Physical Interpretation of Imaginary Time Delay

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 τ_{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 τ_{T} is a well-studied quantity, but the imaginary part of τ_{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. Commun. 7, 13488 (2016)NCAOBW2041-172310.1038/ncomms13488] for the imaginary part of transmission time delay in a nonunitary scattering system. We utilize Gaussian time-domain pulses scattering from a two-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, Im[τ_{T}], from frequency-domain measurements of the subunitary 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.