Gate-tunable Josephson parametric amplifiers based on semiconductor nanowires
September 30, 2025
Abstract
Superconductor-semiconductor hybrid materials have been extensively used for experiments on electrically tunable quantum devices. Notably, Josephson junctions utilizing nanowire weak links have enabled a number of new gate-tunable qubits, including gatemons, Andreev level qubits and spin qubits. Conversely, superconducting parametric amplifiers based on Josephson junctions have not yet been implemented using nanowires, even though such nearly quantum limited amplifiers are key elements in experiments on quantum circuits. Here we present Josephson parametric amplifiers based on arrays of parallel InAs nanowires that feature a large critical current as required for linear amplification. The resonance frequency of the devices is gate-tunable by almost 1 GHz, with a gain exceeding 20 dB in multiple frequencies and noise approaching the quantum-limit. This new platform enables on-chip integration of gate-tunable qubits with quantum limited amplifiers using the same hybrid materials and on any substrate.
2
Superconducting Josephson parametric amplifiers [1] allow the addition of the minimum amount of noise permitted by quantum mechanics in the amplification process [2], [3]. As such, they are now essential in quantum bit readout and microwave quantum optics experiments when the signal of interest is below the noise floor of standard cryogenic amplifiers. They allow for instance fast and accurate single-shot measurement of superconducting quantum bits and nano-mechanical resonators [4], [5] and vacuum noise squeezing [6]. A typical superconducting parametric amplifier, when operated as a resonant amplifier, consists of a microwave cavity in which a tunnel Josephson junction is inserted to introduce non-linearity. As this amplification process relies on a resonant phenomenon, the bandwidth is intrinsically limited to a few MHz. New designs of flux-tunable [7], [8] and wide-band traveling-wave parametric amplifiers [9], [10] have been created to circumvent this issue. Another approach that recently sparked a significant interest, is to introduce a semiconducting weak link as the Josephson junction. The presence of the semiconductor allows for its transport properties, such as non-linearity or critical current, to be tuned using the electric field generated by a gate voltage. Recent implementations of this idea have led to the development of gate tunable Josephson parametric amplifiers (JPAs) using graphene [11], [12] and planar InAs [13], [14]. Among hybrid superconductor-semiconductor (Sc-Sm) devices, InAs nanowires with epitaxial aluminum have been widely used in gate-tunable quantum bits [15], [16] as they provide a defect-free Sc-Sm interface [17] and the nanowire geometry allows for an easy transfer onto a substrate suitable for microwave circuits with low dielectric loss. Recently, kinetic inductance parametric amplifiers [18] and Andreev qubits have also been demonstrated with the same materials [19]–[21]. Additionally hybrid nanowires have been of great interest in search for new bound states and topological features in quantum devices [22]. In a Josephson junction, the amount of non-linearity usually scales inversely with the critical current of the junction. While one-dimensional semiconductor nanowire weak links can provide a large non-linearity (small critical current), suitable for realizing quantum bits, this large non-linearity is a challenge in building a parametric amplifier. This explains why they have not been used in Josephson parametric amplifiers yet.
We demonstrate here JPAs based on Josephson junctions constituted of parallel InAs semiconducting nanowires that feature a large critical current and a reduced non-linearity, not hitherto realized with single nanowires. By integrating the Josephson junctions in a microwave resonator, we demonstrate gate-tunability of the resonance frequency by almost 1 GHz and, in optimal conditions, a parametric gain exceeding 20 dB. The amplifier has a limited compression point, typical in such single junction resonant amplifier. We show that using this amplifier improves the signal-to-noise ratio of the measurement chain compared to a standard cryogenic amplifier, highlighting its potentially quantum-limited behavior.
Arrays of nanowires are grown parallel to each other in order to increase the number of conducting channels (and thus the critical current) while keeping gate tunability (see Supporting Information (SI) section I for pictures of as-grown arrays). To achieve a pristine interface, an aluminum half-shell is subsequently deposited on the nanowires in situ and at low temperature. Part of the aluminum shell is etched away chemically, resulting in a Josephson junction (JJ) with a weak link composed of parallel semiconducting nanowires shown in Fig. 1b. Two different devices were fabricated from the same batch of nanowires using the same nano-fabrication process (SI section I for pictures and fabrication process). The first device (JPA04) is composed of an array of 4 parallel nanowires and the second device (JPA09) features three arrays for a total of 12 nanowires. The parallel nanowire Josephson junction is embedded at the center of a \(\lambda/2\) microstrip resonator (Fig. 1c) with a bare resonance frequency \(\omega_0/2\pi\) = 6.5 GHz (see Methods (and SI) for details on resonator design). The resonator is capacitively coupled to a 50 \(\Omega\) transmission line which is used to probe the reflection scattering parameter \(S_{11}\) of the device. Low frequency lines are galvanically connected to the Josephson junction, near the center of the resonator, in order to perform DC transport measurements. A top-gate is used to tune the carrier density in the semiconducting nanowires. The schematic in Fig. 1a illustrates how the design of the resonator allows to implement four-wave mixing (4WM) parametric amplification with the relation \(2\omega_{\text{pump}} = \omega_{\text{signal}} + \omega_{\text{idler}}\). This corresponds to the annihilation of two pump photons at \(\omega_{\text{pump}}\) to create a photon at \(\omega_{\text{signal}}\) and at \(\omega_{\text{idler}}\).
