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Quantum noise limited microwave amplification using a graphene Josephson junction

Josephson junctions (JJ) and their tunable properties, including their nonlinearities, form the core of superconducting circuit quantum electrodynamics (cQED). In quantum circuits, low-noise amplification of feeble microwave signals is essential and the Josephson parametric amplifiers (JPA) are the widely used devices. The existing JPAs are based on Al-AlOx-Al tunnel junctions realized in a superconducting quantum interference device geometry, where magnetic flux is the knob for tuning the frequency. Recent experimental realizations of 2D van der Waals JJs provide an opportunity to implement various cQED devices with the added advantage of tuning the junction properties and the operating point using a gate potential. While other components of a possible 2D van der Waals cQED architecture have been demonstrated -- quantum noise limited amplifier, an essential component, has not been realized. Here we implement a quantum noise limited JPA, using a graphene JJ, that has linear resonance gate tunability of 3.5 GHz. We report 24 dB amplification with 10 MHz bandwidth and -130 dBm saturation power; performance on par with the best single-junction JPAs. Importantly, our gate tunable JPA works in the quantum-limited noise regime which makes it an attractive option for highly sensitive signal processing. Our work has implications for novel bolometers -- the low heat capacity of graphene together with JJ nonlinearity can result in an extremely sensitive microwave bolometer embedded inside a quantum noise-limited amplifier. In general, our work will open up exploration of scalable device architecture of 2D van der Waals materials by integrating a sensor with the quantum amplifier.

