Mode Structure in Superconducting Metamaterial Transmission-Line Resonators

Superconducting metamaterials are a promising resource for quantum-information science. In the context of circuit QED, they provide a means to engineer on-chip dispersion relations and a band structure that could ultimately be utilized for generating complex entangled states of quantum circuitry, for quantum-reservoir engineering, and as an element for quantum-simulation architectures. Here we report on the development and measurement at millikelvin temperatures of a particular type of circuit metamaterial resonator composed of planar superconducting lumped-element reactances in the form of a discrete left-handed transmission line that is compatible with circuit QED architectures. We discuss the details of the design, fabrication, and circuit properties of this system. As well, we provide an extensive characterization of the dense mode spectrum in these metamaterial resonators, which we conduct using both microwave-transmission measurements and laser-scanning microscopy. Results are observed to be in good quantitative agreement with numerical simulations and also an analytical model based upon current-voltage relationships for a discrete transmission line. In particular, we demonstrate that the metamaterial mode frequencies, spatial profiles of current and charge densities, and damping due to external loading can be readily modeled and understood, making this system a promising tool for future use in quantum-circuit applications and for studies of complex quantum systems.

Figure 1

Metamaterial transmission line schematic: (a) ideal LHTL; (b) composite LHTL including stray reactances; (c) calculated dispersion relations $\omega (k)$ for ideal LHTL (dashed blue line) [Eq. (1)] and composite LHTL (solid black line) [Eq. (2)] using circuit parameters described in the text.

Figure 2

(a) Schematic of a composite LHTL resonator with stray reactances including coupling capacitances $C_c$ at each end. (b) Plot of mode frequencies vs wave number computed for $N-1=41$ modes in a LHTL (${\omega }_{\mathrm{IR}}/2\pi$=5 GHz, ${\omega }_C/2\pi =40\mathrm{GHz}$, ${\omega }_L/2\pi =50\mathrm{GHz}$). The solid line is the disperson relation obtained from Eq. (2).

Figure 3

Optical micrographs of metamaterial resonator device: (a) enlarged image of the entire chip, (b) input coupling capacitor $C_c$ and the first few unit cells, (c) meander-line inductor of the first unit cell, (d) detail of input coupling capacitor and connection between inductor and capacitor of the first unit cell, (e) interdigitated capacitors in several unit cells, (f) detail of interdigitated capacitor.

Figure 4

Measurements of the magnitude of the microwave transmission $|S_{21}(f)|$ on the ADR at two different temperatures: 65 mK (solid black line); 3 K (blue dashed line). The inset shows an enlarged plot in the vicinity of the $n=38$ mode near 5.21 GHz.

Figure 5

(a) LSM reflectivity image with the arrow indicating the direction of 1D line scans. (b) Microwave transmission (not normalized) $|S_{21}(f)|$ measured on LSM. (c) Average LSM photoresponse $\overline{\mathcal{R}}(y_0,f)$ along 1D line scans. (d) 1D line scans $\mathcal{R}(x,y_0,f)$ vs frequency; dashed horizontal lines indicate location of input and output coupling capacitors.

Figure 6

LSM photoresponse compared with Sonnet current and charge density simulation. (a) Reflectivity map from LSM showing details of metamaterial layout in the imaged area. (b) 2D distribution of LSM photoresponse $\mathcal{R}(x,y,3.44\mathrm{GHz})$ for the $n=40$ mode. Bright (dark) regions correspond to large (small) $\mathcal{R}$ signal. Sonnet simulations of the $n=40$ mode (c) charge density and (d) current density.

Figure 7

Array of LSM images of metamaterial for different modes, labeled by mode number and frequency. Bright (dark) regions correspond to large (small) $\mathcal{R}$ signal. Amplitudes of $\mathcal{R}$ signal are normalized for best contrast. Green boxes indicate the pair of $n=34$ and $n=8$ modes, which both have eight antinodes. Blue boxes indicate the pair of $n=22$ and $n=20$ modes, which both have 20 antinodes.

Figure 8

Plot of mode frequency vs number of antinodes in the photoresponse from the corresponding LSM images. Colored boxes indicate the partner modes shown in Fig. 7.

Figure 9

(a) Circuit schematics for ideal ten-cell metamaterial resonator. Waveform for the square of standing-wave current for hypothetical continuous LHTL for mode (b) $n=8$, (c) $n=2$, including red circles corresponding to the location of each unit cell.

Figure 10

Circuit schematics for ideal ten-cell metamaterial resonators with arrows indicating the relative sense of currents flowing in each unit cell along with waveform for standing-wave current for hypothetical continuous LHTL. Red circles correspond to current at location of each unit cell for mode (a) $n=8$, (b) $n=2$.

Figure 11

(a) LSM photoresponse for mode 23 with the arrow indicating the location of subsequent line cut. Bright (dark) regions correspond to large (small) $\mathcal{R}$ signal. (b) Plot of the square root of LSM photoresponse signal along line cut. (c) Standing-wave pattern of voltage across capacitors computed with Eq. ((E3)) for mode 23 of a 42-cell metamaterial resonator.

Figure 12

(a) $|S_{21}(f)|$ from Sonnet simulation (red) offset by 60 dB compared with measured LHTL spectrum from ADR at 65 mK (black). (b) $|S_{21}(f)|$ from the Sonnet simulation plotted out to $45\mathrm{GHz}$ showing a gap beyond the $n=0$ mode near 40 GHz.

