Use of transmission and reflection complex time delays to reveal scattering matrix poles and zeros: Example of the ring graph

We identify the poles and zeros of the scattering matrix of a simple quantum graph by means of systematic measurement and analysis of Wigner, transmission, and reflection complex time delays. We examine the ring graph because it displays both shape and Feshbach resonances, the latter of which arises from an embedded eigenstate on the real frequency axis. Our analysis provides a unified understanding of the so-called shape, Feshbach, electromagnetically induced transparency, and Fano resonances on the basis of the distribution of poles and zeros of the scattering matrix in the complex frequency plane. It also provides a first-principles understanding of sharp resonant scattering features and associated large time delay in a variety of practical devices, including photonic microring resonators, microwave ring resonators, and mesoscopic ring-shaped conductor devices. Our analysis involves use of the reflection time difference, as well as a comprehensive use of complex time delay, to analyze experimental scattering data.

Figure 1

(a) Schematic diagram of a generic ring graph connected to two infinite leads. The two bonds have length $L_1$ and $L_2$. (b) Picture of the experimental microwave ring graph, where a coaxial cable and a coaxial microwave phase shifter are used as the two bonds. (c) Schematic of the experimental setup with the microwave network analyzer included. The two dashed red lines indicate the calibration plane for the $2\times 2S$-matrix measurement.

Figure 2

Transmission spectrum $|S_{21}{|}^2$ vs frequency measured for the first 18 modes of a microwave ring graph. Main figure shows the transmission of nonequal lengths ($L_1\ne L_2$) between the phase shifter and the coaxial cable, while the inset shows the case of equal lengths ($L_1=L_2$). The sinusoidal wiggles come from the shape resonances, while the narrow dips come from the Feshbach resonances. Note that the data in the inset shows no narrow resonances.

Figure 3

Comparisons between the experimental data and the modeling for the complex Wigner time delay (upper plot) and for the complex transmission time delay (lower plot), both normalized by the Heisenberg time ${\tau }_H$, as a function of frequency for a symmetric ($L_1=L_2$) microwave ring graph. The modeling data are plotted on top of the experimental data and are in good agreement.

Figure 4

Comparison between fitted pole location parameters (${\mathcal{E}}_n=E_n-i{\mathrm{\Gamma }}_n$) and predictions for multiple Feshbach modes of the asymmetric microwave ring graph ($L_1\ne L_2$). Inset (a) shows the comparison between fitted real parts of the zeros and the poles, along with the prediction by Eq. (29) shown as a straight purple line. Inset (b) shows such a representative fit to ${\tau }_W\left(f\right)$ for a single Feshbach mode ($n=7$).

Figure 5

Comparison between fitted pole location parameters (${\mathcal{E}}_n=E_n-i{\mathrm{\Gamma }}_n$) obtained from the complex Wigner time delay (blue circles) and the complex transmission time delay (red triangles) for Feshbach modes of the asymmetric ring graph. The lower part of the figure shows a comparison between fitted uniform attenuation ($-\eta$) obtained from the complex Wigner time delay (yellow stars) in Fig. 4 and fitted imaginary parts of the transmission zeros ($\text{Im}{t}_n-\eta$) obtained from the complex transmission time delay (green triangles) on all measured Feshbach modes. Inset (a) shows a representative fit to ${\tau }_T\left(f\right)$ for a single Feshbach mode ($n=7$).

Figure 6

Fitting example of reflection time difference or delay for a single pair of shape and Feshbach resonances for a ring graph with $L_1\ne L_2$. (a) Example of fitting complex reflection time difference ($\delta {\mathcal{T}}_R={\tau }_R^{\left(1\right)}-{\tau }_R^{\left(2\right)}$) experiment data for mode $n=7$. The left feature is due to the Feshbach resonance, while the right one is due to the shape resonance. (b, c) Demonstration of the reconstruction of the individual reflection time delays on both ports, compared to the data, using the fitted reflection zero and Wigner pole (see Fig. 4) information. All time delays are presented normalized by the Heisenberg time ${\tau }_H$ of the loop graph.

Figure 7

Summary of all zeros and poles in the complex frequency plane for shape and Feshbach resonances extracted from Wigner-transmission-reflection time delay analysis for the first 37 modes of the microwave ring graph. The Wigner zeros $z_n^S$ (blue squares) and poles ${\mathcal{E}}_n^S$ (red squares) of the shape resonances are located far from the real axis. The Wigner zeros $z_n^F$ (blue circles) and poles ${\mathcal{E}}_n^F$ (red circles) of the Feshbach resonances are close to, and symmetrically arrayed about, the real axis. The transmission zeros $t_n^F$ (blue crosses) of the Feshbach resonances lie on the real axis. The reflection zeros $r_n^F\text{and}{r}_n^S$ of the Feshbach resonances (dark red triangles) and the shape resonances (green squares) are symmetrically arrayed about the real axis.

Figure 8

Comparison of modeling (red line) and experimental data (blue line) for $detS$ with shape resonances only in a symmetrical ($L_1=L_2$) ring graph. The modeling data is calculated from Eq. (4) using the Wigner zeros and poles for the shape resonances (see the blue and red squares in Fig. 7). The upper plot shows the magnitude of $detS$, while the lower plot shows the phase of $detS$.

