Missing-level statistics in a dissipative microwave resonator with partially violated time-reversal invariance

We report on the experimental investigation of the fluctuation properties in the resonance frequency spectra of a flat resonator simulating a dissipative quantum billiard subject to partial time-reversal-invariance violation (TIV) which is induced by two magnetized ferrites. The cavity has the shape of a quarter bowtie billiard of which the corresponding classical dynamics is chaotic. Due to dissipation it is impossible to identify a complete list of resonance frequencies. Based on a random-matrix theory approach we derive analytical expressions for statistical measures of short- and long-range correlations in such incomplete spectra interpolating between the cases of preserved time-reversal invariance and complete TIV and demonstrate their applicability to the experimental spectra.

Figure 1

(a) The experimental setup. The vector network analyzer Agilent E8364B is connected to the microwave antennas, which are attached to the resonator through the flexible microwave cables. In order to induce $\mathcal{T}$-invariance violation two pieces of ferrite are inserted into the cavity and magnetized by four external magnets placed at the positions of the ferrites below and above the resonator. The latter are marked by ${\mathrm{M}}_1$ and ${\mathrm{M}}_2$. An additional magnet ${\mathrm{M}}_P$ is used to move a metallic perturber inside the cavity alongside its walls to create different realizations of the cavity. (b) Transmission spectra from antenna 1 to antenna 2 (black dotted line) and vice versa (red full line) in three frequency regions.

Figure 2

Nearest-neighbor spacing distributions in the frequency ranges (a) 6.5–8 GHz, (b) 8–9 GHz, and (c) 9.2–11.5 GHz corresponding to a strength of $\mathcal{T}$-invariance violation $\xi =0.19$, 0.35, and 0.49, respectively. The turquoise histograms show the experimental results, and the red (gray) dashed line the RMT curve for the intermediate case between GOE and GUE. The solid red (gray) line shows the result for the intermediate case with a fraction of $\mathrm{\Phi }=0.83$, 0.81, and 0.85 identified eigenfrequencies, respectively. The black dashed and, black solid lines show the corresponding results for GOE and GUE for these values of $\mathrm{\Phi }$, respectively.

Figure 3

Same as Fig. 2 for [(a)–(c)] the number variance and [(d)-(f)] the rigidity. The experimental results are shown as turquoise circles.

Figure 4

Same as Fig. 2 for the power spectrum. The experimental results are shown as turquoise circles.

Figure 5

Nearest-neighbor (black histogram), next-nearest-neighbor (red [gray] histogram), and second-nearest-neighbor (blue [dark gray] histogram) spacing distributions for partial $\mathcal{T}$-invariance violation where the size $\xi$ is indicated in the panels. The dashed curves were obtained from a fit of $\tilde{P}\left(s\right)=\gamma s^{\mu }{e}^{-\chi s^2}$ to the histogram of corresponding color [26, 49].

Figure 6

Comparison of the analytical results (solid lines) for $\mathrm{\Phi }=1$ (red [gray]) and for the values of $\mathrm{\Phi }$ given in the panels (blue [dark gray]) with RMT simulations (histograms and circles) which were performed based on the random matrix given in Eq. (6).

Figure 7

Comparison of the power spectra obtained from RMT simulations based on Eq. (6) for complete spectra (red [gray] dots) and incomplete ones (blue [dark gray] dots) for partial TIV in comparison to the corresponding analytical results (turquoise solid lines) obtained from Eq. (18). The values of the fraction $\mathrm{\Phi }$ and the strength of TIV, $\xi$, are given in the panels.

Figure 8

Comparison of ${\mathrm{\Sigma }}^2\left(L\right)$ obtained from Eq. (16) for $\mathrm{\Phi }=0$ (black dashed line) and incomplete (black solid line) spectra for the values of $\xi$ and $\mathrm{\Phi }$ given in the panels. They are compared to the corresponding analytical results with $20\%$ added to (red [gray] lines) and subtracted (turquoise [light gray] lines) from the corresponding value of $\xi$.