Radio-frequency methods for Majorana-based quantum devices: fast charge sensing and phase diagram mapping

TL;DR

This paper demonstrates RF reflectometry techniques in Majorana nanowire devices for rapid conductance measurement and charge sensing, enabling faster characterization and topological state detection.

Contribution

It introduces two RF-based methods for fast conductance and charge measurements in Majorana devices, improving speed and sensitivity over traditional techniques.

Findings

Conductance measurements are ~40 times faster with RF reflectometry.

RF charge sensing achieves signal-to-noise >3 and 99.8% visibility in 1 μs.

Detection of charge state crossover indicating topological transition.

Abstract

Radio-frequency (RF) reflectometry is implemented in hybrid semiconductor-superconductor nanowire systems designed to probe Majorana zero modes. Two approaches are presented. In the first, hybrid nanowire-based devices are part of a resonant circuit, allowing conductance to be measured as a function of several gate voltages ~40 times faster than using conventional low-frequency lock-in methods. In the second, nanowire devices are capacitively coupled to a nearby RF single-electron transistor made from a separate nanowire, allowing RF detection of charge, including charge-only measurement of the crossover from 2e inter-island charge transitions at zero magnetic field to 1e transitions at axial magnetic fields above 0.6 T, where a topological state is expected. Single-electron sensing yields signal-to-noise exceeding 3 and visibility 99.8% for a measurement time of 1 {\mu}s.

Equations

$$A_N{e}^{-(V_{\mathrm{rf}}^{(S2)}-{\mu }_N{)}^2/2{\sigma }_N^2}+A_{N+2}{e}^{-(V_{\mathrm{rf}}^{(S2)}-{\mu }_{N+2}{)}^2/2{\sigma }_{N+2}^2},$$

Full text

I INTRODUCTION

Solid state quantum computation schemes that involve repeated measurement and feedback, including topological schemes AasenPRX16 ; Vijay ; Plugge ; Karzig with potentially long coherence times Non-Abelian ; Alicea , nonetheless require fast readout of charge or current in order to operate on reasonable time scales Reiher7555 . For topological qubits based on Majorana modes in nanowires (NWs) with proximity-induced superconductivity, quasiparticle poisoning of Majorana modes constrain readout times to microseconds or faster Lossqpp , which has already been demonstrated for superconducting SCq3 ; SCq2 ; SCq1 ; SCq4 and spin qubits Reilley1 ; spin1 ; petta2 ; Vandersypen .

Here, we report the realization of radio-frequency (RF) reflectometry in various configurations of InAs nanowires (NWs) with epitaxial Al, fabricated to form single or coupled Majorana islands, with and without proximal NW charge sensors. Device geometries were inspired by recent theoretical proposals for demonstrating elementary topological qubit operations in these systems AasenPRX16 ; Vijay ; Plugge ; Karzig . Two approaches to fast measurements were investigated in detail. In the first, a resonator made from a cryogenic inductor and capacitor was coupled directly to the leads of the device karl1 ; karl2 ; karl3 , providing conductance measurement similar to what is obtained with a low-frequency (LF) lock-in amplifier, though considerably faster. In the second, a similar resonator was capacitively coupled to a proximal NW charge sensor configured for both LF and RF charge readout. The overall charge sensitivity was investigated as a function of measurement time, and found to yield a signal-to-noise ratio (SNR) for single-charge detection exceeding 3 and visibility of 99.8% for an integration time of 1 $\mu$s, and correspondingly higher for longer integration times. Proximal NW charge sensors were found to be compatible with magnetic fields exceeding 1 T, the range needed to reach the topological regime Mourik ; MT1 ; Quantized_Majorana ; AlbretchNature . All measurements were carried out in a dilution refrigerator (Oxford Instruments Triton 400) with base temperature $\sim$ 20 mK, equipped with a 6-1-1 T vector magnet.