Figure 9

Comparison of modeling (red dashed line) and experimental data (blue line) for $detS$ with both shape and Feshbach resonances in an asymmetrical ($L_1\ne L_2$) ring graph. The modeling data are calculated from Eq. (4) using the Wigner zeros and poles for the shape resonances (see the blue and red squares in Fig. 7) and the Wigner zeros and poles for the Feshbach resonances (see the blue and red circles in Fig. 7). Upper plot shows the magnitude of $detS$, while the lower plot shows the phase of $detS$.

Figure 10

Complex representation of $detS$ evaluated over the complex frequency plane for several modes of an asymmetric ($L_1\ne L_2$) ring graph. $detS$ is calculated from Eq. (4) using complex frequency and the Wigner zeros and poles for the shape resonances (see the blue and red squares in Fig. 7) and the Wigner zeros and poles for the Feshbach resonances (see the blue and red circles in Fig. 7). The 3D plot represents $|detS|$ on a log scale and reveals the zeros (dips) and poles (peaks) at different locations in complex frequency. The base plane shows contour lines of the magnitude of $|detS|$ in the complex frequency plane. The color bar on the right shows the phase of the constructed $detS$. The inset shows a 2D top view of $\text{Arg}[detS]$ for a single pair of shape and Feshbach zeros and poles.

Figure 11

Plot of residue ${\rho }_n^F$ and the “quality factor” of the Feshbach poles vs mode index $n$ for an asymmetric microwave loop graph. The blue filled circles show the absolute magnitude of the residue $|{\rho }_n^F|$ as a function of mode index based on the extracted Feshbach poles and zeros, while the red open diamonds show the associated ratio of $E_n^F/{\mathrm{\Gamma }}_n^F$ of the Feshbach poles.

Figure 12

Complex Wigner time delay ${\tau }_W$ (normalized by the Heisenberg time ${\tau }_H$) determined from measured $S$-matrix data for 37 modes ($0\text{–}10$ GHz) in an asymmetrical ($L_1\ne L_2$) microwave ring graph. The extreme values of ${\tau }_W$ are dominated by Feshbach resonances. Note the sign change of the $\text{Re}\left[{\tau }_W\right]$ extreme values near 7 GHz, which corresponds to the crossover between ${\mathrm{\Gamma }}_n$ and $\eta$ in Fig. 4. Insets (a) and (b) show zoomed-in details of the complex Wigner time delay for individual modes on either side of the crossover.

Figure 13

Complex transmission time delay ${\tau }_T$ determined from measured $S$-matrix data for 37 modes ($0-10$ GHz) in an asymmetrical ($L_1\ne L_2$) microwave ring graph normalized by the Heisenberg time ${\tau }_H$. The extreme values of ${\tau }_T$ are dominated by Feshbach resonances. The nearly sinusoidal variations of $\text{Re}\left[{\tau }_T\right]$ and $\text{Im}\left[{\tau }_T\right]$ with frequency are due to the shape resonances. Insets (a) and (b) show the zoomed-in details of the complex transmission time delay for two individual modes.

Figure 14

Complex reflection time delays ${\tau }_R^{\left(1\right)}$, ${\tau }_R^{\left(2\right)}$ and their difference $\delta {\mathcal{T}}_R={\tau }_R^{\left(1\right)}-{\tau }_R^{\left(2\right)}$ determined from measured $S$-matrix data for 37 modes (0–10 GHz) in an asymmetrical ($L_1\ne L_2$) microwave ring graph, normalized by the Heisenberg time ${\tau }_H$. Insets show the zoomed-in details of the complex reflection time delay or difference for individual sets of shape and Feshbach modes.

Figure 15

Comparison of three different ways to determine the uniform attenuation of the loop graph: by means of direct measurement of insertion loss through $S_{21}$, fitting results to complex time delays ($\eta$), and direct modeling ($\mathrm{\Gamma }$). The blue line shows the data obtained by measuring the $S_{21}$ insertion loss of a serial connection of the coaxial cable and the phase shifter shown in Fig. 1. The yellow stars show the fitting results for $\eta$ from the complex Wigner time delay analysis in Fig. 4. The red line shows the theoretical modeling [Eq. (B1)] of $\mathrm{\Gamma }/2\pi$ in a coaxial cable.

Figure 16

Complex transmission time delay data for a single Feshbach mode ($n=1$) and its contributions from zeros and poles. (a) Total complex transmission time delay data (${\tau }_T$), (b, c) contribution from the zero (${\tau }_T^Z$) and the pole (${\tau }_T^P$), respectively. Here ${\tau }_T$ is from experimental data, while ${\tau }_T^P$ is calculated based on Eqs. (C3) and (C4) with the pole information extracted from the complex Wigner time delay analysis (see Fig. 4). ${\tau }_T^Z$ is obtained by ${\tau }_T^Z={\tau }_T-{\tau }_T^P$.

Figure 17

Comparison between the peak value of $\text{Re}\left[{\tau }_T^Z\right]$ and $-{\eta }^{-1}$ for all 37 modes of the microwave ring graph. Blue circles show the peak value of $\text{Re}\left[{\tau }_T^Z\right]$ from experimental data, while red triangles show $-{\eta }^{-1}$ calculated from the data in Fig. 4. Both quantities are presented normalized by the Heisenberg time ${\tau }_H$ of the loop graph.

Figure 18

Complex representation of $detS$ evaluated over the complex frequency plane for several modes of an asymmetric ($L_1\ne L_2$) microwave ring graph. This 3D plot shows another perspective of Fig. 10.