Missing-level statistics and analysis of the power spectrum of level fluctuations of three-dimensional chaotic microwave cavities

We present an experimental study of missing-level statistics of three-dimensional chaotic microwave cavities. The investigation is reinforced by the power spectrum of level fluctuations analysis, which also takes into account the missing levels. On the basis of our data sets we demonstrate that the power spectrum of level fluctuations in combination with short- and long-range spectral fluctuations provides a powerful tool for the determination of the fraction of randomly missing levels in systems that display wave chaos such as the three-dimensional chaotic microwave cavities. The experimental results are in good agreement with the analytical expressions that explicitly take into account the fraction of observed levels $\phi$. We also show that in the case of incomplete spectra with many unresolved states the above procedures may fail. In such a case the random matrix theory calculations can be useful for the determination of missing levels.

Figure 1

Panel (a): photograph of the 3D microwave cavity showing the antenna positions ${\mathrm{A}}_1$, ${\mathrm{A}}_2$, and ${\mathrm{A}}_3$. For the measurement of the two-port scattering matrix $\widehat{S}$ the vector network analyzer was coupled to the cavity through the HP 85133-616 and HP 85133-617 flexible microwave cables via the two microwave antennas. The inset shows the scatterer which was used to make different realizations of the cavity. Panel (b) shows a sketch of the cavity on the cross-section plane (see the detailed description in the text) and the position of the scatterer inside the cavity.

Figure 2

Measured modules of the elements $|S_{11}|$ and $|S_{22}|$ of the two-port scattering matrix $\widehat{S}$ (ports $A_1$ and $A_3$) of the 3D microwave cavity in the frequency range $7\text{–}8$ GHz and $10\text{–}11$ GHz, respectively.

Figure 3

Spectral properties of the rescaled resonance frequencies of the 3D microwave cavity and the power spectrum of level fluctuations. Panels (a) and (b) show the nearest-neighbor spacing distribution (histogram) and the integrated nearest-neighbor spacing distribution (circles). The insets in panels (a) and (b) show the experimental results for the nearest-neighbor spacing distribution (histogram) and the integrated nearest spacing distribution (circles) obtained in the narrower frequency range 7–9 GHz. In these calculations the complete spectra ($\phi =1$) for nine different realizations of the cavity were used (see the text). The GOE predictions ($\phi =1$) and the theoretical predictions evaluated for the fraction of observed levels $\phi =0.89$ are shown in the insets with full and red broken lines, respectively. The spectral rigidity of the spectrum (triangles) and the average power spectrum of level fluctuations (diamonds) are shown in panels (c) and (d), respectively. The experimental results are compared to those of the eigenvalues of random matrices from the GOE (black full lines). The corresponding missing-level statistics (red broken lines) were calculated for the GOE system for $\phi =0.89$. The insets in panels (c) and (d) show the experimental results for the spectral rigidity (triangles) and the average power spectrum of level fluctuations (diamonds) obtained in the frequency range 7–9 GHz for the complete spectra ($\phi =1$) for nine different realizations of the cavity. The GOE predictions $(\phi =1$) and the theoretical predictions evaluated for the fraction of observed levels $\phi =0.89$ are shown in the insets with full and red broken lines, respectively.

Figure 4

Spectral properties of the rescaled resonance frequencies of the 3D microwave cavity and the power spectrum of level fluctuations calculated for the modified spectra with randomly removed resonances with the fraction of the observed levels $\phi =0.5$. Panels (a)–(d) show the nearest-neighbor spacing distribution (histogram), the integrated nearest-neighbor spacing distribution (circles), the spectral rigidity of the spectrum (triangles), and the average power spectrum of level fluctuations (diamonds), respectively. The experimental results are compared to those of the eigenvalues of random matrices from the GOE (full lines). The corresponding missing-level statistics (red broken lines) were calculated for the GOE system for $\phi =0.5$.

Figure 5

Spectral properties of the rescaled eigenvalues of the 3D microwave cavity and the power spectrum of level fluctuations calculated for the modified experimental spectra from which the closest resonances were removed to get the fraction of the observed levels $\phi =0.7$. Panels (a)–(d) show the nearest-neighbor spacing distribution (histogram), the integrated nearest-neighbor spacing distribution (circles), the spectral rigidity of the spectrum (triangles), and the average power spectrum of level fluctuations (diamonds), respectively. The experimental results are compared to those of the eigenvalues of random matrices from the GOE (full lines). The corresponding missing-level statistics for randomly removed states (red broken lines) were calculated for the GOE system for $\phi =0.7$. The RMT results shown with the green dotted lines are also attained for the incomplete spectra with $\phi =0.7$ but in this case the spectra were obtained by randomly removing 11% of states and then applying the procedure of removing clustering states.