Chaotic scattering in the regime of weakly overlapping resonances

We measure the transmission and reflection amplitudes of microwaves in a resonator coupled to two antennas at room temperature in the regime of weakly overlapping resonances and in a frequency range of $3–16\mathrm{GHz}$. Below $10.1\mathrm{GHz}$ the resonator simulates a chaotic quantum system. The distribution of the elements of the scattering matrix $S$ is not Gaussian. The Fourier coefficients of $S$ are used for a best fit of the autocorrelation function of $S$ to a theoretical expression based on random-matrix theory. We find very good agreement below but not above $10.1\mathrm{GHz}$.

Figure 1

Absolute squares of the scattering matrix elements $S_{ab}$ for signal transmission from antenna 2 to 1 (upper panel) and reflection at antenna 1 (lower panel) between 9.0 and $9.5\mathrm{GHz}$. On the logarithmic decibel scale, $-x$ dB means an attenuation of the microwave power by a factor of ${10}^{x∕10}$. The resonances overlap and create a pattern of fluctuations. Inset: the shape of the two-dimensional microwave resonator used in the experiment. Points 1 and 2 indicate the positions of the antennas.

Figure 2

From left to right: histograms for the scaled distributions of the real and imaginary parts of the reflection amplitude $S_{11}$ and the real part and the phase of the transmission amplitude $S_{12}$, for the two frequency intervals $5–6\mathrm{GHz}$ (upper panels) and $9–10\mathrm{GHz}$ (lower panels). The scaling factors are given in each panel. The solid lines are best fits to Gaussian distributions.

Figure 3

Upper panels: comparison of the autocorrelation function $C_{ab}\left(ϵ\right)$ constructed from the data (points) and the fit using Eq. (2) (solid line), both normalized by the value of $C\left(0\right)$ as given by Eq. (2). Lower panels: Fourier coefficients ${\tilde{C}}_{ab}\left(t\right)$ of the autocorrelation functions (points) and the Fourier transform of the fit of $C_{ab}\left(ϵ\right)$ as given by Eq. (2) to the data (solid line). The elements $S_{12}\left(f\right)$ and $S_{11}\left(f\right)$ were taken from the frequency window $9–10\mathrm{GHz}$.

Figure 4

Same as Fig. 3, but for the scattering function $S_{12}\left(f\right)$ taken in the frequency window $12–13\mathrm{GHz}$.