The dc properties of the Josephson junctions are presented in Fig. 2a. The critical current \(I_c\) is extracted from a differential resistance measurement as a function of the dc bias current and the gate voltage. The top-gated nanowire junction acts as a Josephson field-effect transistor. \(I_c\) saturates towards positive \(V_g\) values while a negative voltage results in a dramatic reduction of \(I_c\) towards pinch-off. The critical current in the parallel nanowire junctions is higher than for a single InAs nanowire/aluminum device (typically \(<\) 50 nA) [23]–[25] showcasing the usefulness of the parallel arrays in achieving the necessary high critical current. Furthermore, by placing several arrays in close proximity and contacting them in parallel, the critical current can be further increased to more than 1200 nA in device JPA09.
We now turn to the high frequency regime. Sending a microwave probe tone allows to extract the resonance frequency \(\omega_r\) of the device (see Methods for details on the measurement). The resonance frequency can be modulated by applying a gate voltage on the nanowires (Fig. 2b) and devices JPA04 and JPA09 are tunable by more than 800 MHz and 150 MHz, respectively. In device JPA09, when the gate voltage is set to obtain a large critical current, the resonance frequency of the device is close to the bare resonance frequency of the resonator \(\omega_0\). However, increasing \(V_g\) until saturation of device JPA04 results in a resonance frequency 800 MHz below \(\omega_0\), a direct consequence of the smaller critical current in this device. In a simple picture the Josephson junction acts as a gate tunable inductor (inset in Fig. 1c) and thus affects the resonance frequency of the resonator \(\omega_r\). We consider the Josephson inductance \(L_J\), \[\label{eq:Lj} L_J = \dfrac{\Phi_0}{2\pi}\left(\dfrac{\partial I}{\partial \phi}\right)^{-1} \approx \dfrac{\Phi_0}{2\pi}\left(\dfrac{1}{I_c}\right)\tag{1}\] with \(\Phi_0 = \dfrac{h}{2e}\), the superconducting flux quantum, and \(\phi\) is the superconducting phase difference. The second equality is only correct in the limit of a sinusoidal current phase relation (CPR), \(I=I_c\sin(\phi)\), near \(\phi=0\). For a small microwave probe power, it is reasonable to consider \(\phi \approx 0\).
In Josephson junctions utilizing a semiconducting weak link with a large transparency, this CPR has been shown to not be accurate [26]. Nevertheless, it allows to grasp the behavior of the resonator frequency when tuning the critical current via the gate voltage. When the critical current \(I_c\) is large, \(L_J\) is small (Eq. 1 ). The resonance frequency \(\omega_r\) of the resonator will therefore be dominated by the geometric inductance \(L_0\) of the microstrip resonator and the effective capacitance \(C\) between the resonator and the ground plane: \[\label{eq:tunability} \omega_r(V_g) = \left[ \sqrt{(L_0 + L_J(V_g))C} \right]^{-1}\tag{2}\]
When lowering \(I_c(V_g)\), \(L_J\) will increase, decreasing \(\omega_r\). This explains the difference in resonance frequency of both devices, even at the same \(V_g\). The effective transparency \(\tau\) [27] of the Josephson junction can be extracted by an indirect measure of the CPR. Biasing the junction with a dc current, the resonance frequency of the device changes with the evolution of the phase across the junction. In other terms, one can access the Josephson inductance, i.e. the local derivative of the CPR and thus reconstruct the CPR of the junction (SI section IV). The evolution of \(\tau\) with gate voltage is presented in Fig. 2b. It is close to unity, which suggests low disorder in the nanowire and at the interface between the InAs nanowires and the aluminum. Such high transparency was already reported in similar systems [23], [28]. This will result in a skewed current-phase relation with a reduced non linearity compared to tunnel junctions made of Al/AlOx [29]. The consequences of a non-sinusoidal CPR for the performance of the amplifier are discussed below.
By fitting the microwave reflection following Ref. [30], we extract the internal loss rate \(\kappa_i\) (Fig. 2c) and the coupling rate \(\kappa_c\) (Fig. 2d) as a function of \(V_g\). Extracting these quantities is critical to understand the behavior of the device and to ensure that the noise performance is not limited by internal losses, i.e.that we are in a regime \(\kappa_c > \kappa_i\). A sharp increase in internal losses is observed at negative gate voltages while the coupling rate decreases around the same gate voltage values, for both devices. While the internal losses can be mainly attributed to dissipation in the junction [14], the coupling rate is dominated by the geometrical coupling capacitance to the transmission line. In Fig. 2c, internal losses are large even at positive \(V_g\) for device JPA04 and decrease towards zero gate voltage. This could indicate that, in this particular device, dissipation is induced by the gate itself. We do not currently understand the microscopic mechanism for the dissipation in the devices. On-chip filtering on the gate or a thicker dielectric could reduce this parasitic phenomenon. Also, the precise gate dependence of the external coupling is not the one expected from simple consideration on the evolution of the resonance frequency and might be due to the existence of spurious microwave modes. The net effect of internal dissipation is to restrain the useful amplification range to a smaller frequency band where losses are not predominant.
