Experimental study of coupled quantum billiards with integrable and chaotic classical dynamics and test of a special Rosenzweig-Porter model

TL;DR

This work experimentally probes the spectral statistics of two coupled quantum billiards—one integrable and one chaotic—via tunable openings in a shared wall and superconducting microwave resonators. The observed spectra are described by a two-block Rosenzweig-Porter–type random-matrix model with coupling strength $\lambda$, enabling quantitative extraction of the mixing between the two parts. Results show that even weak mode overlap alters long-range correlations and breaks symmetry in the integrable component, with stronger coupling driving the system toward fully chaotic statistics, though not perfectly GOE in the explored range. The study demonstrates the applicability of the RP-model variant to capture mixed regular-chaotic dynamics arising from mode overlap and provides a practical method to determine coupling strength in composite quantum systems.

Abstract

We report on the experimental study of the spectral properties of quantum systems consisting of two quantum billiards (QBs), one with chaotic, the other one with integrable classical dynamics, that are coupled to each other via an opening in a common wall. They are compared to those of a special case of the Rosenzweig-Porter model with random matrices composed of two diagonal blocks modeling the spectral properties of the QBs, that are coupled with a tunable parameter. We demonstrate that this model is suitable for the description of the experimental data and thus may be employed to determine the strength of the coupling. It results from the increasing overlap of eigenmodes in the QBs penetrating through the opening into the other one, leading to a mixing of their eigenstates, and the breaking of the symmetry present in the QB with integrable dynamics. This implicates deviations of the spectral properties from those of typical quantum systems with integrable and chaotic dynamics, respectively, and approaches those of a fully chaotic system for sufficiently large coupling strength. In contrast in previous studies the transition from integrable to chaotic dynamics was induced by introducing a random potential of increasing strength into such a QB and applicability of a variant of the Rosenzweig-Porter model was tested.

Figures (22)

• Figure 1: Left: Photograph of the niobium top plate with 10 antenna ports. Right: Photograph of the basin, i.e. the cavity without lid, consisting of a niobium bottom plate and a brass plate with a circular hole, into which a niobium rod is inserted along the diameter, dividing it into a semicircular part and a part which contains four circular niobium scatterers.
• Figure 2: Amplitudes of the transmission matrix element $S_{ba}(f)$ as function of the microwave frequency $f$ from one of the 10 attached antennas $b$ to another one, $a$, in the frequency range from 9 to 10 GHz. From top to bottom the resonance spectra for rod lengths 500 mm, 476 mm and 470 mm, corresponding to weak, moderate and strong coupling of the modes excited in the two parts of the cavity are shown. Shown are the spectra for the transmission from antenna 7 to antenna 10 within the semicircular part.
• Figure 3: Same as Fig. \ref{['Resonance_Spectra']} for rod length 476 mm, corresponding to moderate coupling of the modes excited in the two parts of the cavity are shown. The top panel shows the transmission from antenna 7 to antenna 10 within the empty semicircular part, the middle panel that for transmission from antenna 5 to antenna 2 within the part with scatterers, and the bottom panel for transmission from antenna 2 to 10 between the two parts.
• Figure 4: Intensity distributions of wave functions for the three considered sizes of the opening in the wall separating the cavity into a semicircular apart and a part, which contains four circular niobium disks. The intensity distributions were divided by their maximum value, so that they take values between 0 and 1. The first, second and third column correspond to the cases of weak, moderate and strong coupling, respectively. The resonance frequencies $f_n$ correspond to the same eigenmode number $n$ for the three configurations. They equal 8.5018 GHz, 8.4991 GHz, and 8.4943 GHz in the first row, 10.2945 GHz, 10.2940 GHz, and 10.2946 GHz in the second one, 9.0961 GHz, 9.0955 GHz, and 9.0954 GHz in the third one, and 11.0833 GHz, 11.0794 GHz, and 11.0818 GHz in the fourth row.
• Figure 5: Nearest-neighbor spacing distribution $P(s)$ [(a)-(c)] and the corresponding cumulative distribution $I(s)$ [(d)-(f)] for weak [(a),(d)], moderate [(b),(e)] and strong [(c),(f)] coupling. The red histograms and dots were obtained from the experimental data, the blue ones from the RMT model Eq. (\ref{['RMTModel']}) for coupling strengths (a) $\lambda =0.03$, (b) $\lambda =0.23$, and (c) $\lambda=0.35$, respectively. These are compared with corresponding distributions for Poisson random numbers (black dashed lines) and random matrices from the GOE (black solid lines).