A Physical Interpretation of Imaginary Time Delay

TL;DR

This paper experimentally investigates the imaginary part of transmission time delay in non-unitary scattering systems, confirming theoretical predictions and demonstrating how it influences pulse propagation in non-Hermitian systems.

Contribution

It provides the first experimental validation of the imaginary time delay in non-Hermitian scattering systems and extends theoretical understanding of pulse dynamics in such systems.

Findings

Carrier frequency shift matches the imaginary part of transmission time delay.

Experimental results agree with theoretical predictions for non-unitary systems.

Method enables prediction of pulse behavior in non-Hermitian scattering environments.

Abstract

The scattering matrix $S$ linearly relates the vector of incoming waves to outgoing wave excitations, and contains an enormous amount of information about the scattering system and its connections to the scattering channels. Time delay is one way to extract information from $S$, and the transmission time delay $\tau_T$ is a complex (even for Hermitian systems with unitary scattering matrices) measure of how long a wave excitation lingers before being transmitted. The real part of $\tau_T$ is a well-studied quantity, but the imaginary part of $\tau_T$ has not been systematically examined experimentally, and theoretical predictions for its behavior have not been tested. Here we experimentally test the predictions of Asano, et al. [Nat. Comm. 7, 13488 (2016)] for the imaginary part of transmission time delay in a non-unitary scattering system. We utilize Gaussian time-domain pulses scattering from a 2-port microwave graph supporting a series of well-isolated absorptive modes to show that the carrier frequency of the pulses is changed in the scattering process by an amount in agreement with the imaginary part of the independently determined complex transmission time delay, $\text{Im}[\tau_T]$, from frequency-domain measurements of the sub-unitary $S$ matrix. Our results also generalize and extend those of Asano, et al., establishing a means to predict pulse propagation properties of non-Hermitian systems over a broad range of conditions.