Figure 13

Plot of mode frequency vs normalized wave number for ADR measurements, LSM images, Sonnet simulations, including both staggered and nonstaggered inductor configurations, as discussed in the text; the solid black curve corresponds to the dispersion relation of Eq. (2). The inset shows a closeup of behavior near ${\omega }_{\mathrm{IR}}/2\pi$.

Figure 14

Comparison of internal loss and coupling loss extracted from the measured $S_{21}(f)$.

Figure 15

Coupling losses for different mode numbers extracted from experimentally measured resonances on ADR at 65 mK, AWR simulation of ideal lumped-element LHTL resonator, and Sonnet simulations of both staggered and nonstaggered inductor configurations. The solid curve corresponds to Eq. (3).

Figure 16

Voltage and current in a unit cell of a general discrete transmission line.

Figure 17

Circuit diagram for a $S_{21}$ measurement setup for an $N$-cell metamaterial transmission-line resonator. We assume symmetric coupling so that the source impedance at the input end $Z_s$ and the load impedance at the terminated end $Z_l$ are equal. The total impedance of the LHTL resonator looking from the left end is $Z_{-N}$ given by Eq. ((B10)) and is marked in the red dashed box.

Figure 18

$S_{21}(f)$ calculated for the $n=23$ mode of a 42-cell composite LHTL metamaterial resonator using Eq. ((C3)) (solid red line) and simulated using AWR Microwave Office (blue points). Unit-cell parameters are $C_l=266\mathrm{fF}$, $L_l=0.6\mathrm{nH}$, $C_r=21.806\mathrm{fF}$, and $L_r=0.595\mathrm{nH}$, chosen based on the discussion in Sec. 3 and the Sonnet simulation result of the stray reactances.

Figure 19

Solid line is $1/Q_c$ expression of a continuous transmission-line resonator, as described by Eq. ((D1)); note that this is plotted as a continuous function for clarity, even though $n$ is limited to integer values. The red points are the $1/Q_c$ values for a 40-cell right-handed discrete transmission-line resonator extracted from a circuit simulation using AWR Microwave Office. The parameters for $1/Q_c$ described by Eq. ((D1)) are ${\omega }_0/2\pi =1\mathrm{GHz}$, $Z_0=50\mathrm{\Omega }$, $C_c=1\mathrm{fF}$. The parameters of the discrete RHTL resonator are $L_r=0.625\mathrm{nH}$, $C_r=250\mathrm{fF}$, $C_c=1\mathrm{fF}$.

Figure 20

Coupling-loss comparison for an RHTL resonator for analytic expression from Eq. ((D10)), AWR Microwave Office circuit simulation, and linewidth extraction for $S_{21}(\omega )$ expression from Eq. ((C3)). The parameters for the RHTL are $C=250\mathrm{fF}$, $L=0.625\mathrm{nH}$, $C_c=1\mathrm{fF}$, $N=40$.

Figure 21

Coupling-loss comparison for an LHTL resonator for analytic expression from Eq. ((D13)), AWR Microwave Office circuit simulation, and linewidth extraction for $S_{21}(\omega )$ expression from Eq. ((C3)). The parameters for the LHTL are $C=250\mathrm{fF}$, $L=0.625\mathrm{nH}$, $C_c=10\mathrm{fF}$, $N=40$.

Figure 22

Coupling-loss comparison between calculation for an ideal LHTL resonator using Eq. ((D13)) with an composite LHTL resonator with nonzero stray reactances from linewidth extractions using Eq. ((C3)). The parameters for the LHTL are $C=250\mathrm{fF}$, $L=0.625\mathrm{nH}$, $C_c=10\mathrm{fF}$, $N=40C_r=16.211389\mathrm{fF}$, $L_r=0.0334947\mathrm{nH}$, corresponding to self-resonance frequencies of 50 and 55 GHz.

Figure 23

Standing-wave patterns for modes $n=3$ and 39 of a 42-cell LHTL resonator from the Sonnet simulations of current-density and charge-density distributions compared with expressions for current in inductors (voltage across capacitors) from Eq. ((E4)) [Eq. ((E3))] normalized by the maximum value for each mode of current $i_0(n)$ [voltage $v_0(n)$] so $I_m^{L_l}(n)=|i_m^{L_l}(n)/i_0(n)|$ [$V_m^{C_l}(n)=|v_m^{C_l}(n)/v_0(n)|$].

Figure 24

High-resolution LSM images of 11 cells near the center of metamaterial resonator for (a) $n=40$ and (b) $n=9$ modes. Bright (dark) regions correspond to large (small) $\mathcal{R}$ signal. Plots to the right of the image correspond to line cuts as indicated on the images. For the inductor line cuts, the traces through the top and bottom inductors are added together.

Figure 25

Sonnet simulations for different grounding configurations for a 42-cell LHTL resonator for (a) resonance frequency vs $n$, (b) coupling loss $1/Q_c$ vs $n$.

Figure 26

Sonnet simulations of $S_{21}(f)$ for nonstaggered (orange, with $-40\mathrm{dB}$ offset for clarity) and staggered (blue) inductor layouts. Note that the blue curve is the same as in Fig. 12.