Universal Frequency Correlations and Recurrence Statistics of Complex Impedance Matrices

TL;DR

The paper addresses universal fluctuations in complex impedance matrices for highly scattering wave systems and develops a statistical framework using the Random Coupling Model (RCM) with radiation impedance normalization to obtain a dimensionless impedance $Z$. It provides explicit two-point frequency correlation functions for both diagonal and off-diagonal impedance elements in the high-loss limit, together with a recurrence-time scale $\Lambda(Z^t)$ that links impedance revisit intervals to system parameters $\alpha$, $\Delta$, $\epsilon$, and the impedance PDF peak $Z^{\text{peak}}$. The authors validate the theory through experiments on microwave graphs, billiards, and 3D cavities, and complement it with Random Matrix Theory simulations, demonstrating robust universality that extends to optics and acoustics. These results offer practical insights for sensing and design in complex environments and point toward future exploration of impedance-based exceptional points and related singularities.

Abstract

Linear electromagnetic wave scattering systems can be characterized by an impedance matrix that relates the voltages and currents at the ports of the system. When the system size becomes greater than the wavelength of the fields involved, the impedance matrix becomes a complicated function of the details of the system, in which case a statistical model, such as the Random Coupling Model (RCM) becomes useful. The statistics of the elements of the RCM impedance matrix depend on the excitation frequency, the spectral density of the modes of the enclosed system volume, the average loss factor (Q^{-1}) of the system, and the properties of the coupling ports as given by their radiation impedances. In this paper, properties of the elements of impedance matrices are explored numerically and experimentally. These include the two point frequency correlation functions for the complex impedance of elements and the expected difference in frequencies between which impedance values are approximately repeated. Universal scaling arguments are then given for these quantities, hence these results are generic for all sufficiently complicated scattering systems, including acoustic and optical systems. The experimental data presented in this paper come from microwave graphs, billiards, and three-dimensional cavities with embedded tunable perturbers such as metasurfaces. The data is found to be in generally good agreement with the predictions for the two point frequency correlations and the frequency interval for successive repetitions of impedance matrix elements values.

Figures (8)

• Figure 1: Clockwise path traced by normalized $Z_{12}$ in the complex impedance plane over a small frequency band. The color scale corresponds to the frequency difference $\delta=f_a-f_b$ where $f_b$ is the first frequency at which $Z_{12}$ is within the gray circle of radius $\epsilon=0.015$ centered on $Z_{12}^\text{t}=0.082+0.114i$. $\delta$ is normalized by a characteristic scale corresponding to the typical $Q$-widths of the modes $\gamma=2\alpha\Delta=0.66~\text{MHz}$, where $\alpha$ is the RCM loss parameter and $\Delta$ is the mean mode spacing. The arrows on the color bar mark the frequencies at which $Z_{12}$ enters and leaves the circle. The impedance $Z_{12}$ plotted in this figure is experimentally measured from a three dimensional cavity with absorption $\alpha=5.5$.
• Figure 2: Two point frequency correlation function of the real part of a diagonal impedance element $\langle\tilde{R}(f_a)\tilde{R}(f_b)\rangle_C$ normalized by the zero separation correlation $\langle (\tilde{R}_{pp})^2 \rangle$ as a function of $\delta/\gamma=\frac{|f_a-f_b|}{2\alpha\Delta}$. Comparing prediction of Eq. \ref{['EQN_CorrR']} illustrated by the dashed black curve, and expression derived by Fyodorov, Savin, and Sommers in Ref. Fyodorov2005, represented by the colored curves corresponding to different choices of loss parameter $\alpha$. The green curve which has $\alpha=10$ is indistinguishable from the dashed black curve.
• Figure 3: (a) Vector network analyzer with four total ports. Two of the ports are connected through the red cables to the following experimental systems: (b) a tetrahedral microwave graph ($\mathcal{D}=1$), (c) a ray-chaotic quarter bowtie billiard ($\mathcal{D}=2$), and (d) a three dimensional cavity with various symmetry breaking elements ($\mathcal{D}=3$). Embedded tunable perturbers used for ensemble creation are marked in green.
• Figure 4: Bivariate PDFs of complex impedance $Z$ at port 1 of a $\mathcal{D}=3$ cavity with absorption $\alpha=5.5$. (a) PDF of diagonal impedance $Z_{11}$, projections show the PDF is symmetric along $X_{11}$ but asymmetric along $R_{11}$. (b) Top down view of $\mathcal{P}(Z_{11})$ where the colored lines correspond to trajectories of target impedance $Z^\text{t}_{11}$ away from $Z^\text{peak}_{11}$ along different angles $\theta=\text{Arg}(Z^\text{t}_{11}-Z^\text{peak}_{11})$. (c) PDF of off-diagonal impedance $Z_{12}$, projections show the PDF is symmetric along both $X_{12}$ and $R_{12}$. (d) Top down view of $\mathcal{P}(Z_{12})$ where the colored lines correspond to trajectories of target impedance $Z^\text{t}_{12}$ away from $Z^\text{peak}_{12}$ along different angles $\theta=\text{Arg}(Z^\text{t}_{12}-Z^\text{peak}_{12})$.
• Figure 5: Two point frequency correlation functions of complex impedance normalized by twice $(R^\text{RMS})^2=\langle(R_{12})^2\rangle$, as a function of $\delta/\gamma=\frac{|f_a-f_b|}{2\alpha\Delta}$. Dashed black lines correspond to Eqs \ref{['EQN_CorrR']}-\ref{['EQN_CorrTCross']} while solid colored lines are data results. (a) Experimental data from a three dimensional cavity with absorption $\alpha=5.5$, mean mode spacing $\Delta=0.06~\text{MHz}$ and characteristic frequency scale $\gamma=0.66~\text{MHz}$. (b) RMT numerical simulation with absorption $\alpha=5.5$, mean mode spacing $\Delta=14.2~\text{MHz}$ and characteristic frequency scale $\gamma=156.1~\text{MHz}$.