The presence of the Josephson junction adds non-linearity to the microwave resonator which can be studied by varying the input probe power (SI section II). In Fig. 3a, the evolution of the resonance is apparent and shifts
towards lower frequency when increasing power. This is an evidence of a Kerr-like non-linearity, a critical ingredient to perform parametric amplification [31]. The behavior of the device can be modeled by expanding the simplified non-linear Josephson CPR \(\sin(\phi) \approx \phi -
\phi^3/6\), leading to a non-linear Kerr Hamiltonian [32]
\[\label{eq:hamiltonian} H/\hbar = \left(\omega_r - \dfrac{K}{2}\hat{A}^{\dagger}\hat{A} \right) \hat{A}^{\dagger}\hat{A}\tag{3}\] with \(A^{\dagger}\)(\(A\)), the creation (annihilation) operator for the photons inside the resonator and \(K\), the Kerr coefficient, quantifying the amount of
non-linearity.
The (negative) Kerr term is a direct consequence of the non-linear Josephson potential and will cause a monotonic decrease of the resonance frequency, dependent on the number of photons in the resonator. Above a critical input power \(P_c\) (Fig. 3b, \(P_c\) = -87 dBm), a bifurcation occurs and the device enters a bistable regime where linear amplification is no longer possible [33]. The largest gains are expected by setting the pump just below the bifurcation threshold. Taking line-cuts from
Fig. 3a at various probe powers shows the increase of the resonance dip, while it shifts to lower frequencies due to the Kerr non-linearity (Fig. 3b). This is due to the presence of non-linear losses
and has also been observed in graphene Josephson junctions [11], [34] but is usually not observed in tunnel junctions [35]. The losses are attributed to the dense spectrum of Andreev bound states within the induced superconducting gap of the junction, leading to dissipation in the
junction [36], [37].
We fit the reflection coefficient (dashed lines in Fig. 3b) using the model developed by Yurke and Buks [38] as it takes into account non-linear losses. It allows to extract the Kerr coefficient (\(K\)), whose evolution as function of gate voltage is presented in Fig. 3c for device JPA09 along with the non-linear losses \(\kappa_{nl}\). A weakly non-linear regime is required for parametric amplification such that \(K < \kappa_c\) [39] and typically \(|K|/\omega_0\) \(<\) \(10^{-2}\) to \(10^{-6}\). Kerr non-linearity is extracted by performing the fit at the critical power \(P_c\) (SI section III for details) and decreases with increasing \(V_g\) (Fig. 3c) which is typical of JPAs based on semiconducting weak-links [11], [14] as \(K\) scales with \(1/I_c\). The value of \(K/\omega_0 \sim 10^{-5}\) is significantly smaller than JPAs based on single tunnel JJs and similar to JPAs implemented with a semiconducting weak-link [13]. It has been shown that the Kerr non-linearity derived from the Josephson potential is reduced by the large transparency of Sc-Sm-Sc Josephson junctions [40]. A smaller non-linearity could be desirable to retain a satisfactory dynamic range [41]. The large transparency in our devices allows for a more linear CPR near zero-phase (more details in SI) and could lead to an increased compression point as shown by Hao et al. for InAs 2DEG JPAs [13].
We now present parametric amplification capabilities of the JPA. A strong microwave pump tone with a frequency \(\omega_p\) is sent through the same port as the weaker probe tone of frequency \(\omega_s\) while measuring the microwave reflection \(S_{11}\). The microwave pump tone is periodically modulating the non-linear Josephson inductance leading to parametric gain. Fig. 4a shows the effect of the pump on the device. When the pump is off, a resonance dip in magnitude is present. When the pump is turned on and correctly tuned in frequency and power, the reflected signal amplitude increases,
evidence of parametric amplification. A Kerr-induced frequency shift of the resonance is observed due to the high power of the pump tone.
A careful optimization of the pump frequency (Fig. 4b) and power (Fig. 4c) is necessary and the maximal gain is obtained in a narrow parameters range. In our devices, the
instantaneous-bandwidth is in the range of a few MHz and set by the external quality factor of the resonator. We are able to tune the resonance frequency of the device using the gate voltage, and by modifying accordingly the pump power and frequency, we
are able to perform parametric amplification in a range of more than 100 MHz as shown by the gain profiles in Fig. 4d.
A key metric of an amplifier is its dynamic range. Quantum-limited amplifiers with high dynamic range are of interest as they allow to increase the number of simultaneous qubit readouts, which is essential for scalability [42], [43]. The 1 dB compression point \(P_{\text{1dB}}\) is defined as the input probe power at which the gain is 1 dB lower than the low power gain (Fig. 5a). The device is operated such as to produce 20 dB of gain at the lowest input power. The gate voltage is set at \(V_g\) = -0.5 V and we apply a signal detuned from the pump \(\delta \omega/2\pi\) = 200 kHz where \(\delta \omega = \omega_{\text{signal}} - \omega_{\text{pump}}\). Increasing the signal input power results in a reduction of the gain, giving \(P_{\text{1dB}}\) = -137 dBm. This is a relatively low value, which is expected in such single junction 4WM JPAs [44]. There are several ways to increase the compression point. Implementing an array of junctions in series with larger critical currents has been shown to increase the compression point by diluting the nonlinearity [45]. Another strategy would be to operate the device as a three-wave mixing amplifier, reducing the Kerr effect that can limit the compression point [46].
