Superuniversal Statistics with Topological Origins for non-Hermitian Scattering Singularities

TL;DR

This work studies topological singularities in non-Hermitian scattering by analyzing the zeros of complex scalar functions of the S-matrix in two-parameter spaces. It combines microwave experiments and Random Matrix Theory to reveal a superuniversal statistic: any quantity with a simple-pole divergence at a singularity has a PDF tail of $-3$ when the singularity is a point and $-2$ when it is a curve. The authors show that increasing uniform absorption $oldeta$ generally reduces singularity density (except for certain higher-order composites), while infinitesimal loss creates EP-2s from DPs, and CPAs require finite loss. Higher-order singularities do not exhibit universal tails and depend on system specifics. Overall, the results provide a generic framework for predicting and understanding singularity abundance in diverse wave-scattering contexts, with potential implications for sensing, imaging, and energy transfer.

Abstract

Vortex singularities in speckle patterns formed from random superpositions of waves are an inevitable consequence of destructive interference and are consequently generic and ubiquitous. Singularities are topologically stable, meaning they persist under small perturbations and can only be removed via pairwise annihilation. They have applications including sensing, imaging and energy transfer in multiple fields such as optics, acoustics, and elastic or fluid waves. We generalize the concept of speckle patterns to arbitrary parameter spaces and any complex scalar function that describes wave phenomena involving complicated scattering. In scattering systems specifically, we are often concerned with singularities associated with complex zeros of various functions of the scattering matrix S, such as Coherent Perfect Absorption, Reflectionless Scattering Modes, Transmissionless Scattering Modes, and Exceptional Points. Experimentally, we find that all singularities share a universal statistical property: any quantity that diverges as a simple pole at a singularity has a probability distribution function with a -3 power law tail. The tail of the distribution provides an estimate for the likelihood of finding a given singularity in a generic system. We use these universal statistical results to determine that homogeneous system loss is the most important parameter determining singularity density in a given parameter space of an absorptive scattering system. Finally, we discuss events where distinct singularities coincide in parameter space, which result in higher order singularities that are not topologically protected, and we do not find universal statistical properties for them. We support our empirical results from microwave experiments with Random Matrix Theory simulations and conclude that the statistical results presented hold for all generic non-Hermitian scattering systems.

Figures (13)

• Figure 1: Speckle pattern of scattering singularities in $d=2$ parameter space. The experimental S-matrix data comes from a tetrahedral microwave graph. (a-b) Magnitude of $\text{det}S$, red lines mark locations of $\text{Re}[\text{det}S]=0$ while black lines mark locations of $\text{Im}[\text{det}S]=0$. White symbols highlight intersections which correspond to $\text{det}S=0+i0$ which are CPAs. (c-d) Phase of $\text{det}S$, black circles show locations of phase windings which align exactly with the white symbols in (a-b). Arrows on the circles in (d) show the two nearby CPAs have opposite winding numbers.
• Figure 2: Schematics of experimental systems. (a) A tetrahedral microwave graph ($\mathcal{D}=1$), (b) a ray-chaotic quarter bowtie billiard ($\mathcal{D}=2$), and (c) a three dimensional cavity with various symmetry breaking elements ($\mathcal{D}=3$). Scattering channels connected to the network analyzer (top right) are marked in red, and embedded perturbers are marked in green.
• Figure 3: (a-b) PDFs of $\frac{\langle |\text{Re}[\delta R_{xy}]| \rangle}{|\text{Re}[\delta R_{xy}]|}$ and $\frac{\langle |\delta R_{xy}| \rangle}{|\delta R_{xy}|}$ from 31 ensembles of experimental scattering systems with different values for the four parameters $\mathcal{D},M,\beta,\eta$(see Appendix \ref{['App_ENS']} for an enumeration of the ensembles). Color gradient of curves corresponds to $\eta$ (not uniformly spaced) of the ensembles. Dashed black lines characterize the power law of the tail behavior. (c-d) PDFs of $\frac{\langle|\text{Re}[\mathcal{S}]| \rangle}{|\text{Re}[\mathcal{S}]|}$ and $\frac{\langle |\mathcal{S}|\rangle}{|\mathcal{S}|}$ for various functions $\mathcal{S}$ from two arbitrary ensembles. Dashed black lines characterize the power law of the tail behavior.
• Figure 4: Singularity density as a function of uniform loss in (a) experimental ensembles and (b) RMT simulation. The overall trend of reduced singularity density with increasing loss in this finite range of $\eta$ is easier to see in (b) as the experimental systems have more differences than just $\eta$, which cannot all be normalized out.
• Figure 5: (a) Locations of scattering zeros in the ($\omega,\nu$) parameter space of an RMT simulation with $\eta=0$. (b) Locations of scattering zeros in the ($\omega,\nu$) parameter space of an RMT simulation using the same effective Hamiltonian and mode-channel coupling as in (a) but with $\eta=0.006$. Dashed red and black lines show the $\eta=0$ locations of TSM-2 and $\delta R_{12}=0$ which have become single (orange and black) points with the introduction of finite absorption. (c-d) Phase of $S_{+i}$ and $S_{-i}$ from simulation in (b). Black circles surround EPs (phase singularities) and arrows show direction of phase winding.