Superuniversal Statistics of Complex Time-Delays in Non-Hermitian Scattering Systems

The Wigner-Smith time-delay of flux conserving systems is a real quantity that measures how long an excitation resides in an interaction region. The complex generalization of time-delay to non-Hermitian systems is still under development, and its statistical properties in the short-wavelength limit of complex chaotic scattering systems have not been investigated. From the experimentally measured multi-port scattering ($S$)-matrices of one-dimensional graphs, a two-dimensional billiard, and a three-dimensional cavity, we calculate the complex Wigner-Smith, as well as each individual reflection and transmission time-delays. The complex reflection time-delay differences between each port are calculated, and the transmission time-delay differences are introduced for systems exhibiting non-reciprocal scattering. Large time-delays are associated with scattering singularities such as coherent perfect absorption, reflectionless scattering, slow light, and uni-directional invisibility. We demonstrate that the large-delay tails of the distributions of the real and imaginary parts of each time-delay quantity are superuniversal, independent of experimental parameters: wave propagation dimension $\mathcal{D}$, number of scattering channels $M$, Dyson symmetry class $\beta$, and uniform attenuation $\eta$. The tails determine the abundance of the singularities in generic scattering systems, and the superuniversality is in direct contrast with the well-established statistics of unitary systems, where the distribution tail depends explicitly on the values of $M$ and $\beta$. We relate the statistics to the topological properties of the corresponding singularities. Although the results presented here are based on classical microwave experiments, they are applicable to any non-Hermitian wave-chaotic scattering system in the short-wavelength limit, such as optical or acoustic resonators.