Josephson parametric amplifiers reaching near quantum-limited noise need to feature a sufficiently large gain (typically \(\geq\) 20 dB) in order to overcome the noise of the following amplifier in the measurement chain. To demonstrate this, we assess the amplifier noise using a spectrum analyzer. The following measurements are done with device JPA09 that can feature a gain exceeding 20 dB (Fig. 4). We first compare, in Fig. 5b, the effect of the pump on the measured power spectral density (PSD). The pump power and frequency (\(P_{\text{pump}}\) = -95.5 dBm and \(\omega_{\text{pump}}/2\pi\) = 6.26135 GHz) are tuned to obtain a gain of 22 dB. In the absence of the pump, we see the signal tone 18 dB above the noise floor. When the pump is turned on, we additionally observe the pump and idler tones. We also notice multiples smaller peaks whose origin is unclear at this stage. They may come from higher order non-linearities, neglected in Eq. 3 , that arise from the different harmonics present in the CPR for JJs with a semiconducting weak link [29], or more trivially from noise sources that appear when operating the resonator in the parametric amplifier regime, i.e. close to bifurcation.
The noise floor is higher in the presence of the pump, due to the amplification of the noise by the JPA, but most importantly the signal to noise ratio, i.e.the difference between the signal and the noise floor has improved by 12 dB. This is the key feature of such amplifier, which proves that the noise temperature of the measurement chain has decreased thanks to the JPA operation.
To be more quantitative, we translate in Fig. 5c the measured PSD into an effective added noise temperature \(T_N\) (see SI section V for details). When the pump is off, i.e. when the device shows no gain and the first amplifier of the chain is the 4 K HEMT, we extract an added noise in the range 3 to 8 K, i.e. what is expected for such an amplifier. When the pump is turned on, the added noise temperature is significantly reduced and reaches about three times the quantum-limit [2] (black dashed line).
To conclude this part, we emphasize that this analysis is sensitive to the exact calibration of the chain and that a precise quantitative measurement of the noise performance would require a calibrated noise source at the input of the amplifier.
In conclusion, we have demonstrated a gate-tunable Josephson parametric amplifier based on InAs/Al nanowires. To overcome the typically low critical current of such nanowires, not suitable for realizing a parametric amplifier, we fabricated a device with several nanowires in parallel. The amplification band can be tuned using a gate voltage by more than 100 MHz and a gain over 20 dB can be achieved. We have shown that the Kerr nonlinearity of such nanowire junctions is reduced compared to tunnel junctions due to their large transparency. As this is a key parameter to optimize the performance of an amplifier, this could be leveraged in future designs. We envision that using a three wave mixing scheme, instead of four wave mixing, could allow a larger compression point. In addition, this would spectrally separate the signal and pump tones. Three wave mixing can readily be implemented in such gate tunable junction by pumping through the gate.
The use of semiconducting nanowires allows the integration of larger gap superconductors such as Sn [47] or Pb [48], which would increase the critical current and allow operation of the device under magnetic field.
With the recent developments in nanowire based superconducting qubits, it will be possible in the future to co-integrate, on the same chip, qubits and Josephson amplifiers made using the very same semiconductor-superconductor materials.
Methods↩︎
Device design↩︎
The Josephson junction constituted by parallel InAs nanowires is contacted by two superconducting aluminum electrodes, embedded into a half-wavelength microwave resonator at the voltage node (Fig. 1c). Electromagnetic simulations (Sonnet Software, Inc.) yielded an estimate of the bare resonance frequency \(\omega_0/2\pi\) = 6.5 GHz without the junction. The microwave resonator is capacitively coupled to a 50 \(\Omega\) transmission line. The geometry of the capacitor allows to tune the coupling quality factor \(Q_c = \omega_r/\kappa_c\) and we targeted a value of 100. The internal quality factor \(Q_i\) being typically around few thousands, ensures that the resonator is in the required over-coupled regime. Two bias lines going directly to the Josephson junction (Fig. 1c, “DC lines”) enable dc transport measurements and a gate-line is added to tune the semiconductor (“Top-gate”). Large inductances were added to these lines to filter any unwanted microwave signal.
Device fabrication↩︎
We grow InAs nanowires via the Vapor-Liquid-Solid (VLS) and Vapor-Solid (VS) mechanisms using Molecular Beam Epitaxy (MBE) following Ref. [49]. We use lithographically-defined gold particles as catalysts. Five or ten catalyst particles are arranged in a linear array allowing for the growth of parallel nanowires (Fig. 1b). After loading the pre-patterned substrate inside the MBE chamber, it is annealed at 500°C for 5 min under As\(_4\) overpressure, followed by an axial (VLS) growth at 450°C at In and As beam equivalent pressure (BEP) of 1.2E-7 Torr and 7.0E-6 Torr, respectively, for 80 min. An InAs shell is then grown at 350°C for 60 min around the nanowires (VS) in order to increase their diameter. This two step process has the advantage of growing long but sufficiently thick nanowires that form a dense array. After cooling down the substrate to -50°C, the growth is followed by the deposition of a 35 nm Al half-shell without breaking the vacuum.
