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arXiv:cond-mat/0612421 · Physical Review E

Nodal domains in open microwave systems

Ulrich Kuhl, R. Höhmann, H.‐J. Stöckmann, Sven Gnutzmann∥

Published Mar 9, 2007• 15 citations
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Abstract

Nodal domains are studied both for real ${\ensuremath{\psi}}_{R}$ and imaginary part ${\ensuremath{\psi}}_{I}$ of the wave functions of an open microwave cavity and found to show the same behavior as wave functions 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 ${\ensuremath{\varphi}}_{g}$ according to ${\ensuremath{\psi}}_{R}+i{\ensuremath{\psi}}_{I}={e}^{i{\ensuremath{\varphi}}_{g}}({\ensuremath{\psi}}_{R}^{\ensuremath{'}}+i{\ensuremath{\psi}}_{I}^{\ensuremath{'}})$. This variation can be qualitatively, and the correlation quantitatively explained in terms of the phase rigidity characterizing the openness of the billiard.

DOI
10.1103/physreve.75.036204

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Nodal domains are studied both for real \ensuremathψR{\ensuremath{\psi}}_{R}\ensuremathψR​ and imaginary part \ensuremathψI{\ensuremath{\psi}}_{I}\ensuremathψI​ of the wave functions of an open microwave cavity and found to show the same behavior as wave functions 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 \ensuremathφg{\ensuremath{\varphi}}_{g}\ensuremathφg​ according to \ensuremathψR+i\ensuremathψI=ei\ensuremathφg(\ensuremathψR\ensuremath′+i\ensuremathψI\ensuremath′){\ensuremath{\psi}}_{R}+i{\ensuremath{\psi}}_{I}={e}^{i{\ensuremath{\varphi}}_{g}}({\ensuremath{\psi}}_{R}^{\ensuremath{'}}+i{\ensuremath{\psi}}_{I}^{\ensuremath{'}})\ensuremathψR​+i\ensuremathψI​=ei\ensuremathφg​(\ensuremathψR\ensuremath′​+i\ensuremathψI\ensuremath′​). This variation can be qualitatively, and the correlation quantitatively explained in terms of the phase rigidity characterizing the openness of the billiard.
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.
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In a microwave transmission measurement the Green function $G({r,r}^{\ensuremath{'}},k)$ of a stadium billiard was determined as a function of ${r}^{\ensuremath{'}}$ and kkk for fixed rrr. From this billiard wave functions were obtained including the sign, which was not available earlier. The propagator $K({r,r}^{\ensuremath{'}},t)$ was obtained by Fourier transform of $G({r,r}^{\ensuremath{'}},k)$. It is shown that $K({r,r}^{\ensuremath{'}},t)$ supplies a very suggestive picture of pulse propagation including pulse reconstruction by focusing effects. This can be considered as an experimental verification of a work of Tomsovic and Heller, who found that semiclassical dynamics can account for the quantum mechanical behavior of billiards over surprisingly long times.
Several key advances in the understanding of quantum chaos have resulted from the study of billiard systems, in which a particle is confined in a two‐dimensional potential well. There has been many numerical studies of wave function characteristics, nodal line and nodal point features and current distributions and correlations. One finds that chaotic states may often be represented by Gaussian random waves which consist of a superposition of monochromatic plane waves traveling in different directions.Direct studies of quantum billiards like 2D mesoscopic ballistic semiconductor quantum dots are, however, unfeasible because of their smallness and because they are normally imbedded deeply in a semiconductor structure. The 2D quantum billiard may, however, be mapped onto a flat microwave cavity, which has allowed for direct measurements of wave functions, current distributions etc. Here we discuss an alternative unconventional realization of a quantum billiard. Thus we outline how electrical networks may be used for fundamental studies of wave function properties and transport in general and, more specifically, their mapping onto open quantum dots. In addition to the scientific case the network model has obvious pedagogical merits.