Multimode RF Reflectometry for Spin Qubit Readout and Device Characterization

Abstract

We introduce a multimode superconducting inductor architecture that enables radio-frequency reflectometry at multiple discrete frequencies up to 2 GHz, addressing limitations of conventional single-mode designs. The spiral inductor's distributed inter-turn capacitance yields distinct resonant modes with varied impedance-matching conditions. By probing a quantum dot across several modes, we extract tunneling rates over a broad frequency range and identify signatures of nearby charge defects. Using one of the higher-order modes, we demonstrate single-shot spin readout via a radio-frequency single-electron transistor (RF-SET), achieving singlet-triplet readout with an integration time of 8 us and a readout fidelity of 98%. These results establish multimode inductance as a scalable and flexible component for fast spin-qubit readout and device-quality characterization.

Figures (5)

• Figure 1: (a) Discretized transmission-line model of the spiral, consisting of a ladder of series inductances $L_n = 323~\mathrm{nH}$ and shunt capacitances $C_n = 60~\mathrm{fF}$, terminated by $(C_s \parallel R_s)$ and connected in parallel with $C_L = 100~\mathrm{fF}$. (b) Simulated $|S_{11}|$ (dB) for two loads. Top: $R_s=50~\Omega$, $C_s=1~\mathrm{pF}$; bottom: $R_s=1~\mathrm{k}\Omega$, $C_s = 0.9~\mathrm{pF}$ showing multimode notches and load-dependent matching. (c) NbN spiral and cryogenic setup for 4 K characterisation measurement. (d) Measured $|S_{11}|$ (dB) at 4 K for the same loads, in qualitative agreement with (b) and highlighting mode-dependent matching.
• Figure 2: Schematic of the double quantum dot (DQD) device. Quantum dots are formed beneath gates P1 and P2, with lateral confinement provided by gate $CG$. The barrier gate P1 controls the interdot coupling, while B2 tunes the coupling between the dot and the reservoir $R$. The single-electron transistor (SET) is defined using two barrier gates ($LB$ and $RB$) and a top gate (shown in grey). Current flows through the ohmic contacts labeled Drain ($D$) and Source ($S$). For spin readout measurements of the quantum dots under P1 and P2, the RF line is connected to the drain contact $D$ of the SET (rf-SET configuration). For dispersive charge sensing experiments, the RF line is instead connected to the SET top gate.
• Figure 3: Maximum values of $\Delta C$ (blue) and $\Delta G =1/\Delta R$ (orange) from Eqs. \ref{['eq1']} and \ref{['eq2']}, evaluated at $\Delta\mu = 0$, as a function of tunnel rate $\gamma/2\pi$ for a probing frequency of 200 MHz.
• Figure 4: (a) Stability diagrams of a quantum dot located beneath the SET, measured via reflectometry using three resonant modes of the multimode inductor. The derivative of the reflected signal phase with respect to gate voltage is shown as a function of the barrier gate voltages $LB$ and $RB$ for modes 2, 3, and 4, corresponding to frequencies of 222.5 MHz, 360.2 MHz, and 523.8 MHz, respectively (see device layout in Fig. \ref{['fig:Device']}). Multiple charge transitions are visible, including one highlighted for detailed analysis. (b) Maximum signal amplitude along the selected transition in (a), plotted as a function of $V_{RB}$ for the three modes. The data points (blue circles for 222.5 MHz, green triangles for 360.2 MHz, and red diamonds for 523.8 MHz) are fitted using Eq. \ref{['eq1']}, demonstrating the dependence of reflectometry sensitivity on the tunnel rate relative to the probing frequency.
• Figure 5: Singlet-Triplet Spin Readout with Higher Inductance Mode. (a) Stability diagram of the (4,0)–(3,1) transition, captured using pulsed gate measurements in video mode, revealing a clear readout window. (b) Fast traces along the transition showcasing the distinct signals of the singlet and triplet states. (c) Histogram of the singlet-triplet readout.(d) $S_{12}$ measurement of the resonance at 245 MHz with a fit used to extract the loaded quality factor ($Q\approx870$).