Quantum point contact galvanically coupled to planar superconducting resonator for ultra-sensitive broad-band electrical amplification

Probing single charge dynamics in solids can give insights into various quantum transport phenomena, most of which are fragile and short-time-scaled. Detection of these events in real-time requires a mesoscopic electrical amplifier with unprecedented sensitivity and operational bandwidth. In this work, we explore a hybrid electrical amplifier consisting of a semiconducting quantum point contact galvanically coupled to a superconducting λ/2 transmission-line resonator for ultra-fast and ultra-sensitive charge amplification. The resonator, made of Aluminium with a coplanar waveguide geometry, is designed to operate at its first harmonic resonant mode ~ 3.4 GHz , where the reflected power from the resonator is amplitude-modulated by the conductance changes in the quantum point contact channel. From the sidebands of the amplitude modulated reflected signal we extract a conductance sensitivity of 2.85 × 10 −7 (𝑒 2 /ℎ)/√Hz ( 11.05 pS ∕ √Hz ). This sensitivity translates to a unit signal-to-noise measurement time ~ 1.62 ns for a variation of 0.01 ( 𝑒 2 /ℎ) in the conductance. The optimization of various operational parameters of the device reveals a bandwidth of ~ 155 MHz which corresponds to a rise-time ~ 2.2 ns . Both the sensitivity and bandwidth that we obtain are greater by an order compared to the existing reports. In addition, the device also exhibits very good sensitivities ~ 3.58 × 10 −4 (𝑒 2 /ℎ)/√Hz up to a measurable frequency of 240 MHz . The extremely high sensitivity, ultra-fast operation reaching the nanosecond timescales, and the circuit QED architecture makes this scheme an attractive choice for single charge detection and counting experiments for spin-qubit readout and quantum electrical metrology. Methods: The QPC devices are fabricated using photo- and electron-beam lithography followed by metallization by physical vapour deposition technique. The Ohmic contacts to the 2DEG are realized using Indium alloying technique. The CPW resonator is fabricated on a sapphire substrate using photolithography followed by Aluminium metallization. The RF signal from the QPC drain is grounded to the resonator ground-plane using a 50 nF SMD capacitor, while the DC and low frequency AC signal are return to the room-temperature for a direct measurement of the QPC current. All the electrical measurements are performed in a cryogen-free dilution refrigerator. Schematic diagram of the measurement setup is shown in Supplementary Information SI-01. The input RF signal to the resonator is attenuated at various temperature stages of the dilution refrigerator using coaxial inline attenuators. The reflected and the


Introduction
The efforts to push the limits of precision and sensitivity of measurements have always been driven by the quest to probe and understand new phenomena, validate theoretical predictions, and conquer new technologies; for example, the detection of feeble signals of gravitational waves using laser-interferometric techniques nearly a century after its prediction 1 . Extremely weak signal levels combined with the fragile and ultra-fast nature make the detection of single electrons in solids a challenging task. Conventional transport measurements probe timeaveraged collective response of electrons while the key to understanding various quantum phenomena generated by many-body effects, electron-electron correlations, dephasing mechanisms in qubits, etc., require spatial and temporal access to individual charge carriers and the higher moments of their distribution 2 which are not accessible during conventional transport measurements. Though, optical spectroscopy techniques can probe collective charge dynamics down to attosecond or even shorter timescales 3 , not sensitive to probe the spatial response of individual electrons. Time-resolved detection of single-electrons in solid circuits require a mesoscopic and ultra-sensitive broad-band charge/electrical amplifier with unprecedented spatial and temporal resolution.