II Experimental setup

Reflectometry signal is optimized by matching the circuit impedance $Z$, including device resistance $R_{\rm dev}$ to the characteristic impedance of the transmission line, $Z_{0}=50\,\Omega$. Near matching, the reflection coefficient of the resonant circuit, ($Z-Z_{0}$)/($Z+Z_{0}$), is sensitive to small changes in $R_{\rm dev}$ qpc_rf ; impedance_matching . To enable multiple simultaneous measurements, four RF resonant circuits with different discrete inductances in the range $L=1.2$ - 4.7 $\mu$H, were coupled to a single directional coupler via coupling capacitor, C. One such resonant circuit is depicted in Fig. 1(a). It consists of a ceramic-core chip inductor, parasitic capacitance, $C_{P}$, from bond wires and on-chip metal electrodes, and the device, with $R_{\rm dev}$ tuned by gate voltages. Parasitic capacitance was found to be unchanged over several cool-downs.

LF lock-in measurements of differential conductance $g$ = d$I/$d$V|_{V_{\rm bias}}$ of either the device or the sensor were carried out in a two-wire voltage-bias configuration using a transimpedance (current to voltage) amplifier amp connected to the drain of the device, providing voltage input to a lock-in amplifier (Stanford Research SR830). The voltage bias consisted of a DC component, $V_{\rm bias}$, and a LF component in the range of 4 - 10 $\mu$V at frequencies below 200 Hz.

Reflectometry measurements of either the device or the sensor were performed as follows. An RF carrier at frequency $f$ with amplitude $V_{\mathrm{TX}}$ was applied to the source lead following a series of attenuators at various temperature stages [Fig. 1(a)], giving a total of 21 dB of attenuation, with an additional 15 dB of attenuation from the directional coupler, mounted below the mixing chamber plate. After reflection from the device, the signal passed back through the directional coupler into a cryogenic amplifier (Caltech CITLF3; noise temperature $T_{n}$ = 4 K from 10 MHz to 2 GHz) with +40 dB of gain. The output signal, $V_{\mathrm{RX}}$, was then detected using one of three methods: (1) using a network analyzer to measure $S_{21}\equiv 20\log(V_{\mathrm{RX}}/V_{\mathrm{TX}})$ [Fig. 1(c)]; (2) using discrete analog components to demodulate by standard homodyne detection, followed by a fast-sampling oscilloscope (see Appendix B for details); (3) using an RF lock-in amplifier (Zurich Instruments UHFLI zi ). Each method has its advantages. Method (1) is convenient for quickly determining if a change in device resistance has an effect on circuit impedance, which shows up as a change in the magnitude of $S_{21}$. Method (2) provides fast acquisition of phase maps at different gate configurations, particularly if the device is tuned into the regime of small charging energies. For these applications, Methods (2) and (3) are comparable. Method (3) has advantages in simultaneously measuring phase and magnitude of the reflected signal, and was used to quantify SNR of the proximal NW sensors and to detect charge occupancy of Majorana islands tuned to low barrier transmission.

Figures 1(b-d) show a comparison of LF lock-in measurement and reflectometry measurement, $S_{21}(f)$, of conductance $g^{(S)}$ of a charge sensor as it is pinched off using electrostatic gates. In the reflectometry measurement, $V_{RX}$ varies rapidly near the resonance frequency $f_{\rm res}\sim$ 30 MHz, yielding a dip in $S_{21}(f)$ that depends on the common gate voltage. Line cuts of $S_{21}$ at different $V_{G^{*}}$ values are shown in Fig. 1(d). The depth of the resonance changed by approximately 21 dB as the sensor conductance, $g^{(S)}$, was decreased from 0.5 $e^{2}/h$ to 0.02 $e^{2}/h$. In this case, an increasing $R_{\rm dev}$ moves the resonator impedance toward matching.

III Conductance: LF lock-in versus RF reflectometry

Figure 2(a) shows a hybrid InAs/Al island (Device A) defined by Ti/Au gates that wrap around the NW, isolated by HfO2 dielectric. Gate voltages $V_{L}$ and $V_{R}$ control coupling of the island to the leads, while three additional gates tuned the chemical potential and density on different parts of the island (see Appendix D). Only the gate marked $V_{LP}$ in Fig. 2(a) was used, with the others fixed at zero volts. DC voltage $V_{\mathrm{bias}}$ was applied to the left lead while the right lead was connected to the RF circuit ($L$ = 2.7 $\mu$H, $f_{\rm res}$ $\sim$ 52 MHz) using method (2), described above. Simultaneous LF and RF measurements of pinch-off characteristic of the right barrier with left barrier fixed at $V_{L}\sim 1$ V is shown in Fig. 2(b). At positive $V_{R}$, the right side of the island was open and showed positive demodulated voltage $V_{\rm rf}$, while at negative $V_{R}$, the right junction is closed and no current could flow through the island. Overall, $V_{\rm rf}$ was found to be proportional to $g$ measured with an LF lock-in, as shown in the inset of Fig. 2(b).