2D materials-based JPAs ideal for applications such as quantum amplitude sensing [21][22][23], single-shot qubit readout [24], dispersive magnetometry [8] and detection of dark matter axions [25]. Integration of 2D materials with unique electrical and thermal properties have further provided an avenue to explore novel detectors like the single-photon bolometers [26][27][28][29]. Here we report for the first time, the observation of quantum-limited microwave amplification in a graphene-based Josephson parametric amplifier (Gr-JPA), where we implement a gr-JJ to form the nonlinear LC resonator.
We fabricate the JJ on a few-layer graphene flake encapsulated between two flakes of hexagonal boron nitride (h-BN) and edge contacted with a type-II superconductor molybdenum-rhenium (MoRe) to realize the MoRe-graphene-MoRe planar SNS junction [3,30]. We start with a pre-patterned coplanar waveguide (CPW) made of MoRe, where we make two MoRe-Al 2 O 3 -Al parallel plate capacitors of identical dimensions (60 × 60 µm 2 ) in series. At the last fabrication step, a JJ is fabricated on the hBNgraphene-hBN (hBN-gr-hBN) stack. The JJ is patched parallel to the capacitors to realize a parallel LC oscillator. We use a thick graphite flake below the hBN-gr-hBN stack to set the local back gate potential of the gr-JJ (see methods for details of the fabrication). Fig. 1b shows a schematic representation of the whole device on a SiO 2 /Si substrate. Fig. 1c shows an optical micrograph of the device, the junction has length ∼ 350 nm and width ∼ 4 µm. For microwave frequencies, the effective lumped element circuit model is shown in Fig. 1d. In series with L J there is a finite stray inductance (L stray ) present in the circuit, which is a combined contribution of the geometric and kinetic inductance of the superconducting leads.
Minimizing L stray in comparison to L J is crucial for the stable operation of the JPA as we will discuss later in this manuscript.
Prior to microwave measurements, we perform a few basic DC characterizations of the junction to know the critical current (I c ) value and its tunability with electrostatic doping. Fig. 2a shows a 4-probe differential resistance (dV /dI) map of the junction as a function of dc-bias current (I dc ) and applied gate voltage (V g ), measured at 20 mK temperature in a dilution fridge using standard lock-in detection technique (see methods for details of the setup). The dark blue region in the resistance map indicates the superconducting state of the junction where it has zero resistance and the boundary of the sharp peak in the normal state resistance indicates the value of the critical current (I c ). The critical current is zero near the charge neutrality point (CNP) of graphene (V g ∼ 0 V) and increases up to 1.5 µA as the electrostatic doping is increased on the positive V g side. There is a strong asymmetry of I c on the electron and hole sides which is similar to the previously reported results on gr-JJs [3,6]. We attribute this primarily to the low transparency factor for holes arising from device fabrication-related steps. We also measure the Fraunhofer modulation of I c as a function of the applied magnetic field (see supplementary Fig. S7).
Following past work [31] on modeling the supercurrent in gr-JJs, we model the CPR using a simplified relation, based on the formation of Andreev bound states in the normal metal region. In Eq. (1), I s is the supercurrent, φ is the phase difference across the junction, ∆ 0 is the induced superconducting energy gap in graphene, R n is the normal state resistance of the junction and τ is the averaged transparency factor for N conducting channels in graphene. We use the CPR shown in Eq. (1) to numerically extract the evolution of transparency (τ) as a function of V g (see supplementary Fig. S8). Given a CPR one can estimate the junction inductance (L J ) as, (2) Combining, Eq. (1) and Eq.
(2) we numerically calculate the nonlinear junction inductance of the form L J (I) = L 0 + L 1 I + L 2 I 2 + L 3 I 3 + .. where L , i s are the nonlinear inductor coefficients and I is the drive current. We use this form of L J (I) to simulate a nonlinear phase response of the resonator which we discuss later in this manuscript. In the next section, we discuss the microwave response of our Gr-JPA.
The value of the I c based on DC measurements gives us an estimation of the junction inductance.
The device parameters were chosen to target the maximum resonant frequency around 6 GHz. We probe the resonator with low-power microwave signals and measure the reflected signals using a vector network analyzer (VNA). The measurements were done in a dilution fridge at 40 mK temperature (see supplementary section-I for details on the setup). In Fig. 2b, the reflected phase (∠S 11 ) of the resonator is plotted as a function of signal frequency ( f s ) and applied gate voltage (V g ). As the junction inductance (L J ) is a function of electrostatic doping, the linear resonance frequency ( f lin res ) gets modulated from 2 GHz up to 5.5 GHz with gating; this wide frequency tuning is a key aspect of our device. The 2π phase change at resonance is the indication of over-coupled (Q int Q ext ) state of the resonator, which means the internal loss rate of the resonator is less than the external loss rate. The resonator is over-coupled all along on the positive V g side. However, there is a frequency band of critically/under-coupled region between V g =−6 to 0 V where the resonator is lossy. This can be understood from the DC resistance map ( Fig. 2a) as the junction has very weak superconducting proximity (low I c ) in that gate voltage region.