The demand for charge sensors with better sensitivities and broader bandwidths is not only for capturing new physical phenomena but also is driven by developments in quantum enabled technologies such as solid-state quantum computing 4 and, quantum electrical sensing and metrology 5 . The recent paradigmatic change of redefining the international system of units purely based on fundamental constants, the charge of the electron , the Planks constant ℎ and, the speed of light in vacuum has a profound effect on the way the unit of current the Ampere is defined 6 . Realization of quantum current standard and the closure of the quantum electrical metrology triangle invariably involves measuring current in terms of the transfer rate of electrons in single-electron pumps which requires precise, ultra-fast, and highly sensitive charge detection and counting techniques. Ampere corresponds to an electron transfer rate of 6.24 × 10 18 (exa) Hz, while the latest gated quantum dot based single-electron pumps operate in the GHz regime 7 , both of which are far beyond the reach of the current generation of singleelectron sensors 8,9 . The solid-state qubit technology has now reached a stage where the researchers are solving problems related to the interaction of the qubits with each other and the environment such as the de-coherence effects and the error-correction 10 . A major challenge in the practical implementation of quantum error correction protocols is that the state of the qubit needs to be determined via minimally invasive non-demolition measurements. The proposed ancilla-based repetitive quantum non-demolition measurement requires probing the qubits at rates a couple of orders faster than the dephasing rates of the states 11,12 . Though the fidelity of these measurements depends on the repetition rate, it has also been reported that the visibility contrast is strongly affected by the readout process 13,14 . All these points demand measurements at timescales much faster than the decoherence times and the requirement of a highly sensitive mesoscopic charge amplifier with large bandwidth.
Time-averaged single-charge detection has been accomplished using mesoscopic charge amplifiers, the single electron transistors (SET) 15 and the quantum point contacts (QPC) 16 .
Ever since the discovery of electrons, it took more than a century to detect its motion in realtime in solid 17 where the single-electron charging-discharging events in a gated quantum dot are recorded using a radio-frequency superconducting single-electron transistor. Though sensitivities close to the theoretical limit have been reported for radio-frequency-SETs 18,19 the operational bandwidth has been limited to a few tens of MHz. The ease of integration with semiconducting quantum circuits such as gated quantum dot qubits and fast single-electron pumps 7,20,21 , and the operational simplicity makes QPCs a preferred choice over the SETs.
Moreover, unlike the SETs which work on single-electron charging-discharging events imparting higher detector noise onto the system, QPCs are less noisy and have the potential to reach the quantum limit of detection [22][23][24] . Single-shot readout of electron tunnelling events in a quantum dot using an integrated QPC was achieved by the use of a cryogenic transconductance amplifier, but the bandwidth was limited to ~100 kHz 4,21 . Detection bandwidths ~20 MHz and charge sensitivities of ~2 × 10 −4 /√Hz has been reported for radio-frequency QPCs 9,25 . Dispersive readout of charge configuration of a few-electron double quantum dot with a 10 MHz bandwidth gate-sensor was also reported recently 26 . The inherent bandwidth of QPCs extends into the THz range 27 , nevertheless, uncontrollable circuit reactance owing to the lumped element nature of the RF tank circuit used in these experiments pauses serious limitation to the performance limiting the operation to sub-GHz frequencies with bandwidths ~ a few MHz in frequencies.
In contrast, distributed element planar transmission-line resonators such as the coplanar waveguide (CPW) resonators 28,29 provide convenient alternatives while maintaining superior control over the operational parameters. Planar geometry also makes integration with other nanoscale devices such as gated quantum dots and superconducting circuits seamless. CPW resonators are widely used for circuit-QED applications in superconducting qubit circuits and lately have also been explored for coupling spin-states to the cavity modes in quantum-dot circuits [30][31][32][33] . Recently, QPCs coupled to CPW resonators has been proposed for the study of electron-photon interaction 34 and non-classical light driven photon assisted tunnelling in quantum conductors 35 .
In this work, we explore a hybrid device consisting of a semiconducting QPC galvanically coupled to a superconducting CPW resonator as an ultra-fast broadband mesoscopic electrical amplifier for various charge detection and counting applications. The resonator made of Aluminum (Al) is designed to operate ~ 3.4 GHz whose reflected power at the first-harmonic Our device also exhibits excellent sensitivity in the range of ~10 −4 ( 2 /ℎ)/√Hz up to a signal frequency of 240 MHz. We note here that the operational frequency, bandwidth, and sensitivity achieved in this work are superior by an order to the reported values for similar sensors.