Setting both barriers into the tunneling regime using $V_{L}$ and $V_{R}$ created a Coulomb blockaded island. A two-dimensional map of Coulomb diamonds as a function of $V_{\mathrm{bias}}$ and left plunger gate, $V_{LP}$, is shown in Figs. 2(c,d). At finite bias, $V_{\mathrm{bias}}$ $\geq$ 0.2 mV, above the superconducting gap of Al, conductance oscillations with period half the zero-bias period was found, characteristic of a superconducting island. At low bias, transport is via Cooper pairs yielding 2e periodicity; at biases above the superconducting gap, 1e transport is available, halving the period.

The similarity of LF lock-in and RF reflectometry data exhibited in Figs. 2(c,d) indicates that RF reflectometry yields essentially equivalent results to LF conductance, though with a dramatic reduction of data acquisition time. For instance, a two-dimensional map of $V_{L}$ vs. $V_{LP}$ consisting of 3000 $\times$ 1500 points (Appendix A) required roughly 1 hour of acquisition time, including data processing. Acquiring comparable data using LF lock-in methods with a 30 ms integration time would require $1500\times 3000\times 30$ ms $\sim$ 38 hours to achieve comparable SNR and resolution.

IV Charge sensing

Charge sensing of a Majorana island was accomplished by placing a second NW (sensor wire), without a superconducting layer, next to the hybrid-NW Majorana device, and capacitively coupling the two NWs with a floating metallic gate charge_sensing1 . Charge sensing complements conductance and is the basis of parity readout in several theoretical proposals, for instance Ref. AasenPRX16 . The approach is similar to schemes used for spin qubit readout long_distance_charge_sensing ; charge_sensing2 . In the context of topological qubits, one can generalize the idea used in spin qubits known as “spin-to-charge conversion,” where a well-isolated quantum variable (spin) is read out projectively by mapping the relevant qubit state onto charge and then detecting charge petta2 ; Vandersypen . In a similar way, the parity of a Majorana island grounded via a trivial superconductor, a well-isolated quantum state, can be read out projectively as a charge state if the island is gated into isolation, forming a topological Coulomb island AasenPRX16 , a process we denote “parity-to-charge conversion.”

IV.1 LF charge sensing

A Majorana island formed from a gated segment of InAs/Al NW, with extended leads made from the same wire (Device B), is shown in Fig. 3(b). Regions with tunable carrier density and conductance, made by removing the Al shell, are aligned with electrostatic gates deposited in a subsequent lithography step. Local depletion of the charge carriers in these regions (tuned by $L$ and $R$ gates) creates two superconductor-insulator-superconductor tunnel junctions with a semiconductor-superconductor island in between. A T-shaped floating gate couples the superconducting island to the charge sensor NW, which was operated in the Coulomb blockade regime by depleting its barriers with gate voltage $V_{L}^{(S)}$ and $V_{R}^{(S)}$ (see Appendix D for details).

LF lock-in measurement of conductance through the InAs/Al NW island as a function of $V_{\mathrm{bias}}$ and compensated gate voltage $V_{P}^{*}$ is shown in Fig. 3(a). Compensation means that whenever the device plunger voltage $V_{P}$ is swept, the sensor plunger $V_{P}^{(S)}$ is also varied to prevent $V_{P}$ from affecting the sensor charge state via capacitive coupling, allowing the sensor to remain on a single Coulomb peak as $V_{P}^{*}$ is swept. Compensation is illustrated in Fig. 3(c), where the green dashed line shows a compensated trajectory through the space of the two plunger voltages.

