Long-range correlations in rectangular cavities containing pointlike perturbations

We investigated experimentally the short- and long-range correlations in the fluctuations of the resonance frequencies of flat, rectangular microwave cavities that contained antennas acting as pointlike perturbations. We demonstrate that their spectral properties exhibit the features typical for singular statistics. Hitherto, only the nearest-neighbor spacing distribution had been studied. In addition we considered statistical measures for the long-range correlations and analyzed power spectra. Thereby we could corroborate that the spectral properties change to semi-Poisson statistic with increasing microwave frequency. Furthermore, the experimental results are shown to be well described by a model applicable to billiards containing a zero-range perturbation [T. Tudorovskiy et al., New J. Phys. 12, 123021 (2010)].

Figure 1

Schematic view of the rectangular microwave billiard. The length of its shorter side was kept fixed at $L_2=20.2$ cm, while the longer one took five different values differing by 0.2 cm between $L_1=45.9$ cm and $L_1=46.7$ cm. For the transmission and reflection measurements, two microwave antennas were introduced into the cavity at two of the positions marked by $A_1,A_2,A_3$, and $A_4$.

Figure 2

Transmission measurement between antennas positioned at $A_1$ and $A_3$ (see Fig. 1) for the frequency range $13.5\mathrm{GHz}\le \nu \le 16.5$ GHz. The spectrum consists of overlapping resonances. Therefore, in order to determine the resonance frequencies, we had to compare all reflection and transmission measurements for each resonator geometry.

Figure 3

Experimental results for the NNSD [red (gray) histograms in upper panel], the ${\mathrm{\Delta }}_3$ statistic [red (gray) circles in the middle panels], and the power spectrum [red (gray) dots in lower panel]. The corresponding curves for Poisson and GOE statistics are shown as full and dashed lines, respectively.

Figure 4

Asymptotic behavior of the power spectra, shown in the lower panels of Fig. 3, in a log-log plot. The experimental results and the straight line best fitting the data are plotted as red (gray) circles and as dashed lines, respectively. The slopes of the latter are close to $-2$ in all frequency intervals.

Figure 5

Same as Fig. 3. The experimental results are compared to the curves deduced from the Porter-Rosenzweig model, shown as green (light gray) lines in the upper and lower panels and as black crosses in the middle panel. The chaoticity parameter equals $\kappa =0.6$ in the frequency range 3.8–8.0 GHz, $\kappa =1.5$ for 8.0–11.3 GHz, and $\kappa =2.25$ in the interval 13.5–16.7 GHz.

Figure 6

Same as Fig. 3. The experimental results are compared to the curves obtained for the corresponding singular billiards, shown as green (light gray) lines in the upper and lower panels and as black crosses in the middle panels, for the scattering length of the perturbation $\beta =0.9$ in the frequency range 3.8–8.0 GHz, $\beta =0.5$ for 8.0–11.3 GHz, and $\beta =0.01$ in the interval 13.5–16.7 GHz. The corresponding results for Poisson and semi-Poisson statistics are shown as full and dashed lines, respectively. In the uppermost frequency interval the curves for semi-Poisson and the experimental and RMT results lie on top of each other.