Measurements and Results
All measurements discussed in this manuscript are performed in a cryogen-free dilution refrigerator with a base temperature of 10 mK. The measurements utilize a radio-frequency reflectometry set-up as shown in Supplementary Information SI-1. The device consists of an Aluminium superconducting CPW resonator galvanically coupled to a QPC defined on a GaAs/AlGaAs two-dimensional electron gas 34,36 as shown in Fig. 1 (a) (top) while the equivalent circuit of the resonator-QPC system is shown in the bottom. We operate the resonator in its first-harmonic resonant mode. The DC and low-frequency source-drain voltages to the QPC have been introduced at the low-impedance points located at distances λ/4 from both ends of the resonator using inductor-terminated CPW feedlines of length λ/2. It has been shown elsewhere 36 , that this configuration allows the introduction of DC bias onto the central conductor without altering the first harmonic resonance characteristics, our operational regime of interest. Fig. 1 (b) shows an optical image of a representative Aluminium CPW resonator fabricated on a Sapphire substrate using standard micro-fabrication techniques. The inductors on the DC feed lines, shown in the upper inset, prevents the leakage of the RF-signal into the DC lines. We estimate an inductance ~61.73 nH for the inductor from the lithographic dimensions 37 . The centre conductor of the resonator has a lithographic length ~ 3.7 cm and width ~100 μm. The separation between the centre conductor and the ground planes  Fig. 1 (c) shows the simulated two-port reflectance S11 at the input port of a resonator with identical dimensions as the fabricated one, without the DC feed lines from which we obtain the fundamental 0 and first harmonic 1 resonant frequencies at 1.56 GHz and 3. 44 GHz respectively. The inset shows the simulated electric field distribution for 1 = 3.44 GHz, our operating frequency, from which we confirm that the DC feed lines are introduced at locations of the minimum electric field. The coupling capacitor, Ck, calculated from the geometry ~3.17 fF puts the resonator-QPC system in the weak coupling regime with the external circuitry 29,38 . Fig. 1 (e) shows the measured S11 of the resonator-QPC system around 1 = 3.34 GHz at 4 K(red) and 10 mK(black  39 . In support of this, we observe a reduction in the from 116 to 49 as the resonator is taken from the superconducting state to the normal state [ Fig. 1 (d)]. We also note here that the introduction of DC bias to the centre conductor via the λ/2 feed lines does not affect the first-harmonic resonance of the cavity; refer Supplementary Information SI-2 for details.