Parallel InAs nanowires are transferred onto an intrinsic silicon substrate and processed using e-beam lithography and positive resist. We deposit Ti/Al (5 nm/450 nm) using an e-beam evaporator to electrically contact the nanowires and form the microwave circuit (resonator, transmission line and dc lines). Ar\(^+\) milling is used beforehand to remove the native aluminum oxide. The Josephson junction is formed by selectively etching part of the in situ grown aluminum using Transene aluminum etchant type D for 12 seconds at 50°C. We use atomic layer deposition to deposit around 10 nm of Hf\(_2\)O\(_3\) as a top-gate dielectric. A final step consists of depositing the top-gate electrode using Ti/Al (5 nm/450 nm). A detailed description of the fabrication steps is available in SI section I.
DC and microwave measurements↩︎
We perform measurements in a wet dilution fridge at a base temperature of 25 mK. The dc transport measurements are done using standard lock-in techniques at a frequency of 17.17 Hz. The device JPA04 is measured in a quasi four-probe configuration while device JPA09 is measured in two-probes, so a constant line resistance of 1180 \(\Omega\) is subtracted. The microwave resonator is probed in reflection (\(S_{11}\)) using a Vector Network Analyzer (VNA). Power-dependent measurements are performed by setting a constant output power at the VNA and changing the attenuation using a variable attenuator. Another microwave source is used to send the pump tone through the same microwave port as the VNA using a beam splitter. The JPA output is amplified by a HEMT at 4 K.
Acknowledgments↩︎
We thank Frederic Gay, Thomas Kanne and Laurent Cagnon for experimental assistance. The samples were prepared at the Nanofab clean room facility of the Néel Institute. We are grateful for the Chair of Excellence established by the Labex LANEF (CEA, CNRS and University of Grenoble). The work was also supported by the Carlsberg Foundation, and the Novo Nordisk Foundation SolidQ project, the French National Research Agency (ANR) in the framework of the Graphmon project (ANR-19-CE47-0007) and a French government grant managed by the ANR agency under the ‘France 2030 plan’ (ANR-22-PETQ-0003).
Supporting information↩︎
1 Detailed fabrication of the device↩︎
Nanowire arrays were grown according to the procedure described in the Methods section of the paper. We present SEM images of as-grown arrays of InAs nanowires with in situ aluminum in Fig. 6. The yield of successful growth is large but some arrays present various differences, in the length or shape notably, necessitating to deposit several arrays per chip which allows to select the best one to contact.
The backside of a highly resistive (\(>\) 20 k\(\Omega\).cm) intrinsic silicon substrate (Si-Mat) is coated with Ti/Au (5 nm/150 nm) using e-beam evaporation to define a ground plane. Parallel InAs nanowires with an in situ aluminum half-shell are transferred onto the silicon substrate using a tungsten probe (American Probe & Technologies, 72X-G3/035) and a micro-manipulator under optical microscopy. We use e-beam lithography (80 kV Nanobeam Ltd. NB5) and positive resist (poly(methyl methacrylate) AR-P 679) to process the nanowires in 5 steps. Typically, N\(_2\) gas is blown on the substrate after deposition of the nanowires to increase their adhesion to the substrate and prevent them from moving when dipped in solvent. We have found that the arrays of nanowires may still move even after this step. Before etching the Josephson junction, we add a first exposure to define a strip at each end of the nanowires followed by the deposition of Ti/Al (5 nm/400 nm) using an e-beam evaporator (Fig. 7a). This prevents any movement during subsequent lithography steps. A second lithography step defines a small opening at the center of arrays along the width of the nanowire array to selectively etch part of the in situ aluminum to form a Josephson junction. We use Transene aluminum etchant type D for 12 seconds at 50°C to etch the aluminum. We target a dimension along the nanowires length of 30 nm by design (electron-beam dose 20 \(C/m^2\)) but in practice the junction is always over-etched to a final width of around 100 nm. The next step allows to expose the transmission line, microwave resonator and dc lines (Fig. 7b). The sample is then subjected to Ar milling to remove the native Al\(_2\)O\(_3\) prior to the deposition of Ti/Al (5 nm/450 nm). A window is opened around the Josephson junction by e-beam lithography and Hf\(_2\)O\(_3\) is deposited using atomic layer deposition (Cambridge NanoTech Savannah S100) for 100 cycles resulting in a roughly 10 nm thick top-gate oxide. A final lithography step consists of exposing the top-gate electrode and depositing Ti/Al (5 nm/450 nm) as shown in Fig. 8. The finished device is presented in Fig. 7c. The resonator is capacitively coupled to a 50 \(\Omega\) transmission line via a capacitor while the dc lines and a gate electrode are highlighted in pink and blue respectively.
