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

Abstract

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, in particular, its statistical properties in the short-wavelength limit of complex chaotic scattering systems has 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 ($\tau_{WS}$), as well as each individual reflection ($\tau_{xx}$) and transmission ($\tau_{xy}$) time-delays. The complex reflection time-delay differences ($\tau_{\delta R}$) between each port are calculated, and the transmission time-delay differences ($\tau_{\delta T}$) are introduced for systems exhibiting non-reciprocal scattering. Large time-delays are associated with 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 of these time-delay quantities are superuniversal, independent of experimental parameters: uniform attenuation $\eta$, number of scattering channels $M$, wave propagation dimension $\mathcal{D}$, and Dyson symmetry class $\beta$. This superuniversality is in direct contrast with the well-established time-delay statistics of unitary scattering systems, where the tail of the $\tau_{WS}$ distribution depends explicitly on the values of $M$ and $\beta$. Due to the direct analogy of the wave equations, the time-delay statistics described in this paper are applicable to any non-Hermitian wave-chaotic scattering system in the short-wavelength limit, such as quantum graphs, electromagnetic, optical and acoustic resonators, etc.