The QPC is formed by the conventional Schottky split-gating technique 40  shutdown of the transport. The inset to Fig. 2 (a) shows a scanning electron microscope image of a QPC device with dimensions similar to the one studied in this manuscript. Fig. 2 (b) shows S11 vs. frequency for the resonator around 1 = 3.34 GHz for various QPC gate-voltages corresponding to different conductance values along the pinch-off curve. As inferred from the experiment, the reflectivity of the resonator-QPC system at resonance is a strong function of the QPC conductance. We also observe that the of the resonance at 1 undergoes a substantial change, while the QPC channel is pinched-off, as shown in Fig. 2 (c). At resonance, the variation in the QPC conductance modulates the amplitude of the reflected RF signal. This aspect of the resonator-QPC system allowing one to capture the variations in the QPC resistance at timescales prescribed by the resonance-characteristics has important consequences in (i) probing ultra-fast charge transport across the device and (ii) charge readout process; CPW resonators being routinely used in circuit-QED applications makes this technique an attractive and compatible choice for qubit readout using QPC charge sensors. The rest of the studies described in this manuscript are based on the measurement and analysis of the amplitude-modulated reflected RF signal from the resonator-QPC system at resonance, 1 , unless otherwise specified. For this an input RF signal (carrier-wave) with frequency of 1~ 3.34 GHz with a power (Prf) applied onto the resonator input port while a small AC signal, at a frequency , inducing a small variation in the QPC conductance is applied to one of the Reflectance S11 of the resonator-QPC system for various QPC conductance values, shown in red-squares in panel (a), along the pinch-off characteristics. (c) Qvalue of the resonator-QPC system as a function of QPC conductance extracted from the S11. (d) Reflected signal from the resonator-QPC system at a carrier frequency 1 ~ 3.34 GHz with a power ~ -79 dBm while an excitation signal of amplitude 1 mV RMS at 10 kHz applied onto one of the gates. The charge sensitivity is calculated from the SNR of the sidebands in the reflected power spectrum. (e) SNR vs. temperature with an excitation of 1mV rms at 10kHz applied to the QPC gate while an RF signal with a frequency 1 ~ 3.34 GHz with a power of ~ -59 dBm is applied to the resonator. Inset: a power spectrum of the ( 1m ) side-band at 4K (red) and 10mK(black) showing a change of 7 dB while the resonator was taken across the superconducting transition temperature of Aluminium ~ 1.2 K. All SNR measurement performed at 100 Hz resolution band width.
QPC gates in addition to the DC gate voltage. The power spectrum of the resulting amplitude modulated reflected signal is acquired with the help of an RF spectrum analyser as function of various device operating parameters.
First, we inspect the small-signal response of the device from which we extract the conductance sensitivity of our device. For this, we bias the QPC at a point along the steep portion of the pinch-off curve ( g = −0.735 V) with a gate excitation signal ~1 mV RMS at = 10 kHz causing an RMS conductance oscillation ∆ = 0.092 μS (~0.002 2 /ℎ) in the channel while maintaining Prf = -69 dBm. Fig. 2 (d) shows a representative power spectrum of the reflected signal. The QPC conductance oscillation resulting from the gate excitation modulates the at the frequency causing amplitude modulation of the reflected RF signal manifested as sidebands at frequencies 1 ± , as shown in Fig. 2 (d). From the SNR of these sidebands, one can extract the conductance sensitivity of the device using the formula = (1/2)∆ (BW) −1/2 10 −SNR/20 where BW is the resolution bandwidth of the external RF circuitry 25 . We extract a conductance sensitivity of 5.19 × 10 −6 ( 2 /ℎ)/√Hz [0.2 nS/√Hz] from the data. We note here that the typical conductance changes experienced by QPC sensors against single-electron tunnelling events in nearby quantum dots are ~0.01 ( 2 /ℎ)or greater 9,21,25,41 in amplitude compared to which, the conductance oscillations we induce for these measurements are lower by an order in amplitude 9,25,26,42 .
Now we explore the role of the superconducting resonator in attaining a high SNR and sensitivity. Fig. 2 (e) shows the SNR as a function of the device temperature with a 10 kHz 1 m RMS gate excitation and Prf ~− 59 dBm. As the sample temperature is increased from 10 mK through 4K, the SNR shows little variation until a temperature of ~1.1 K is reached beyond which the SNR drops sharply by ~7 dB. The inset to Fig. 2 (e) shows a power spectrum of the sidebands at 10 mK(black) and 4K(red) for reference. This temperature corresponds to the superconducting transition temperature for Aluminium 43 . The dissipation in the normal metal resonator results in a lower above the transition temperature [as shown in Fig. 1 (d)] and consequently a reduced SNR.
