Compact and Tunable Forward Coupler Based on High-Impedance Superconducting Nanowires

Developing compact, low-dissipation, cryogenic-compatible microwave electronics is essential for scaling up low-temperature quantum computing systems. In this paper, we demonstrate an ultracompact microwave directional forward coupler based on high-impedance slow-wave superconducting-nanowire transmission lines. The coupling section of the fabricated device has a footprint of $416\mu {\mathrm{m}}^2$. At 4.753 GHz, the input signal couples equally to the through port and forward-coupling port (50:50) at $-6.7\mathrm{dB}$ with $-13.5\mathrm{dB}$ isolation. The coupling ratio can be controlled with dc bias current or temperature by exploiting the dependence of the kinetic inductance on these quantities. The material and fabrication processes are suitable for direct integration with superconducting circuits, providing a practical solution to the signal distribution bottlenecks in developing large-scale quantum computers.

Figure 1

Superconducting-nanowire directional forward coupler: device and model. (a) Optical and scanning electron micrographs (SEMs) of the fabricated device: (i) device die mounted in the radio-frequency (rf) testing box; (ii) SEM showing the full four-port device before the fabrication of the dielectric spacer and the top ground; (iii) closeup SEM of the impedance-matching taper; (iv) closeup SEM of the coupled nanowire transmission lines. (b) Geometry, model, and implementation of coupler: two nanowire transmission lines brought together for a coupling length $l$. (i) Analytical model: coupled $LC$ ladder, where we explicitly separated the geometric (Faraday) component of the total line inductance per unit length (p.u.l.), ${\mathcal{L}}_F$, from the kinetic inductance p.u.l. ${\mathcal{L}}_k$; $\mathcal{C}$ is the capacitance p.u.l. of each line; $\mathcal{E}$ is the coupling capacitance p.u.l.; $\mathcal{M}$ is the mutual inductance p.u.l. (ii) Implementation as side-coupled striplines. In the analytical model, the gold layers ($\mathrm{Au}$) have been treated as perfect electric conductors. (c) Simulation of the port voltages versus coupler length at $5\mathrm{GHz}$ for 50:50 forward coupling ($L_k=80\mathrm{pH}$ per square).

Figure 2

Microwave response of the directional forward coupler. (a) Analytical modeling of the coupling section (without taper) and simulation of the impedance-matched coupler. Above $3\mathrm{GHz}$ the curves match, showing that the impedance-matching taper, interfacing the high-impedance coupling section to the $50\mathrm{\Omega }$ rf electronics, does not disturb the coupling operation. The 10% bandwidth is $4.4$ to $5.8\mathrm{GHz}$. (b) Measured microwave response of the fabricated device at $1.3\mathrm{K}$. We compare the data to simulation, corrected for fabrication nonidealities, and including normal conductor losses from the $\mathrm{Au}$ ground plane and CPW feedlines. The balanced forward coupling is observed at $4.753\mathrm{GHz}$.

Figure 3

Nonlinear dependence of the kinetic inductance $L_k(x^{\prime }=I_b/I_{\mathrm{sw}},t=T/T_c)$ and coupler response tunability at $=4.75\mathrm{GHz}$ with bias current and temperature. (a) Kinetic inductance’s dependence on current, for $I_{\mathrm{sw}}/I_d=0.7$, according to the fast relaxation model with $\alpha =2.27$. (b) Tunability of the transmission ($S21$) and coupling ($S41$) responses as a function of the current (normalized by $I_{\mathrm{sw}}$) supplied through the isolation port. The measured data (symbols) well match our analytical model (lines) based on a coupled-mode formulation with current-dependent kinetic inductance. (c) Kinetic inductance’s dependence on temperature calculated from the explicit solution of the temperature dependence of the superconducting gap $\mathrm{\Delta }(T)$, with $N(0)V=0.32$ [49, 50, 51] and $T_c=8\mathrm{K}$ (see the Supplemental Material [48]). (d) Modulation of transmission and coupling parameters, as a function of the cryostat temperature (normalized by $T_c$). The measured data (symbols) match our analytical model (lines) based on the coupled-mode formulation with temperature-dependent kinetic inductance.

Figure 4

Analytical calculation of the ports voltage for a coupling section made of normal conductor (lossless with $L_k=0$). The physical dimensions are the same as in Fig. 1.

Figure 5

Simulation of the nanowire stripline parameters. (a) Characteristic impedance and phase velocity fraction versus conductor width for a sheet kinetic inductance $L_k=80\mathrm{pH}$ per square. (b) Characteristic impedance and effective index for a $300\mathrm{nm}$ wide stripline, as a function of the kinetic inductance.

Figure 6

Transmission simulation and measurement for a structure made of two coplanar waveguide feedlines, two impedance-matching tapers, and a straight $520$-$\mu \mathrm{m}$-long nanowire in between. Inset: in-band ripple.

Figure 7

Transmission and coupling $S$ parameters for symmetric and asymmetric coupler designs. The asymmetric coupler achieves a wider operation bandwidth.