Distribution of resonance strengths in microwave billiards of mixed and chaotic dynamics

A new measure for statistical properties of the wave function components of quantum systems, the distribution of the product of two partial widths, is introduced. It is tested with data obtained in analog experiments with microwave billiards, where the product of two partial widths equals the resonance strengths in the microwave spectra. The billiards are from the family of the Limaçons, one with chaotic and two with mixed classical dynamics. For completely chaotic systems the partial widths generically obey a Porter-Thomas distribution. We show that in this case the distribution of their product equals a $K_0$ distribution. While we find deviations of the experimental strength distribution from the $K_0$ distribution for the billiards with mixed dynamics, the distributions agree perfectly for the chaotic billiard, when taking into account the experimental threshold of detection in the theoretical description. Hence, the strength distribution provides another stringent test for the connection between statistical properties of systems with classical chaotic dynamics and random matrix theory.

Figure 1

Photograph of the three desymmetrized Limaçon microwave billiards $(\lambda =0.125,0.150,0.300)$. The degree of chaoticity of the corresponding classical billiard has been calculated to 0.55, 0.66, and 1.00, respectively.

Figure 2

Part of the transmission spectrum of the superconducting $\lambda =0.300$ Limaçon microwave billiard. The upper part shows magnifications of the transmission spectrum in the vicinity of two singlets (a) and (c) and a doublet (b). For all three cases the resonance formulas of [30] have been fitted to the measured spectrum.

Figure 3

The total widths (upper figure) and the square root of the strengths (lower figure) versus the resonance frequency of the $\lambda =0.300$ Limaçon billiard for one antenna combination. The full line is the polynomial of fifth order obtained from a fit to the total widths. All strengths are rescaled with this polynomial in order to remove their secular dependence on the frequency

Figure 4

The Distribution of the resonance strengths of the three Limaçon billiards. For each histogram the strength distributions of all six transmission spectra have been superimposed. While expected deviations from the GOE behavior (full line) are clearly visible for the billiards showing mixed dynamics ($\lambda =0.125$ and $\lambda =0.150$), the agreement between the RMT prediction, i.e., the $K_0$ distribution given in Eq. (12), and the measured strength distribution is good for the fully chaotic billiard $(\lambda =0.300)$ over more than six orders of magnitude.

Figure 5

The NNSDs for the three billiards from the family of the Limaçons. For the chaotic billiard $(\lambda =0.300)$ the distribution coincides with the Wigner distribution, i.e., with that of random matrices from the GOE, while the two other billiards show a behavior in between GOE and Poisson statistics, see [17].

Figure 6

A comparison of the experimental strength distribution of the resonances for one antenna combination in the chaotic Limaçon billiard (histogram) with the theoretical prediction (thick line) including a threshold of detection [Eq. (16)]. The threshold parameters and the parameter for the smooth cutoff were determined by a fit of the theoretical to the experimental curve to $x_a=x_b=0.0057$ and $ϵ=0.0075$. The thin line shows the $K_0$ distribution Eqs. (7, 12), i.e., the RMT prediction for chaotic systems with no threshold of detection.