2 Experimental setup↩︎
A schematic of the experimental setup is shown in Fig. 9. The devices are cooled down in a dilution fridge at a base temperature of 25 mK. The samples were measured using a VNA (Anritsu MS46522B) and a spectrum analyzer (SignalHound SA124B) with an additional microwave source (Rohde & Schwarz SMB 20 GHz) used as a pump. The JPA is followed by a circulator, a HEMT (LNF-LNC1 12A) and room temperature amplifiers are used to amplify the outgoing microwave signal.
The dc setup uses standard lockin techniques (SRS 830) and low noise amplification to amplify the voltage drop measured across the device. The lockin amplifier sine wave and output of the dc source \(V_{DC}\) are combined and fed through a large resistor to current bias the Josephson junction. A large resistor is also used in series with the topgate voltage to prevent any unwanted current leakage.
3 Microwave analysis of Kerr non-linearity↩︎
3.1 Normalization of the microwave background↩︎
As the elements constituting the wiring to the device (wire-bonds etc.) are not exactly matched to 50 \(\Omega\), standing waves appear in the microwave spectrum, resulting in a non-flat microwave background. This makes it more difficult to have a reliable fit, as such it needs to be subtracted. We set \(V_g\) = -3 V and record the microwave reflection of the device for a range of frequencies. At this gate voltage, the resonance is shifted away from the frequency window of interest and we only record the microwave background. This recording serves as a normalization trace for all subsequent measurements that require fitting.
3.2 Fitting procedure↩︎
The fit of \(S_{11}\) is performed based on the model developed in Ref [38] and allows to extract \(K\), \(\kappa_{i}\), \(\kappa_{c}\) and \(\kappa_{nl}\). The
schematic in Fig. 10 presents the system constituted of the JPA coupled to three ports: one linear loss, one nonlinear loss (two-photons loss) and the probe port. This port is used to send the signal and pump tones and measure
the microwave reflection. The Hamiltonian \(H_0\) of the JPA is given by Eq. 3 where \(\hat{A}\) (\(\hat{A}^{\dagger}\)) are the operators for the
annihilation (creation) of photons inside the cavity and \(\omega_r\) the resonance frequency. Using input-output formalism [50], the model defines the coupling to the transmission line as \[a_c^{\text{out}} - a_c^{\text{in}} = -i\sqrt{2\kappa_{c}}A\] with \(a_c^{\text{in}}\) (\(a_c^{\text{out}}\)) the incoming (outgoing) bath mode and the cavity mode \(A\).
The linear loss rate is defined with a port \(a_i\) as \[a_i^{\text{out}} - a_i^{\text{in}} = -i\sqrt{2\kappa_{i}}A\]
A non-linear loss rate, depending on \(A^2\) is defined as \[a_{nl}^{\text{out}} - a_{nl}^{\text{in}} = -i\sqrt{2\kappa_{nl}}A^2\]
By defining an incoming classical field \(a_c^{\text{in}} = b^{\text{in}}e^{-i(\omega t + \Psi)}\) with a frequency \(\omega\) and phase \(\Psi\), the microwave reflection reads \[\label{eq:s1195kerr} \dfrac{b^{\text{in}}}{b^{\text{out}}} = S_{11} = 1 - \frac{2\kappa_c A}{[i(\omega_r - \omega) + \kappa]A + (iK + \kappa_{nl})A^3}\tag{4}\] where \(\kappa =\kappa_c + \kappa_i\).
An additional cubic equation is obtained in order to calculate the cavity mode \(A\) \[A^{6} + \frac{2[(\omega_{r} - \omega)K + \kappa \kappa_{nl}]}{K^{2} + \kappa_{nl}^{2}}A^{4} + \frac{(\omega_{r} - \omega)^{2} + \kappa^{2}}{K^{2} + \kappa_{nl}^{2}}A^{2} - \frac{2\kappa_{c}}{K^{2} + \kappa_{nl}^{2}}(b^{\text{in}})^{2} = 0\]
Above a critical power \((b_c^{in})^2\), the system will bifurcate from a single real physical solution to two bi-stable solutions. The critical power depends on \(K\) and \(\kappa_{nl}\) as well
\[\label{eq:critical95power} \left(b_{c}^{in}\right)^{2} = \dfrac{4}{3\sqrt{3}} \dfrac{\kappa^{3} (K^{2} + \kappa_{nl}^{2})}{\kappa_{c} \left(\left|K\right| - \sqrt{3}\kappa_{nl}\right)^{3}}\tag{5}\]
As Eq. 4 depends on a large number of free parameters, it is not fitted directly. After performing the normalization procedure, we perform a circle fit [30] at the lowest input power to extract \(\kappa_i\), \(\kappa_c\) and \(\omega_r\). The critical power \((b_c^{in})^2 = P_c/\hbar\omega_r\) is attained when the slope of \(|S_{11}|\)
becomes infinite. We thus locate \(P_c\) by manually choosing the power at which \(d|S_{11}|/d\omega\) is the largest.