These lumped element resonators are by construction low Q in nature, which is desired to get a large gain-bandwidth product as discussed later in this manuscript.
To explore the nonlinear aspects of the resonator, we fix the gate voltage V g at 4 V which sets the linear resonant frequency at 5.185 GHz. Fig. 3a shows the experimental nonlinear phase diagram, where we measure the reflected phase (∠S 11 ) of the resonator as a function of microwave signal frequency ( f s ) and power (P s ). At low powers, we observe the linear behavior with a smooth 2π phase shift where a phase of zero (white) marks the effective resonant frequency. As the drive power is increased, we observe the resonant frequency shift to lower values as expected for an oscillator with a CPR shown in Eq. (1).
We also observe that the phase shift through resonance becomes sharper which indicates the onset of nonlinear behavior. The inset of Fig. 3a shows a line slice of the phase response as a function of signal power at a frequency of 4.749 GHz which also shows a sharp transition. This is another characteristic feature of a driven nonlinear resonator where one can cross the effective resonant frequency by changing either drive frequency or power. At higher drive powers, the oscillator first becomes bistable and then unstable [9]. However, the region of interest for observing parametric amplification is shown in the inset ( Fig. 3a), where the response is stable and shows a phase change with drive power. Fig. 3b shows a numerically simulated phase diagram at the same bias point of V g = 4 V, which qualitatively captures the main features of a driven nonlinear resonator and agrees well with the experimental data (simulations details are provided in supplementary section-V).
We now discuss the central results of this manuscript demonstrating parametric amplification. At the same gate bias (V g = 4 V) discussed above, we add a pump tone at f p = 4.749 GHz using a separate microwave source and set the pump power to bias the resonator at the sharp transition shown in the inset of Fig. 3a. A second weaker (by ∼ 40 − 50 dB) probe tone is added using the VNA to measure the amplification. We extract the gain by comparing the output at the probe frequency with and without the pump tone turned on. We then optimize the gain by a fine adjustment of the pump frequency and pump power. At the optimal pump frequency and power, we observe parametric amplification with 24 dB gain in a ∼ 10 MHz bandwidth as shown in Fig. 4a. The noise properties of the Gr-JPA are calculated by measuring the signal-to-noise improvement in the presence of the Gr-JPA and using the system noise temperature of the rest of the amplification chain measured in a separate experiment (see supplementary section-II). The total system noise temperature (T sys,Gr−JPA n ) including the Gr-JPA is plotted in Fig. 4a (right axis) and shows quantum-limited noise performance. The evolution of the gain curve with pump power (P p ) is shown in Fig. 4b and we observe that the gain is maximized at a particular value of pump power. This point is the preferred bias point for amplifier operation as the gain remains stable against small drifts in the pump power. The maximum gain achievable at a particular pump frequency depends on the detuning between the pump and linear resonance frequency [9] with larger detunings providing larger gain as shown in Fig. 4e. Here we fix the pump tone ( f p ) and change the detuning (∆ d = f lin res − f p ) between the pump and linear resonance frequency by pushing the linear resonance up in frequency by increasing V g , shown in the schematic Fig. 4d. As ∆ d increases the gain starts to increase, reaches a maximum as it approaches the threshold for critical behavior, and then drops rapidly. The variation of the gain with pump power, pump detuning, and signal frequency described above confirms that our Gr-JPA shows the nonlinear behavior expected from its CPR.
The observed 24 dB gain along with quantum-limited noise temperature demonstrates that our device can be used as an amplifier in quantum circuits. Another important characteristic of an amplifier is the 1-dB compression point which we show in Fig. 4c. We plot the amplifier gain curve for increasing signal power (P s ) to observe the power at which the gain drops by 1-dB from its maximum value and obtain a value of −130 dBm. This is on par with the best single oscillator based JPAs using SIS junctions [2,8,10]. We also measure the amplifier performance at different V g and obtain at least 20 dB maximum gain (see supplementary Fig. S5).
In addition, for the stable operation of the JPA a careful optimization of the device parameters, stray inductance (L stray ) and capacitance (C), is crucial. The gain-bandwidth product of a conventional JPA follows the relation B √ G ∝ 1/Q. Having a low Q resonator helps in increasing the gain-bandwidth product. However, one cannot reduce Q arbitrarily in order to increase both the gain and bandwidth together. With a low Q the JPA needs to operate at high pump powers; hence with high power, higherorder nonlinear processes become important and prevent the stable operation of the JPA. An approximate condition for the stability is Qp 5 [32], where p = L J /L tot is the participation ratio of the Josephson inductance to the total inductance. There is always some stray inductance present in the circuit as discussed earlier which constraints p < 1. In our Gr-JPA, at the operating point, we estimate Q ≈ 11 and p ≈ 0.93 which makes the product Qp ≈ 10.
In summary, we report 24 dB quantum-limited amplification in a gate tunable graphene JPA. Our work is the first implementation of a 2D van der Waals-based parametric amplifier and is an essential Methods:

I Device fabrication
We make multiple Gr-JPA devices. Our fabrication process involves four major steps. First, we fabricate a CPW of MoRe on a SiO 2 /Si substrate following standard e-beam lithography. The CPW is designed to have a characteristic impedance (Z 0 ) of 50 Ω. The MoRe films are deposited using DC magnetron sputtering in a high vacuum chamber (∼ 4 × 10 −7 mbar); with sputtering pressure ∼ 2 × 10 −3 mbar.
Second, we make two MoRe-Al 2 O 3 -Al parallel plate capacitors (main Fig.1c)

II Measurement technique
We perform low-temperature DC and microwave measurements on our Gr-JPA to characterize its DC and microwave properties. The DC and microwave measurements were done in two different dilution fridges at base temperatures of 20 mK and 40 mK respectively; the measurement techniques are discussed below

II.1 DC measurements:
The DC measurements primarily involve the 4-probe differential resistance measurements of the JJs using standard lock-in detection technique with SRS 830 lock-in amplifier. We send small (∼ 50 nA I c ) low frequency (17 Hz) current through the parallel LC resonator and measure the voltage across it as a function of DC bias current (I dc ) and applied back gate voltage (V g ); due to the frequency of the AC current being very small the capacitor can effectively be thought as open. Hence the voltage across the parallel LC we assume to be the response of the JJ only. We take precautions to filter any undesired high-frequency signals which may travel from room temperature to the device. Three-stage filtering of the DC lines is done using low pass RC filters that are kept at different temperature plates of our dilution fridge (room temperature, 4 K plate, 20 mK plate) and they all have cut-off frequencies ∼ 70 kHz.
The line used for the back gate (V g ) also passes through the aforementioned three-stage RC filtering; additionally, we put a 10 Hz low pass RC filter at room temperature. In addition to the RC filters we also have copper powder filters (at 20 mK) and eccosorb filters (at 4 K) in our measurement lines to attenuate any higher frequency noise.

II.2 Microwave measurements:
The microwave measurements involve the standard reflectometry technique. Similar to DC setup in microwave measurements we take precautions for filtering any undesired noise that may reach the device.
The microwave set up along with the wiring diagram is discussed in detail in supplementary section-I.

