Radio-Frequency Capacitive Gate-Based Sensing

Developing fast, accurate, and scalable techniques for quantum-state readout is an active area in semiconductor-based quantum computing. Here, we present results on dispersive sensing of silicon corner state quantum dots coupled to lumped-element electrical resonators via the gate. The gate capacitance of the quantum device is placed in parallel with a superconducting spiral inductor resulting in resonators with loaded $Q$ factors in the 400–800 range. We utilize resonators operating at 330 and 616 MHz, and achieve charge sensitivities of 7.7 and $1.3\mu e/\sqrt{\mathrm{Hz}}$, respectively. We perform a parametric study of the resonator to reveal its optimal operation points and perform a circuit analysis to determine the best resonator design. The results place gate-based sensing on a par with the best reported radio-frequency single-electron transistor sensitivities while providing a fast and compact method for quantum-state readout.

Figure 1

Device and resonator. (a) A scanning electron micrograph of a NWFET showing source (s), drain (d), and top-gate (TG) terminals. (b) Optical image of a superconducting NbN spiral inductor. Spiral track width and track spacing are both $8\mu \mathrm{m}$. (c) Circuit diagram for rf reflectometry. The NbN inductor $L$ is connected in parallel with the top gate of the NWFET. The circuit has a parasitic capacitance of $C_p$ to the ground. $V_{\mathrm{DS}}$ and $V_{\mathrm{TG}}$ are the bias voltages. $C_{\mathrm{LG}}=100\mathrm{pF}$. (d) Model for the parallel resonator coupled to external line impedance $Z_0=50\mathrm{\Omega }$ through $C_c$. The resistor $R_d$ represents the losses in the resonator. (e) Magnitude $|\mathrm{\Gamma }|$ and (f) phase $\phi =\arg (\mathrm{\Gamma })$ as a function of the carrier frequency $f_c$. Data in blue and fit in dashed red.

Figure 2

Dispersive regime. (a) Energy diagram of a fast driven TLS, $E_0$ and $E_1$, across a charge-degeneracy point. The energy is normalized to charging energy $E_c$ of the QD. (b) Data (blue dots) and Lorentz fit (red dashed curve) for the phase change $\mathrm{\Delta }\phi$ of the resonator as a function of $V_{\mathrm{TG}}$. (c) $|\mathrm{\Gamma }|$ as a function of carrier frequency $f_c$ and $V_{\mathrm{TG}}$. Experimental data for (d) $|\mathrm{\Gamma }|$ and (e) $\phi$ at two different $V_{\mathrm{TG}}$ voltages—away from and at the charge degeneracy, blue and red traces, respectively.

Figure 3

Measuring and optimizing charge sensitivity. (a) Reflected power $P_r$ spectrum showing the sidebands at $f_c\pm f_m$ due to gate-voltage modulation. (b) SNR in dB as a function of $V_{\mathrm{TG}}$ and $P_c$ ($V_{\mathrm{offset}}=472.23\mathrm{mV}$). (c) Sideband SNR as a function of $P_c$. (d) Modulation index $M$ (blue dots) and fit (red dashed curve) as a function of input rf carrier voltage $V_c$. (e) Sideband power SNR in linear scale vs rf carrier frequency $f_c$ measured at $P_c=-115\mathrm{dBm}$. (f) Measured charge sensitivity as a function of the magnetic field $B$. (g) Noise temperature $T_n$ as a function of $V_{\mathrm{TG}}$ measured at $f_0$ with $\mathrm{RBW}=300\mathrm{kHz}$.

Figure 4

Resonator optimization. (a) Equivalent circuit at resonance. (b) $|\mathrm{\Delta }\mathrm{\Gamma }|$ (black) and $|\mathrm{\Gamma }|$ (blue) as a function of $C_c$ for the experimental values $R_d=800$ $\mathrm{k}\mathrm{\Omega }$, $C_p=0.48\mathrm{pF}$, $L=405\mathrm{nH}$, and $Z_0=50$ $\mathrm{\Omega }$. We consider $\mathrm{\Delta }C=1\mathrm{fF}$. (c) Dependence on $R_d$. $|\mathrm{\Delta }\mathrm{\Gamma }|$ as a function of $C_c$ for $R_d=200$ $\mathrm{k}\mathrm{\Omega }$ (blue), $800\mathrm{k}\mathrm{\Omega }$ (red), and $2\mathrm{M}\mathrm{\Omega }$ (black). Inset: Maximum $|\mathrm{\Delta }\mathrm{\Gamma }|$ as a function of $R_d$. (d) Dependence on $C_p$. $|\mathrm{\Delta }\mathrm{\Gamma }|$ as a function of $C_c$ for $C_p=0.8\mathrm{pF}$ (blue), 0.48 pF (red), and 0.2 pF (black). Inset: Maximum $|\mathrm{\Delta }\mathrm{\Gamma }|$ as a function of $C_p$.

Figure 5

High-frequency resonator. Magnitude (blue dots) and a Lorentz fit (red dashed curve) (a) and phase (b) of the reflection coefficient $\mathrm{\Gamma }$ as a function of carrier frequency $f_c$. (c) Fast data acquisition of a dot to reservoir transition. $V_{\mathrm{DS}}$ is ramped at 4 kHz while $V_{\mathrm{TG}}$ at 7 Hz. Each trace is averaged five times. The total measurement time is 700 ms.

Figure 6

(a) Transmission electron micrograph (TEM) of silicon NWFET cross section perpendicular to the direction of transport of a narrower device similar to the ones measured in this paper. (b) Measurement PCB showing the NbN spiral inductor on sapphire and the NWFET chips.