distinct difference between experiments and theory is observed at higher net flow, which could be explained using a Gaussian random field, where the net current was taken into account by an additional plane wave with a preferential direction and amplitude.
Flat microwave cavities can be used to experimentally simulate quantum mechanical systems. By coupling two such cavities, we study the equivalent to the symmetry breaking in quantum mechanics. As the coupling is tunable, we can measure resonance strength distributions as a function of the symmetry breaking. We analyze the data employing a qualitative model based on Random Matrix Theory (RMT) and show that the results derived from the strength distribution are consistent with those previously obtained from spectral statistics.