III Noise calibration and measurements
To extract the noise temperature of the Gr-JPA, we first measure the frequency-dependent system noise temperature (T [11] Larsen, T. et al.  With an increase in drive, the oscillation amplitude increases, and the nonlinear terms start to play a role. Consequently, the resonance peak starts to shift towards the left side. The sign of the first nonlinear term in the equation of motion of a Duffing oscillator determines the direction of this frequency shift. When the drive is increased at some point the amplitude response becomes very sharp (orange curve). This state of a Duffing oscillator is called the bifurcation regime where further increase in drive makes the oscillation amplitude a multi-valued function of frequency (red curve) and the oscillator becomes unstable. The orange curve is a bias point of the oscillator where it operates as a parametric amplifier that amplifies an input signal, as shown in the figure. b, Schematic shows our experimental implementation of a Duffing oscillator. We make a coplanar waveguide (CPW) of MoRe, where the central line is terminated to the ground plane through a lumped element parallel LC resonator. We have two parallel plate MoRe-Al 2 O 3 -Al capacitors in series. We fabricate a MoRe-graphene-MoRe Josephson junction on an hBN-gr-hBN stack and patch the JJ to the capacitor leads, which results in a parallel LC resonator. We have a thick graphite flake below the hBN-gr-hBN stack which is connected to an electrode to implement a local back gating for carrier density modulation in gr-JJ. c, Shows a zoomed optical micrograph of our device where the Josephson junction is in parallel to a series combination of two parallel plate capacitors C-1 and C-2. Both the capacitors have an identical dimension of 60 × 60 µm 2 .
In the image S, D, G implies respectively the source, drain, and gate electrodes of the gr-JJ in FET terminology. Inset shows a zoomed image of the junction where the scale bar is 5 µm. The junction has length ∼ 350 nm and width ∼ 4 µm. d, Shows the equivalent lumped element circuit model of the resonator where L J is the junction inductance, L stray is the stray inductance (L stray = L geometric + L kinetic ) originating from the superconducting leads, C-1 and C-2 are the two parallel plate capacitors in series, R int represents the internal loss of the resonator in the form of a resistor, and V g is the applied gate potential to the gr-JJ. Eventually, the resonator is connected to a microwave port through a 50 Ω matched environment for reflection-based measurements. . a, Shows a 4-probe differential resistance (dV /dI) map of the gr-JJ as function of dc-bias current (I dc ) and applied gate voltage (V g ), measured at 20 mK temperature. The dark blue region in the resistance map indicates the superconducting state of the junction where it has zero resistance. We extract the critical current (I c ) value from the boundary of the sharp peak in the normal state resistance. The critical current becomes zero near the charge neutrality point (CNP) of graphene (V g ∼ 0 V) and increases up to 1.5 µA as the doping is increased on the electron side. There is a strong asymmetry of I c with respect to gating. Primarily we attribute this to the low transparency factor for holes arising from device fabrication. The inset shows a line slice of the differential resistance map at V g = 8 V. b, Shows reflected phase (∠S 11 ) of the resonator plotted as a function of signal frequency ( f s ) and applied gate voltage (V g ). Since the junction inductance (L J ) is related to its critical current (I c ) as Eq.
(2), the linear resonance frequency of the resonator ( f lin res ) gets modulated as a function of gating. V g tunes the linear resonance in a large frequency band of 2 GHz to 5.5 GHz. The 2π phase change at resonance indicates that the resonator is over coupled (Q int Q ext ) all along the electron doping side, which means the internal loss is less than external loss. There is a frequency band of critically/under coupled region between V g =−6 to 0 V where the resonator is lossy. This can be understood from the DC resistance map (Fig. 2a) as the junction has very weak superconducting proximity (low I c ) in that gate voltage region. The inset shows a line slice of the phase plot at V g = 8 V. Here we fix the V g at 4 V, which fixes the linear resonance ( f lin res ) at ∼ 5.185 GHz at low enough power (−115 dBm). However, with increasing signal power the resonator moves into the nonlinear regime and the resonance starts to shift towards the left side of linear resonance. Resonance shifting towards the left is the indication of the negative sign of the first nonlinear term in the SNS junction CPR. An increase in power pushes the resonator towards the bifurcation regime beyond which further increase in power makes the resonator unstable. We mark a frequency point at 4.749 GHz near the bifurcation regime (yellow dashed circle) where we observe parametric amplification while introducing a pump tone at that frequency; this result is discussed in detail in Fig. 4. The inset shows a line slice of the phase response as a function of signal power at 4.749 GHz. The amplification is expected to occur at the power where the phase falls sharply as indicated in the inset. b, Shows a simulated nonlinear phase diagram at the same bias point of V g = 4 V. We see resonance shifting with increasing signal power which qualitatively captures the main feature of a nonlinear resonator and agrees well with the experimental data. The simulation is performed in Microwave Office based on a nonlinear inductor model (L J (I) = L 0 + L 1 I + L 2 I 2 + L 3 I 3 + ..) that we construct using Eq. (1) and Eq. (2). Gain (dB) Fig. 4. Gr-JPA as a quantum limited amplifier. a, Shows parametric amplification (left axis) at a bias point of V g = 4 V. V g fixes the linear resonance ( f lin res ) at ∼ 5.185 GHz. Based on the nonlinear phase diagram (Fig. 3a) we add a pump tone near the bifurcation regime. Carefully adjusting the pump frequency ( f p ) and pump power (P p ) we see 24 dB amplification of signals around the pump frequency having ∼ 10 MHz amplification bandwidth.
Here we set f p at 4.749 GHz and P p at −94.08 dBm. The measured system noise temperature of the amplification chain (T sys,Gr−JPA n ) including the Gr-JPA, shown on the right axis is very close to the standard quantum limit (SQL). The frequency dependent SQL is plotted in the green dashed line (T SQL n = h f s /k B ). The T sys,Gr−JPA n being close to SQL indicates the quantum-limited noise performance. b, Shows the gain evolution as a function of probe signal frequency ( f s ) and pump power (P p ). The maximum gain region roughly follows an ellipse-like domain beyond which the gain starts to drop rapidly. The inset shows a line slice of the same gain plot as a function of signal frequency ( f s ) at a fixed pump power −94.08 dBm. c, Shows the 1-dB compression point characterization of the amplifier. Here we plot gain as a function of probe signal frequency ( f s ) and power (P s ). The maximum gain is stable and constant for low powers; with the increase in signal power, the maximum gain starts to drop. The inset shows the 1-dB compression point of ∼ −130 dBm at a signal frequency 4.748 GHz. d, e, Shows frequency detuning effect on gain. Here we fix the pump tone ( f p ) and change the detuning between the pump and linear resonance frequency (∆ d = f lin res − f p ) by pushing the linear resonance up in frequency by increasing V g . As the ∆ d increases the gain starts to increase and eventually reaches a maximum. However, further increase in detuning drives the resonator into the bistable region and the gain starts to drop.