The resonance shift between \(\omega_r\) and \(\omega_c\), the resonance at the critical power is given by \[\omega_r - \omega_c = -\kappa \frac{K}{|K|} \left[
\frac{4\kappa_{nl} |K| + \sqrt{3}(K^2 + \kappa_{nl}^2)}{K^2 - 3\kappa_{nl}^2} \right]\]
in the limit \(K \gg \kappa_{nl}\), it reduces to
\[\label{eq:approx95gamm3} \omega_{r}-\omega_{c}=-\sqrt{3}\kappa\dfrac{K}{|K|}\tag{6}\]
After the fit, the shift estimated by the model was not correct and since it can be measured experimentally by comparing \(\omega_r\) and \(\omega_c\), we used it to correct our
estimation of \(\kappa_i\). Indeed, when \(\kappa_c > \kappa_i\) (\(Q_c\) is fixed by design), the internal loss rate becomes difficult to estimate with
accuracy.
The increase in the resonance dip as the input power increases (Fig. 3b in the main text) is only due to the presence of non-linear losses. As such, at \(P = P_c\), the minimum of \(S_{11}\)
only depends on a single free parameter \(\kappa_{nl}\). The fit is performed iteratively, that is \(K\) is computed at each iteration until maximum matching is found.
4 Microwave measurement of the current-phase relation↩︎
As explained in the main text, sending dc current through the Josephson junction modulates its inductance and therefore varies the resonance frequency of the microwave resonator (Fig. 11a) as \[\label{eq:Lj95supp} L_J = \dfrac{\Phi_0}{2\pi}\left(\dfrac{\partial I}{\partial \phi}\right)^{-1}\tag{7}\] The above equation shows that \(L_J\) depends on the variation of the applied current with respect to the superconducting phase difference of the Josephson junction i.e. its current-phase relation (CPR). The CPR in SNS junctions deviates from the typical sinusoidal dependence observed in tunnel junctions and the analytical expression will change depending on the regime of the junction (short/long – diffusive/ballistic).
To get a correct expression of the CPR, we first estimate the coherence length of our InAs nanowires. To first calculate the mean-free-path of InAs \(l_{\text{mfp}} = \mu m^*v_f/e\), we use the bulk value for the
effective mass of InAs and Fermi velocity, \(m^*_{\text{bulk}} = 0.026m_e\) and \(v^{\text{bulk}}_f\) = 1.3\(\times\)10\(^6\) m/s. The mobility \(\mu = 5000\) cm\(^2\)/Vs is taken from Ref. [51]. This gives \(l_{\text{mfp}} \approx 96\) nm which is around the junction length \(l_{\text{JJ}}
\approx\) 100 to 150 nm. We thus consider the junction to be in a diffusive regime. This is also backed up by the absence of Fabry-Perot like interference in the critical current modulation.
The coherence length reads \(\xi = \sqrt{\hbar D/\Delta} \approx\) 700 nm with \(D = v^{\text{bulk}}_f l_{\text{mfp}}\) in the one-dimensional limit [52]. Given that \(l_{\text{mfp}} < l_{\text{JJ}} \ll \xi\), we can conclude
that the junctions is in the short diffusive regime.
We therefore use the current-phase relation of a short SNS Josephson junction [53]:
\[\label{eq:cpr} I(\phi) = I_0\frac{\sin(\phi)}{\sqrt{1-\tau\sin^2(\phi/2)}}\tag{8}\] where \(I_0\) is a scaling factor and \(I_c = \underset{\phi}{\max} \left[ I(\phi) \right]\).
As we do not know the number of conduction channels \(N\) present in the JJ, we only extract information on \(\sum_{i=1}^{N} \tau_i = N\tau\)
We compute the derivative of the CPR from Eq. 8 : \[\label{eq:didphi}
\frac{dI}{d\phi}=\frac{I_0}{\sqrt{1-\tau\sin^2(\phi/2)}}\Big(\cos(\phi)+\frac{1}{4}\frac{\tau\sin^2(\phi)}{1-\tau\sin^2(\phi/2)}\Big)\tag{9}\]
Since there is no phase bias, we do not know the phase dependence at any time. To this end, we solve Eq. 8 for \(\phi\) to get a bias current dependence: \[\label{eq:phi} \phi=2\arcsin\left[\sqrt{\dfrac{1}{2}\left(\dfrac{4I_0^2+\tau I^2}{4I_0^2}-\sqrt{\left(\dfrac{4I_0^2+\tau I^2}{4I_0^2}\right)^2-\dfrac{I^2}{I_0^2}}\right)}\right]\tag{10}\] By inserting the above equation into 9 , we now have a direct dependence of \(\tau\) and \(I_c\) on the bias current \(I\). We then insert this equation into Eq. 7 . We record the microwave reflection of the device for a range of dc currents and perform a fit (Fig. 11) using the usual resonance frequency relation with \(L_J\) derived from the above equations : \[\label{eq:tunability95supp} \omega_r(I_{DC}) = \left[ \sqrt{(L_0 + L_J(I_{DC}))C} \right]^{-1}\tag{11}\] In the fit, the effective transparency \(\tau\) and the critical current \(I_c\) of the Josephson junction are free parameters. The bare inductance \(L_0\) and the capacitance \(C\) are extracted using electromagnetic simulations.