Sensitivity and operational bandwidth are two crucial parameters, defining the merits and limits of a sensor. In this session, we discuss the optimization of the operational parameters, the QPC conductance, amplitude and frequency of the gate excitation and, carrier-wave power and frequency to demonstrate the high sensitivity and broadband amplification characteristics of the device. The QPC conductance and carrier wave frequency are optimized by inspecting the small-signal response of the device. We apply a gate excitation, = 10 kHz, of amplitude 1 m RMS (~0.002 2 /ℎ) while maintaining Prf ~− 69 dBm. Fig. 3 (a) Fig. 2 (a). The resonator to QPC power coupling factor is given by 2 = 0 / where is the dynamic QPC resistance and 0 ≈ 50 Ω is the characteristic impedance of the resonator 34 . Fig. 3 This is beneficial especially in multiqubit charge readout schemes where gate compensations are needed to nullify the effects of parasitic coupling of various qubit gates on to the sensor to maintain a highly sensitive operating regime. Fig. 3 (c) shows the SNR vs the excitation amplitude taken with = 10 kHz and Prf ~ −69 dBm. We observe that the SNR dependence on the excitation amplitude is consistent with the amplitude modulation picture. We note here that we are able to detect excitation signals as low as 0.2 m RMS (~0.0005 2 /ℎ) on the gate, an important aspect for an electrical amplifier aspiring to reach the quantum limit. where QPC is the average power coupled to the QPC from the cavity and is the QPC resistance. Ps varies linearly with RF power as expected for amplitude-modulated signals.
We note here that we have limited the applied RF power and the resulting RF voltage across the QPC to lower values such that the heating effects and subsequent dissipation in the resonator are negligible. We also note here that the RF voltage-range used in this experiment is comparable to those used elsewhere 9,25 .
To investigate the bandwidth, first, we inspect the SNR as a function of carrier-wave frequency while maintaining a gate excitation of frequency and amplitude, = 10 kHz and 1 mVRMS respectively and, Prf ~ −69 dBm. Fig. 3 (e) shows a plot of the SNR as a function of carrier-wave frequency in the vicinity of the resonance. The SNR attains a maximum at the

Conclusions
Quantum point contacts are electrical amplifiers with inherent bandwidths extending into the THz regime 27 . The detection bandwidths and sensitivity has so far been limited to a few tens of MHz, owing to the frequency-response characteristics of the associated electrical circuits. In this work, we realize a hybrid electrical amplifier consisting of a semiconducting QPC and a superconducting CPW resonator to achieve ultra-fast and ultra-sensitive detection.
The QPC is galvanically coupled to the superconducting CPW resonator whose resonance characteristics are influenced by minute variations in the QPC conductance, enabling detection of these changes at frequencies and bandwidths dictated by the resonator.

Methods:
The QPC devices are fabricated using photo-and electron-beam lithography followed by metallization by physical vapour deposition technique. The Ohmic contacts to the 2DEG are realized using Indium alloying technique. The CPW resonator is fabricated on a sapphire substrate using photolithography followed by Aluminium metallization. The RF signal from the QPC drain is grounded to the resonator ground-plane using a 50 nF SMD capacitor, while the DC and low frequency AC signal are return to the room-temperature for a direct measurement of the QPC current. All the electrical measurements are performed in a cryogen-free dilution refrigerator. Schematic diagram of the measurement setup is shown in Supplementary Information SI-01. The input RF signal to the resonator is attenuated at various temperature stages of the dilution refrigerator using coaxial inline attenuators. The reflected and the input signals are de-coupled using a directional-coupler (ZHDC-16-63-S+, Mini-Circuits) mounted at the 600 mK stage. The reflected signal is further amplified by a 36 dB gain cryogenic HEMT amplifier (CITL F4, Cosmic Microwave Technology, Inc.) mounted at 4 K and a 9 dB gain room-temperature amplifier (ZJL-6G+, Mini-Circuits). Total attenuation of the input RF signal down to the resonator ~ 56 dB while the total gain of the output line starting from the resonator to room temperature ~ 42 dB. AC gate excitation signals in the kHz range along with the DC voltage are applied using a summing amplifier, while in the MHz range with the help of home-made Bias-Tee whose frequency response is shown in Supplementary Information SI-03.