Rapid Detection of Coherent Tunneling in an $\mathrm{In}\mathrm{As}$ Nanowire Quantum Dot through Dispersive Gate Sensing

Dispersive sensing is a powerful technique that enables scalable and high-fidelity readout of solid-state quantum bits. In particular, gate-based dispersive sensing has been proposed as the readout mechanism for future topological qubits, which can be measured by single electrons tunneling through zero-energy modes. The development of such a readout requires resolving the coherent charge tunneling amplitude from a quantum dot in a Majorana-zero-mode host system faithfully on short time scales. Here, we demonstrate rapid single-shot detection of a coherent single-electron tunneling amplitude between $\mathrm{In}\mathrm{As}$ nanowire quantum dots. We realize a sensitive dispersive detection circuit by connecting a sub-GHz, lumped-element microwave resonator to a high-lever arm gate on one of the dots. The resulting large dot-resonator coupling leads to an observed dispersive shift that is of the order of the resonator linewidth at charge degeneracy. This shift enables us to differentiate between Coulomb blockade and resonance—corresponding to the scenarios expected for qubit-state readout—with a signal-to-noise ratio exceeding 2 for an integration time of $1\mu \mathrm{s}$. Our result paves the way for single-shot measurements of fermion parity on microsecond time scales in topological qubits.

Figure 1

Dispersive sensing on an $\mathrm{In}\mathrm{As}$ nanowire double quantum dot. (a) Schematic of the experiment measurement setup. One of the quantum dots is capacitively coupled to a resonant circuit that is probed in reflectometry. Inset: false-colored electron micrograph of a nominally identical device. The sensing top gate is colored red. (b) Charge-stability diagram measured with the gate resonator. The dashed lines are guides to the eye. The triangle marker denotes charge degeneracy while the square marker denotes Coulomb blockade. The arrow denotes the detuning from charge degeneracy, $\delta$. (c) Sketch of the energy levels and resulting quantum capacitance vs detuning. Solid lines, ground state; dashed lines, first excited state; dotted line, case of no interdot tunneling.

Figure 2

Charge-resonator coupling. (a) Right panel: resonator reflection spectrum measured from the difference between injected ($P_{\mathrm{rf}}$) and reflected rf power ($P_r$) corrected for estimated attenuation and gain in the setup, as a function of detuning $\delta$. T2-gate voltage is $-0.768$ V for this data. Left panel: line cuts in blockade (orange; square) and at degeneracy (blue; triangle) together with fits (black) to Eq. (2). (b) Resonator spectroscopy at charge degeneracy for different tunneling rates together with the fit to Eq. (2). Traces are offset for clarity. Tunnel rates $t_C/h$ extracted from the fit are indicated on the right.

Figure 3

Evolution and modeling of the dispersive shift. (a) Frequency shift as a function of tunneling rate, extracted from fitting spectroscopy data to Eq. (2). We estimate an accuracy of 5% for extracting the tunnel coupling and of 40 kHz in the extraction of $\mathrm{\Delta }f$. Solid line: independent theoretical prediction from Eq. (4). Inset: charge-stability diagrams for tunneling rates corresponding to the yellow markers. (b) Resonator response as a function of frequency and power. Power is given at the sample level; this is attenuated by a total of approximately $79\mathrm{dB}$ after the generator. Top left panel: resonator spectroscopy as a function of rf power. Bottom left panel: calculated steady-state population in the excited state. Right panel: resonator shift in blockade (orange), and on degeneracy (blue). Red: prediction from the excited-state population by assuming that the net shift is given by the population-weighted average between ground- and excited-state shifts.

Figure 4

Readout SNR. (a) Histogram of resonator reflection measurements in Coulomb blockade (square marker) and charge degeneracy (triangle marker). This data is taken with a tunnel coupling of 4.3 GHz and a readout power of $P_{\mathrm{rf}}=-105\mathrm{dBm}$. Each count corresponds to an integration time of $1\mu \mathrm{s}$. The SNR is defined as $\mathrm{\Delta }/2\sigma$. The dashed line is the threshold used for state identification. (b) Attained SNR as a function of readout power (left panel) and tunnel coupling (right panel). For the tunnel coupling dependence, readout power and frequency are optimized for every data point individually.