Coulomb blockade diamonds are visible in Fig. 3(a). The suppression of conductance for $|V_{\mathrm{bias}}|$ $<$ 0.4 mV, independent of $V_{P}^{*}$, reflects the presence of a superconducting gap in both leads, and is consistent with the gap of Al, assuming the induced gap $\Delta_{I}$ $\sim$ 0.2 meV is roughly equal in the three NW segments. Charging energy $E_{C}$ $\sim$ 0.7 meV was extracted from Coulomb diamonds of Fig. 3(a). The large charging energy, $E_{C}/\Delta_{I}>1$ is consistent with suppressed conductance of Cooper pairs at $V_{\mathrm{bias}}$ = 0 Joyez1994 ; Matveev1993 ; Lotkhov2003 . The large $E_{C}$ results from the small capacitance between device island and the metal back-gate due to thick (500 nm) SiO2. By comparison, Device A had 200 nm of SiO2, reducing the charging energy to below the induced gap, leading to $2e$ Cooper-pair transport between Coulomb valleys.

The sensor conductance, $g^{(S)}$, at zero DC bias, $V_{\mathrm{bias}}^{(S)}$ = 0, as a function of plunger gate voltages $V_{P}^{(S)}$ and $V_{P}$, is shown in Fig. 3(c). Conductance oscillations along the $V_{P}^{(S)}$ axis indicate that the sensor island is tuned into the Coulomb blockade regime, whereas discontinuities along $V_{P}$ reflect charge transitions in the main hybrid device. We emphasize that charge transitions are not visible in zero-bias conductance of the device [Fig. 3(a)] but are visible as plateaus in sensor conductance $g^{(S)}$ as the device charge changes by two between adjacent Coulomb valleys [Figs. 3(c,d)].

IV.2 RF charge sensing

A double-Majorana-island device motived by Ref. AasenPRX16 (Device C) is shown in Fig. 4(a). Near the main device, two bare InAs NWs, capacitively coupled to each of the islands via floating gates, serve as independent charge sensors of the two islands. Each sensor is part of an independent RF circuit, with $L_{1}$ = 3.3 $\mu$H ($f_{\rm res}$ $\sim$ 60 MHz) and $L_{2}$ = 4.7 $\mu$H ($f_{\rm res}$ $\sim$ 40 MHz). Data acquisition used method (3), described above. Gates $V_{L}$, $V_{M}$, and $V_{R}$ were each set to the tunneling regime. Voltages applied to plunger gates $LP$ and $RP$ affect both the carrier density in the semiconductor and the charge offset of each island (see Appendix D). Figure 4(b) shows the charge sensing signal of a 2e-2e periodic superconducting double-island at $B$ = 0, measured using the right charge sensor (S2), with a plane subtracted to remove cross-coupling of the plungers to the three barrier gates, $V_{L}$, $V_{M}$ and $V_{R}$. Periodic 1e-1e double-island plane-fitted data, measured using the left charge sensor (S1) at finite magnetic field ($B$ = 0.8 T) parallel to NW axis, is shown in Fig. 4(c). A hexagonal pattern, characteristic of a double-island devices, is readily seen at both zero field and $B$ = 0.8 T [Figs. 4(b,c)]. Magnetic field $B$ evolution of the right 2e periodic island into the 1e periodic island regime, with the left island tuned into a Coulomb valley, is shown in Fig. 4(d). The data is differentiated along $V_{RP}$ to improve visibility of the charge transitions.

Previous works AlbretchNature ; sherman investigated nearly 1e periodic island charge occupancy, consistent with an emerging topological phase, using conductance. Using reflectometry and charge instead has the advantage of not require electron transport through the device itself. As seen from Fig. 4(d), sensing is consistent with these previous transport studies AlbretchNature . We will not focus on peak spacing and motion here, to keep the focus on measurement methods.

IV.3 Fast charge measurement and

signal-to-noise ratios in 1e regime

The signal-to-noise ratio (SNR) for detecting the transfer of a single electron between islands of the double-island device in Fig. 4(a) was investigated as a function of measurement time using the pulsed gate sequence shown in Fig. 5(a). Measurements were done in an applied axial magnetic field $B$ = 0.6 T, where the charge-stability diagram shows 1$e$-1$e$ hexagons. However, in contrast to the tuning in Fig. 4(c), $V_{L}$ and $V_{R}$ were set to isolate the double-island, with negligible coupling to the source and drain. Only inter-island transitions [white and red dashed lines in Fig. 5(a)] were measurable in this configuration.