The fit is shown as a dashed orange line in Fig. 11b. The internal losses are extracted by circle fitting the microwave reflection at each \(I_{DC}\). The increase of \(\kappa_i\) with \(I_{DC}\) has been observed before in graphene Josephson junctions [36] and can be attributed to phase slips that appear as the tilt in the washboard potential becomes larger when \(I_{DC}\) increases. When moving away from \(\phi\) = 0, we expect more dissipation due to the presence of Andreev levels as pointed out in the main text for non-linear losses [37].
The CPR can then be inferred with \(\tau\) and \(I_c\) from the fit (Fig. 11d). As suggested by Eq. 8 , it deviates from the standard purely sinusoidal CPR in tunnel junctions and the dispersion is more linear close to zero-phase.
5 Noise analysis of the JPA↩︎
The system is composed of the Josephson parametric amplifier followed by a classical measurement chain, including in particular the HEMT amplifier at the 4K stage. To obtain the power spectrum at the input of the HEMT, we subtract from the raw measured signal the contributions of the room temperature amplifiers, the gain of the HEMT and losses from the cables, totalizing 74.7 dB. By dividing this measured spectrum with the bandwidth of the spectrum analyzer (\(B\) = 150 Hz), we obtain the power spectral density (PSD) presented in Fig. 5b in the main text. The effect of the losses between the JPA and the HEMT amplifier, for instance due to the directional coupler, filter and isolator at the mixing chamber stage as well as an additional attenuator, can be taken into account with a beam-splitter like model, see Fig. 12. For simplicity, we will neglect the effect of components after the HEMT amplifier on the noise.
The noise \(N_{\star}\) refereed to point \(\star\) is related to the input noise, the noise of the HEMT and vacuum fluctuations \(V = \hbar \omega/2\): \[\label{eq:hemt95noise} N_{\star} = N^{\text{OFF}}_{\text{tot}} - \eta N_{\text{in}} + (1 - \eta)V + T_{\text{add,4K}}\tag{12}\] where \(N^{\text{OFF}}_{\text{tot}}\) is the total output signal measured at the spectrum analyzer pump off, after subtracting the contribution of 74.7 dB mentioned before. \(N_{\text{in}}\) is the input noise thermalized at the mixing chamber (25 mK), \(\eta\) is the transmission of the beam splitter and \(T_{\text{add,4K}}\) the added noise temperature due to the HEMT being thermalized on the 4 K stage of the fridge.
The noise referred to the input of the JPA is then \[N_{\star}/\eta = N_{\text{in}} + \frac{1-\eta}{\eta}V + \frac{1}{\eta}T_{\text{add,4K}}.\] The value of \(\eta\) is determined by taking into account the insertion loss of each aforementioned component between the JPA input and the HEMT input giving a total loss of 4.9 dB which yields \(\eta = 0.32\). When the amplifier is on, a gain \(G\) is present from the JPA yielding a transmission \(\eta_{\text{on}} = G\eta\).
The added noise of the JPA is then \[\label{eq:JPA95noise} N_{\text{add,JPA}} = N^{\text{ON}}_{\text{tot}} - N_{\text{in}} + \frac{1-\eta}{\eta G}V + \frac{T_{\text{add,4K}}}{\eta G}.\tag{13}\] with \(N^{\text{ON}}_{\text{tot}}\), the total output signal measured at the spectrum analyzer pump on, after subtracting the contribution of 74.7 dB mentioned before and \(N_{\text{add,JPA}}\) the added noise of the JPA.
We retrieve the added noise of the HEMT in the absence of gain from the JPA (pump off) that is referred to point \(\star\). We use Eq. 12 where we assumed the input noise off resonance to be \(N_{\text{in}} = V\) and \(T_{\text{add,4K}} =\) 4 K. The noise temperature is subsequently obtained using \[\label{eq:noise95temperature} N_{\star} = Bk_BT_{\rm HEMT}\tag{14}\] with \(B\) = 150 Hz, the bandwidth of the spectrum analyzer and \(T_{\rm HEMT}\) is the noise temperature of the HEMT.
When parametric gain is present (pump on), Eq. 13 is used to retrieve the added noise of the JPA, \(N_{\text{add,JPA}}\). The value of the signal gain of the JPA, \(G\), is first estimated from the power spectral density graph of the signal tone when the pump is on and off (Fig. 5b in the main text) and finely tuned as to match the peak of the signal tone for both the HEMT and the JPA noise temperature curves (\(G\) = 22.7 dB). In the high gain limit, the last two terms on the right hand side of Eq. 13 are small and can, in principle, be neglected. The signal is then converted to noise temperature using \[\label{eq:noise95temperature95JPA} N_{\text{add,JPA}} = Bk_BT_{\rm JPA}G\tag{15}\]
The quantum-limit (black dashed line in Fig. 5c-d in the main text) sets the minimum added noise of the amplifier to be \(\hbar\omega/2\) or equivalently in noise temperature \(T_{\text{QL}} = \dfrac{\hbar(\omega_{\text{pump}}+\delta\omega)}{2k_B} \approx 0.150\) K with the pump frequency \(\omega_{\text{pump}}\).