Supplementary Information I Microwave measurement setup
Full characterization of the Gr-JPA involves measurement of its gate tunability, nonlinear response, gain, and noise performance. We measure the noise performance from the signal-to-noise ratio improvement due to the Gr-JPA. The calibration of the system noise temperature (T sys n ) and gain (G sys ) of the amplification chain was done using the noise generated by a voltage-biased shot noise tunnel junction (SNTJ) in a separate cooldown. Next, during the measurements on the Gr-JPA, the SNTJ was replaced by the Gr-JPA to keep the amplification chain the same.   Supplementary Fig. S1. Microwave setup for SNTJ calibration and Gr-JPA noise measurements. a, Shows the wiring diagram of the setup for the SNTJ calibration. We send a 10 Hz triangular voltage wave to bias the SNTJ through the DC port of a bias tee. The input signal goes through two 1.9 MHz low pass filters one at room temperature and another at 40 mK. The bias tee directs the noise power spectrum of the SNTJ to the output line. The noise spectrum is analyzed using a spectrum analyzer. The presence of the circulator is just to keep the output signal path identical for the SNTJ calibration and Gr-JPA noise measurement. b, Shows the wiring diagram for the Gr-JPA measurements. The DC line for the gate passes through a 10 Hz low pass RC filter followed by a 1.9 MHz low pass filter at room temperature and a eccosorb filter at 40 mK plate. The presence of the bias tee is just to keep the output signal path identical for the SNTJ calibration and Gr-JPA noise measurement.

II Details on noise measurements
We measure the system noise temperature (T sys n ) of our amplification chain with an SNTJ as a calibrated source of the noise. The SNTJ is a standard Al/AlO x /Al tunnel junction with ∼ 60 Ω normal state resistance (300 K value) which is embedded in a 50 Ω CPW transmission line. A strong rare earth magnet is used to suppress superconductivity in Aluminium to realize a normal tunnel junction at 40 mK with an estimated normal state resistance of ∼ 67 Ω. The noise power coupled to a matched load by a voltage biased SNTJ at temperature T and frequency f is given by [1] P where G sys is the gain of the amplification chain, k B is the Boltzmann constant, h is Planck's constant, B is the bandwidth over which the noise spectra is measured, T sys n is the system noise temperature of the amplification chain, and V is the voltage applied across the tunnel junction. According to Eq. (S1), measuring the noise power P n ( f , V ) one can determine the system temperature T , gain G sys , and noise temperature T sys n of the amplification chain. The high voltage limit of Eq. (S1) results in a linear relation between P n and V given by Using Eq. (S2) we extract G sys and T

II.1 SNTJ calibration
We measure the SNTJ noise spectrum as a function of bias voltage. To do that we voltage bias the SNTJ using a 10 Hz triangular wave; we use a spectrum analyzer in time-domain mode and capture the noise power in the half-cycle (50 millisecond) of the biasing triangular wave. Eventually, discarding the "time", we parametrically get the noise power as a function of bias voltage P n (V ). We measure the noise spectrum at different frequencies (in a band where the Gr-JPA works); hence, we get the frequencydependent noise spectra P n (V, f ) of the SNTJ. Supplementary Fig. S2a shows a noise data at f s = 4.76

II.2 Analysis of system noise temperature
We fit the measured SNTJ noise power with a straight line using Eq. (S2); shown in supplementary

III Additional amplification data at different V g values
Here we show the Gr-JPA characterization data at different points V g = 3, 5, 7, 9, 11 V. We explore the nonlinear phase response followed by the amplification of the Gr-JPA. We observe a doubly peaked gain profile at some gate biasing V g = 7, 9 V (see supplementary Fig. S6). However, at other points, V g = 3, 5, 11 V the gain profile is singly peaked and clean as the one at V g = 4 V shown in the main manuscript.

III.1 Nonlinear phase diagram at different V g points
In supplementary Fig. S4 we show the nonlinear phase diagram of the Gr-JPA at V g = 3, 5, 11 V. The reflected phase (∠S 11 ) is measured as a function of microwave signal frequency ( f s ) and power (P s ). III.2 Gr-JPA amplification at different V g points In supplementary Fig. S5 we show the Gr-JPA amplification at V g = 3, 5, 11 V respectively. At all these V g points we have obtained at least 20 dB amplification. In supplementary Fig. S6a, we show gain along with the noise temperature for the bias point V g = 3 V. The noise performance varied from the SQL. We attribute this noise performance variation to be due to the variations in the environmental impedance. This is evident in supplementary Fig. S4 where the effective resonance (white) doesn't vary as smoothly as in the simulated data in main manuscript Fig. 3b which uses a constant 50 Ω impedance. Due to the same reason, we observe a double peak gain profile at V g = 7, 9 V (supplementary Fig. S6b, c) Gain at V g = 11 V 5.10 5.12 5.14 5.16 5.18 5.20   Fig. S6. Gr-JPA gain with noise temperature at V V V g = = = 3 3 3 V and doubly peaked gain profile at V V V g = = = 7 7 7, , , 9 9 9 V. a, Shows gain along with the noise temperature (T sys,Gr−JPA n ) at V g = 3 V. The noise performance varies from the standard quantum limit (SQL) at some frequencies; primarily we attribute this due to environmental impedance mismatches. b, Shows a bias point at V g = 7 V where the gain profile is doubly peaked around pump tone (black dashed vertical line) and the maximum gain is also low (∼ 15 dB). c, Shows a bias point at V g = 9 V where the gain profile is doubly peaked around pump tone (black dashed vertical line) however, the maximum gain reaches ∼ 25 dB at some signal frequencies.