A cyclic pulse sequence was applied to gates $LP$ and $RP$ using an arbitrary waveform generator (Tektronix 5014c), placing the system in three configurations, Initialization (I) for 150 $\mu$s, Preparation (P) for 200 $\mu$s, and Measurement (M) for a range of times from 1 $\mu$s to 50 $\mu$s [see Fig. 5(a) inset and Appendix C for details]. The preparation position and duration were chosen to yield roughly equal populations of relaxed and exited populations, which also depended sensitively on the inter-island barrier gate voltage, $V_{M}$. Results of the measurement, integrated over the measurement time, were then binned to form histograms showing the distinguishability of $N$ and $N+2$ charge-difference states ($N$ = $N_{L}-N_{R}$ is the charge difference, where $N_{L}$ and $N_{R}$ are the occupancies of the left and right islands). Note that the number of cycles used to gather histogram statistics does not affect the distinguishability of the two states. More cycles yield a convergence of the histogram to a stable, smooth bimodal distribution. On the other hand, distinguishability of the two populations is affected by the duration at the measurement (M) point. We note that only during the measurement point ($M$) readout was done by triggering the waveform digitizer card [see Appendix C for details].

The resulting histogram after $10^{8}$ cycles was fit with a sum of two gaussians,

[TABLE]

where $A$, $\mu$, $\sigma$ are the amplitudes, means, and standard deviations of the $N$ and $N+2$ charge differences. Measured distributions and best fits to Eq. (1) for measurement times $\tau=1\,\mu$s and $\tau=5\,\mu$s are shown in the Fig. 5(b). Separation of the two peaks, \Delta$$V, reflects the sensitivity of the charge sensor, while peak widths $\sigma_{N}$ and $\sigma_{N+2}$ result from measurement noise. We define {\rm SNR}=~{}\Delta$$V/$\sigma$, where $\sigma^{2}$ = $\sigma_{N}^{2}+\sigma_{N+2}^{2}$. Note that Eq. (1) does not include relaxation from $N$ to $N$+2 during the measurement. A more complicated form that includes relaxation during measurement was investigated in Ref. spin2 . In the present case, where $\tau$ is much shorter than the charge relaxation time, as set by $V_{M}$, Eq. (1) is valid. The measured SNR as a function of measurement time $\tau$ is shown in Fig. 5(c) (left axis). SNR $>$ 3 with an integration time of 1 $\mu$s was achieved.

Figure 5(c) shows that SNR increased with measurement time, $\tau$, as expected. The simplest model of this dependence, assuming uncorrelated noise spin1 , is ${\rm SNR}(\tau)$ = $[\Delta V/\sigma(1\,\mu{\rm s)}][(\tau+\tau_{0})/1\,\mu{\rm s}]^{1/2}$. By using fit parameter $\Delta V$ = $175.3$ mV, $\tau_{0}$ = $1.5$ $\mu$s and $\sigma(1\,\mu{\rm s)}$ = $74.8$ mV, the model yields the curve shown in Fig. 5(c), which compares well with the experimentally measured ${\rm SNR}(\tau)$ in the range 1 - 10 $\mu$s. Another quantity that characterizes the quality of detection is the visibility, $V$, defined as the probability of correctly identifying excited and ground states ($N$ and $N+2$) and is expressed as $V$ = $F_{N}+F_{N+2}-1$, where $F_{N}$ and $F_{N+2}$ are the fidelities calculated following spin2 (see Appendix C for details). The resulting dependence of visibility on measurement time, $V(\tau)$, is shown in Fig. 5(c), where again effect of relaxation during measurement are neglected. We find $V(1\,\mu{\rm s})=0.998$. These results are comparable or better than previously reported charge detection studies gatesens ; sens1 ; sens2 ; sens3 ; sens4 .

