Rapid Microwave-Only Characterization and Readout of Quantum Dots Using Multiplexed Gigahertz-Frequency Resonators

Superconducting resonators enable fast characterization and readout of mesoscopic quantum devices. Finding ways to perform measurements of interest on such devices using resonators only is therefore of great practical relevance. We report an experimental investigation of an $\mathrm{In}\mathrm{As}$ nanowire multiquantum dot device by probing gigahertz resonators connected to the device. First, we demonstrate accurate extraction of the dc conductance from measurements of the high-frequency admittance. Because our technique does not rely on dc calibration, it could potentially obviate the need for dc measurements in semiconductor qubit devices. Second, we demonstrate multiplexed gate sensing and the detection of charge tunneling on microsecond timescales. The gigahertz detection of dispersive resonator shifts allows rapid acquisition of charge stability diagrams, as well as resolving charge tunneling in the device with a signal-to-noise ratio of up to 15 in $1\mu \mathrm{s}$. Our measurements show that gigahertz-frequency resonators may serve as a universal tool for fast tuneup and high-fidelity readout of semiconductor qubits.

Figure 1

Experimental setup and resonator response. (a) Schematic of the device layout. (b) False-colored electron micrograph of the nanowire and the surrounding gates. (c) The rf equivalent circuit diagram of the device. The five topgates are coupled to resonators as is the source electrode of the nanowire that can be dc-biased by $V_B$ with a bias-T. The topgates are separated by six tunnel gates such that the nanowire can be pinched off at various positions and quantum dots can be defined. The charge on the quantum dots can be controlled by the sidegates. (d) Transmission through the feedline without magnetic field and at 1 T applied parallel to the plane of the resonators. The arrows $L$ (left), $R$ (right), and $B$ (bias) mark the resonators used here.

Figure 2

Pinchoff measurements. (a),(b) Response of the conductance resonator to the tunnel gate voltage T2 and linecuts at the indicated gate voltages in (b) offset for clarity. The quantity $|A_B/A_0|$ denotes the ratio of measured signal to input signal. (c) Frequency shift $\mathrm{\Delta }{\omega }_0$ and internal resonator decay ${\kappa }_d$ extracted from individual resonator line traces of (b). (d) Schematic of the nanowire for the experiment in (b) with the corresponding lumped-element model used to convert between resonator admittance and conductance $G_{\mathrm{rf}}$. (e) Conductance $G_{\mathrm{dc}}$, measured with standard voltage-biased current measurements, together with the conductance $G_{\mathrm{rf}}$ extracted from (c). The inset shows the conductance $G_{\mathrm{rf}}$ as a function of conductance $G_{\mathrm{dc}}$, for the gate response of all tunnel gate voltages T1 through T6. The dashed line indicates $G_{\mathrm{dc}}=G_{\mathrm{rf}}$. The individual traces are included in the Supplemental Material [34]. All measurements in this figure are taken at $V_B=10\mathrm{mV}$ while unused gates are held at 0 V such that only the active tunnel gate can deplete the nanowire.

Figure 3

Coulomb blockade diamonds measured in a single quantum dot. (a) Single-frequency response of the resonator. (b) $G_{\mathrm{dc}}$ measurements obtained with standard lock-in techniques. (c) Frequency shift $\mathrm{\Delta }f$ and resonator decay rate ${\kappa }_d$ extracted from frequency traces. (d) Conductance $G_{\mathrm{rf}}$ extracted from resonator data in (c).

Figure 4

Charge stability diagram measured using multiplexed gate-based readout in the double dot regime. (a),(b) Amplitude response of the resonators coupled to the two rightmost quantum dots. Readout power in the feedline is $-105$ dBm per multiplexed resonator with an integration time of $3\mu \mathrm{s}$. The dimensions of this dataset are $101\times 101$ points yielding a total integration time of 30 ms excluding overhead from gate settling time, set by low-pass filters on the gate wiring. The dashed lines are guides to the eye delineating the different charge configurations of the double dot and are identical in (a) and (b).

Figure 5

Readout SNR. (a) Amplitude response as a function of detuning $\delta$ of the resonator coupled to the right dot for the two different tunnel coupling regimes. (b) Linecuts for Coulomb blockade (square marker) and on resonance (circle marker) together with fits to the theoretical model. (c) Histograms of the resonator responses in Coulomb blockade and charge degeneracy, with pulse length of $1\mu \mathrm{s}$. Responses are acquired with a probe frequency tuned to resonance for the Coulomb blockade case, at approximately 3.826 GHz. (d) Attained SNR on the right dot’s resonator, defined as $\mathrm{\Delta }/(2\sigma )$, as a function of measurement pulse length. Optimized with excitation voltage as a free parameter (red, up triangles) and optimized at fixed excitation voltage of $5\mu \mathrm{V}$ (aqua, down triangles).