IV Additional data from another device
We measured a few more devices. In supplementary Fig. S7a, b, c we show response of another device (Gr-JPA-2). We performed DC and microwave characterizations; which showed good tuning of critical current and hence linear resonance frequency. Supplementary Fig. S7. Additional data from another Gr-JPA device. a, Shows differential resistance map (dV /dI) of Gr-JPA-2 device as a function of DC bias current (I dc ) and applied gate voltage (V g ). This device also show good tuning of critical current with gating. b, Shows Fraunhofer modulation of the of Gr-JPA-2; differential resistance map (dV /dI) is plotted as a function of DC bias current (I dc ) and applied perpendicular magnetic field. c, Shows the |S 11 | 2 (dB) of the Gr-JPA-2 as function of signal frequency ( f s ) and applied gate voltage (V g ). Note: during this Gr-JPA-2 microwave measurements the circulator in the reflection measurement setup went bad and it was unable to provide desired isolation between through and isolated signal paths, and due to signal interference the ripples in the response appeared. However, the bending feature confirms that the linear resonance of the device is tuning as a function of gating. d, Shows Fraunhofer modulation of the of Gr-JPA-1; differential resistance map (dV /dI) is plotted as a function of DC bias current (I dc ) and applied perpendicular magnetic field.

V Details on microwave office simulations
We extract the critical current dependence on gate voltage I c (V g ) from the dV /dI map of main manuscript Where I s is the supercurrent, φ is the phase difference across the junction, ∆ 0 is the induced superconducting energy gap in graphene, R n is the normal state resistance of the junction and τ is the averaged transparency factor for N conducting channels in graphene. Eq. (S4) has three parameters ∆ 0 , R n , τ.
The maximum value of I s (φ ) with respect to φ is the critical current (I c ) of the junction. We know I c , R n from experimental data. We estimate the ∆ 0 from I c R n product. We use the limiting values of ∆ 0 /e = I c R n /2.08 at CNP (V g ∼ 0 V) and ∆ 0 /e = I c R n /2.44 at far away from CNP (V g 0 V) for short ballistic junctions [31]. In between this two limits we assume a linear approximation of the prefactor (p f ) and assume the form ∆ 0 /e = I c (V g )R n (V g )/p f . We numerically calculate τ by equating the maxima of Eq. (S4) to I c for each V g values using a search and match method. As a result we extract τ(V g ) as shown in supplementary Fig. S8a. Once we know τ(V g ) the CPR at each V g is known. Next we take numerical derivative of the CPR with respect to φ to calculate the L J (φ ) as,  Fig. S8. Numerical simulations. a, Shows the JJ transparency (τ) as a function of V g that we extract from the main manuscript Fig.2a data. The τ extraction is based on the CPR (Eq. (S4)). b, Shows a numerically extracted CPR and junction inductance (L J ) as a function of phase (φ ) at V g = 4 V. c, Shows a plot of junction inductance as a function of normalized current (I s /I c ) at V g = 4 V. We extract this from the Fig.S8b; next we fit a polynomial and extract the nonlinear inductor coefficients of L J (I) = L 0 + L 1 I + L 2 I 2 + L 3 I 3 + ... d, Shows a circuit schematic in the microwave office that we use for the nonlinear phase diagram simulation. The "NLIND" element is the nonlinear inductor L J (I) = (L 0 + L stray ) + L 1 I + L 2 I 2 + L 3 I 3 + ..