V Conclusions

In summary, we have investigated RF charge sensing and readout of various InAs/Al nanowire devices relevant for Majorana qubits. Two readout types were studied: First, resonant circuits were directly coupled to the device lead, yielding an improvement in measurement time by a factor of 40 compared to conventional lock-in measurements. Second, charge sensing via a second nanowire capacitively coupled via floating gate to the device allowed charge occupancy in the device to read-out non-invasively and even when visible transport is suppressed through the device. As an application, we followed the evolution of Coulomb charging from 2e periodicity to 1e periodicity as an axial magnetic field was increased from 0 to 0.6 T, complementing previous conductance measurement of Majorana signatures, without needing to run current through the device. Sensor quality as a function of measurement time was investigated using a pulse sequence that cycled the charge occupancies of the islands. Signal to noise ratio exceeding 3 can be achieved for integration times of 1 $\mu$s with visibility $V=99.8$%. Presented results show that rf resonant circuits, both directly coupled to the device, or to proximal capacitive sensors, can be used for fast and detailed characterization that conventional low-frequency techniques are not able to provide.

VI Acknowledgments

We thank Shivendra Upadhyay for help with fabrication, and Wolfgang Pfaff and David Reilly for valuable discussion. Research is supported by Microsoft, the Danish National Research Foundation and by the Australian Research Council Centre of Excellence for Engineered Quantum Systems (project ID CE170100009). PK acknowledges support from ERC starting grant no. 716655. CMM acknowledges support from the Villum Foundation.

Appendix A Lead reflectometry

Figure 6(a) shows a 2D map of $V_{\rm rf}$ as a function of $V_{L}$ and $V_{P}$ with 3000 $\times$ 1500 = 4.5 million points. The effective time constant per point $\tau$ = 200 $\mu$s. The data was acquired over $\sim 1$ h using RF lead reflectometry.. The estimated time of completion is about 40 hours using lock-in techniques with 30 ms integration time. Such gate maps at the moment are necessary for locating the topological regime in NW devices and as a result any technique that can speed up acquisition of such data sets can give a big advantage in experimental research.

Appendix B Instruments

Reflectometry measurements presented in Fig. 2 and Fig. 6 were performed with the customised demodulation circuit presented in Fig. 7. Below we list other electronic equipment used in the experiments.

Demodulation unit used for reflectometry measurements in Fig. 4 and Fig. 5: Zurich Instruments, Ultrafast Lock-in Amplifier (600 MHz) zi 2. 2.

Current-to-voltage converter: University of Basel, Electronics Lab, Low Noise/High stability I/V converter, SP 983 with IF3602 3. 3.

Voltage sources: 48-channel QDAC, custom digital-to-analog converters, QDevil ApS qdevil 4. 4.

Lock-in: Stanford Research SR830 DSP Lock-in amplifier 5. 5.

Waveform generator: Keysight 33500B 6. 6.

Arbitrary waveform generator: Tektronix 5014 C, 1.2 GS/s 7. 7.

Vector network analyser: Rohde $\&$ Schwarz - ZVB8 8. 8.

Directional coupler: Minicircuits ZEDC-15-2B (1 MHz - 1 GHz) 9. 9.

Microwave switch Minicircuits ZASWA-2-50DR+ (DC - 5 GHz) 10. 10.

Cryogenic 4 K amplifier: Caltech Weinreb CITLF3 11. 11.

Digitizer: AlazarTech ATS9360 - 12 bit, 1.8 GS/s

Appendix C Signal-to-noise ratio and visibility

The extraction of signal-to-noise ratio (SNR) and visibility was accomplished with the following pulse sequence cycle [Fig. 5(a) inset]. The pulse sequence starts with a fixed amplitude voltage pulse on gates $RP$ (positive voltage pulse) and $LP$ (negative voltage pulse) bringing the system to a point I for a duration of $\tau_{I}$ = 150 $\mu$s for initialization into a relative charge state $N$+2. Then, the gates $LP$ (positive voltage pulse) and $RP$ (negative voltage pulse) bring the system into a relative charge $N$ state (point P) for a time $\tau_{P}$ = 200 $\mu$s. Finally, gates $LP$ (negative voltage) and $RP$ (positive voltage) bring the system close to intra-island degeneracy point M (between $N$ and $N$+2 relative charge states) which we denote as measurement position. $V_{TX}$ excitation was controlled with microwave switch (ZASWA-2-50DR+), in order to avoid disturbances in the system during the manipulation phase ($I$ and $P$). The readout was performed only at the measurement point (M) by triggering the ATS9360, 12 bit waveform digitizer card for a total time duration of $\tau$ = 50 $\mu$s. To build statistics $N_{\rm cycles}$ = $10^{8}$ experimental runs of the pulse sequence were performed. From histograms of $V^{(S2)}_{rf}$ measurements (with 2 mV bin size), the probability, $P_{V^{(S2)}_{rf}}$ of singe-shot outcomes can be estimated for each value of measurement time $\tau$.

