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arXiv:0805.2345 · Physical Review Letters

Superscars in Billiards: A Model for Doorway States in Quantum Spectra

Sven Åberg, Thomas Guhr, M. Miski-Oglu, A. Richter

Published May 23, 2008• 36 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.100.204101

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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.
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.
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.
In the preceding chapters we have learnt that random matrix theory is perfectly able to explain the universal properties of the spectra of chaotic systems, and this in spite of the oversimplifying assumptions applied. On the one hand it is very satisfactory that one single theory can cope with such a variety of systems as nuclei, mesoscopic structures, or microwave billiards, on the other hand this is a bit disappointing. If there is no possibility of discriminating between the spectra of a nucleus and a quantum dot, then there is little hope of learning anything of relevance about it. Fortunately, random matrix theory is only one side of the coin. We have already come across some examples demonstrating its limits of validity. Remember the spectral level dynamics where bouncing balls disturbed the otherwise universal Gaussian velocity distribution (see Section 5.2.2). Another example is the scarring phenomenon observed in many wave functions. Here obviously closed classical orbits have left their fingerprints in the amplitude patterns. We cannot expect that the universal random matrix theory can correctly account for individual features such as periodic orbits. We now come to an alternative approach to analysing the spectra. As we know from the correspondence principle, in the semiclassical limit quantum mechanics eventually turns into classical mechanics. That is why classical dynamics must be hidden somewhere in the spectra, at least in the limit of high quantum numbers. In the introduction we have already discussed this connection for a particle in a one-dimensional box.
Flat microwave cavities can be used to experimentally simulate quantum mechanical systems. By coupling two such cavities, we study the equivalent to 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 by employing a qualitative model based on random matrix theory and show that the results derived from the strength distribution are consistent with those previously obtained from spectral statistics.