Generalization of Wigner time delay to subunitary scattering systems

We introduce a complex generalization of the Wigner time delay $\tau$ for subunitary scattering systems. Theoretical expressions for complex time delays as a function of excitation energy, uniform and nonuniform loss, and coupling are given. We find very good agreement between theory and experimental data taken on microwave graphs containing an electronically variable lumped-loss element. We find that the time delay and the determinant of the scattering matrix share a common feature in that the resonant behavior in $\text{Re}\left[\tau \right]$ and $\text{Im}\left[\tau \right]$ serves as a reliable indicator of the condition for coherent perfect absorption (CPA). By reinforcing the concept of time delays in lossy systems this work provides a means to identify the poles and zeros of the scattering matrix from experimental data. The results also enable an approach to achieving CPA at an arbitrary frequency in complex scattering systems.

Figure 1

(a) shows a schematic of the graph experimental setup. The lumped loss ${\mathrm{\Gamma }}_A$ is varied by changing the applied voltage to the variable attenuator. (b) and (c) show experimental data of both real and imaginary parts of Wigner time delay $\text{Re}\left[\tau \right]$ and $\text{Im}\left[\tau \right]$ (normalized by the Heisenberg time ${\tau }_{\text{H}}$) as a function of frequency under different attenuation settings for a single isolated mode. For each attenuation setting, the data are plotted from 2.645 to 2.665 GHz. For clarity, plots with a higher attenuation setting are shifted 0.01 GHz from the previous one. The inset shows the entire range of $\text{Re}\left[\tau \right]$ for an attenuation setting of 2.35 dB. (d) and (e) demonstrate the two-Lorentzian nature of the real and imaginary parts of the Wigner time delay as a function of frequency. The fitting parameters in these two plots are $\text{Re}{z}_n=2.6556$ GHz, $E_n=2.6544$ GHz, $\text{Im}{z}_n-\gamma =-7.1065\times {10}^{-4}$ GHz, and ${\mathrm{\Gamma }}_n+\gamma =0.0110$ GHz. The constants used in the $\text{Re}\left[\tau \right]$ and $\text{Im}\left[\tau \right]$ fits are $C_{\text{R}}=0.26$ and $C_{\text{I}}=-0.0018$ in units of ${\tau }_{\text{H}}$. A detailed discussion about the fitting constants and degree of isolation of the modes can be found in Supplemental Material Sec. IV [70].

Figure 2

(a) Fitted parameters $\text{Im}{z}_n-\gamma$ and ${\mathrm{\Gamma }}_n+\gamma$ for the complex Wigner time delay from graph experimental data. Also shown is the evolution of $|det(S\left)\right|$ at the specific frequency of interest, $f_{\text{CPA}}$, which reaches its minimum at the zero-crossing point. The inset shows the evolution of $\text{Re}{z}_n$ and $E_n=\text{Re}{\mathcal{E}}_n$ with attenuation. (b) Evolution of complex zero and pole of a single mode of the graph in the complex frequency plane as a function of ${\mathrm{\Gamma }}_A$. The black crosses are the initial state of the zero and pole at the minimum attenuation setting. The insets show the details of the complex zero and pole migrations.