For the sake of simplicity, all denoted $V_{\rm rf}$ here will refer to demodulated voltage with the right charge sensor ($V_{\rm rf}^{(S2)}$). Visibility is defined as $V$ = $F_{N}$ + $F_{N+2}$ - 1 spin2 , where $F_{N}$ and $F_{N+2}$ are the fidelities of relative charge state $N$ and $N+2$, respectively. Fidelity of a charge state $N$ is defined by $F_{N}$ = 1 - $err_{N}$, where $err_{N}$ is an error of having a pure $N$ charge state. $N+2$ state fidelity is similarly expressed as $F_{N+2}$ = 1 - $err_{N+2}$. This error is calculated by cumulative normal distribution function, which for $N+2$ state is $\int_{-\infty}^{V_{T}}n_{N+2}dV_{rf}$, where $V_{T}$ is the threshold voltage calculated by two mean Gaussian fit peak position [($\mu_{N}+\mu_{N+2})/2$], and $n_{N+2}$ is the probability density for relative charge state $N+2$ which is expressed as $e^{{(V_{\rm rf}-\mu_{N+2})^{2}}/2\sigma_{N+2}^{2}}/\sqrt{2\pi}\sigma_{N+2}$. Error of having a pure $N$ state can be expressed as $\int_{V_{T}}^{\infty}n_{N}dV_{rf}$, with probability density $e^{{(V_{\rm rf}-\mu_{N})^{2}}/2\sigma_{N}^{2}}/\sqrt{2\pi}\sigma_{N}$. Minimizing the function of two errors ($err_{N}$ and $err_{N+2}$) and then inserting found fidelities we calculate the visibility $V=1-err_{N}+1-err_{N+2}-1$. This yields a visibility $V$ = 99.8$\%$ for an integration time of 1 $\mu$s.

Appendix D Fabrication

All devices presented have nanowire (NW) diameter $\sim$100 nm. NWs were grown using the vapor-liquid-solid technique in a molecular beam epitaxy system with the InAs [111] substrate crystal orientation Krogstrup . Following the NW growth, Al is deposited epitaxially in situ on several facets of the NW with an average thickness of 10 nm Krogstrup ; MT1 . The NW is then positioned on a chip with a homebuilt micro-manipulator tool (Zaber xyz theta stage with Eppendorf micromanipulator (model 4r) and large working distance Leica microscope) which allows micrometer precision in placement. The Al was selectively etched using wet etchant Transene D. All patterning was performed using an Elionix ELS-7000 EBL. Next we present the details specific to fabrication of all three devices:

Device A: The InAs/Al NW has Al shell on two of its facets and is fabricated on Si chip covered with 200 nm of $\text{SiO}_{\text{2}}$. The Ti-Au contacts (5 nm + 150 nm) were evaporated after performing RF milling to remove the oxide from the NW. Then, 7 nm of $\text{HfO}_{\text{2}}$ was deposited by atomic layer deposition. Finally the last set of Ti-Au gates (5 nm + 150 nm) was evaporated. 2. 2.

Device B: The InAs/Al NW has Al shell on two of its facets and is fabricated on Si chip covered with 500 nm of $\text{SiO}_{\text{2}}$. Then first set of Ti-Au contacts (5 nm + 100 nm) were evaporated after performing RF milling to remove the oxide from the NW. Finally the last set of Ti-Au gates (5 nm + 100 nm) was evaporated. 3. 3.

Device C: The InAs/Al NW has Al shell on two of its facets and is fabricated on Si chip covered with 200 nm of $\text{SiO}_{\text{2}}$. The Ti-Au contacts (5 nm + 150 nm) were evaporated after performing RF milling to remove the oxide from the NW. Then, 5 nm of $\text{HfO}_{\text{2}}$ was deposited by atomic layer deposition. Finally the last set of Ti-Au gates (5 nm + 150 nm) was evaporated.