Introduction

The Josephson effect describes the dissipationless current flow between two weakly coupled superconductors. Today, numerous technologies are based on this fundamental quantum phenomenon, ranging from superconducting qubit devices1,2,3,4,5 to parametric amplifiers6,7,8.

Regardless of whether the weak link consists of an insulator or a normal conducting material, the supercurrent is a periodic function of the phase difference φ between the superconductors9. In a Josephson tunnel junction, the supercurrent arises from the coherent tunneling of individual Cooper-pairs through the insulating barrier, each carrying a charge 2e10. The current-phase relation (CPR) in this case is given by \(I(\varphi )={I}_{{{{{{{{\rm{c}}}}}}}}}\sin (\varphi )\), with Ic being the critical current. However, when the superconductors are separated by a conducting weak link, such as a semiconductor or a metal, coherent transport of multiple Cooper-pairs can also occur, resulting in a non-sinusoidal CPR11,12,13,14,15. In general, the CPR of the junction can be expanded in a Fourier series as:

$$I(\varphi )=\mathop{\sum }\limits_{m=1}^{\infty }{c}_{m}\sin (m\varphi ).$$
(1)

The \(\sin (m\varphi )\) terms correspond to processes involving the simultaneous, coherent transport of m Cooper-pairs carrying a charge m × 2e. The amplitude of the higher harmonic terms cm, m > 1, reflects the probability of multi-Cooper-pair transport and decreases with higher harmonics, indicating that transport across the junction arises mainly from individual Cooper-pairs. Often, the CPR can be described by the junction transparency τ, defined as the transmission probability of electron in the weak link. The more transparent a junction is, the higher the ratio between successive Fourier coefficients ∣cm+1(τ)/cm(τ)∣.

Several theoretical proposals16,17,18,19,20,21,22,23,24 have investigated the possible advantages of using a so-called \(\sin (2\varphi )\) Josephson junction (JJ) for the realization of a parity-protected superconducting qubit. In this case, the parity of the Cooper pairs is protected by using a Josephson element with a dominant second harmonic term c2 in Eq. (1), corresponding to the supercurrent being carried by pairs of Cooper pairs with charge 4e. The qubit states can be therefore encoded into the even and odd parity of the number of Cooper-pairs on a superconducting island.

Important steps towards realizing a parity-protected qubit have been taken with superconducting quantum interference devices (SQUIDs) made of tunnel junctions arranged in a rhombus geometry25,26. By designing the loop inductances and the junction’s position, it is possible to engineer a CPR with a large second harmonic component ∣c2/c1∣ ~ 0.527, corresponding to an effecting transparency τ* ~ 128,29. When the magnetic flux through the SQUID is tuned to half a flux quantum Φ0/2, the first harmonic is suppressed due to destructive interference, leaving a dominant second harmonic term. This method relies on the fabrication of identical junctions, and departures from symmetry spoils parity protection.

A promising alternative approach is based on gate-tunable hybrid superconducting-semiconducting materials with high transparency channels. In ref. 30 the authors realize a \(\sin (2\varphi )\) element with a SQUID made of proximitized InAs nanowires, where local gate control of each junction allows precise balancing of the first harmonics. They show that the qubit relaxation time increases by an order of magnitude when the qubit is tuned close to the protected regime. However, for practical use of the parity-protected qubit, the Josephson energy of the second harmonics in the balanced configuration must be at the same time much larger than the residual Josephson energy coming from the first harmonics (for a long relaxation time) and much larger than the island charging energy (for small dephasing rate). The few conduction channels in the nanowires limit the maximum obtainable critical current and make the last requirement difficult to satisfy.

Hybrid two-dimensional materials have seen in recent years a great improvement in growth techniques that allow up-scaling and offer the advantage of wide gate tunability and top-down fabrication31,32. In this work, we report the observation of a 4e supercurrent in a SQUID consisting of two planar Josephson junctions formed in an InAs two-dimensional electron gas proximitized by an epitaxial Al layer33,34. Even if the operation of superconducting qubits has already been shown in this material platform35, the realization of high-quality resonators on III-V substrates remains a challenging task. Therefore, here we investigate the contribution of the 4e supercurrent by measuring the evolution in frequency of the ac Josephson radiation emitted by the SQUID as a function of a dc bias voltage. The high transparency of these JJs13 allows us to engineer an effective CPR in which the first harmonic is suppressed due to destructive interference, leaving a dominant second harmonic term. To achieve this, we balance the critical current of the junctions with local gate voltages and tune the magnetic flux through the SQUID loop to half a flux quantum Φ0/2. In the balanced configuration, radiation measurements reveal a pronounced suppression of emission at the fundamental Josephson frequency in favor of a strong ac signal at twice this frequency. We corroborate this finding by additionally detecting fractional half-Shapiro steps, characteristic of a \(\sin (2\varphi )\) junction.

Results and Discussion

Device and procedures

A simplified schematic of the device is shown in Fig. 1a. A superconducting loop, threaded by an external magnetic flux Φext, is interrupted on each arm by a section where the superconductor has been selectively removed. The Josephson junctions are formed in an InAs two-dimensional electron gas (green) which is proximitized by the close vicinity to an epitaxial Al layer (blue) grown on top. By locally removing the Al top layer with etching techniques that are detailed in the Methods section, we form InAs weak links. Local gate electrodes, VG1 and VG2, allow us to tune the electron density in the weak links and, consequently, adjust the critical currents of the JJs. The hereby formed Josephson junctions are symmetric by design, but the wet etching step produced two different widths: ~3 μm for JJ1 and ~2.5 μm for JJ2. Despite fabrication-related asymmetries, we were still able to tune the SQUID into a symmetric configuration by leveraging the gate tunability of the semiconducting weak link. Junctions this wide typically contain many conduction channels with a bimodal distribution of transparency values distributed between zero and one36,37,38,39. Earlier experiments on the same material platform have shown that the CPR in these junctions can be described by a single channel short diffusive junction model13,40 with an effective transparency τ* ~ 0.86.

Fig. 1: Device description and measurement setup.
figure 1

a Circuit schematic of a dc superconducting quantum interference device (SQUID) formed by two gate-tunable Josephson junctions with effective transmission probabilities \({\tau }_{1}^{* },{\tau }_{2}^{* }\), threaded by the external flux Φext. b False-color electron micrograph of the device and experimental setup. Each junction is fabricated by selectively removing the epi-Al layer (blue) over 150 nm long stripes. The charge carrier density in the exposed InAs two-dimensional electron gas (green) is tuned by a set of electrostatic gates (VG1, VG2, and VFG) shown in yellow and orange, which are galvanically isolated from the loop by 15 nm of HfO2 (light blue). dc and ac current bias is defined through the voltage drop over a bias resistor Rb = 1 MΩ. The SQUID is shunted to the ground with Rs = 10 Ω. We send a microwave tone to the device and also detect photon emission. The scale bar in the main figure is 1 μm, and the scale bar in the zoom-in is 300 nm. c Individual components I1 (blue) and I2 (orange) and total current (green) flowing through a symmetric SQUID as a function of the phase drop φ1 at Φext = Φ0/2. The current phase relation of both junctions is plotted using a single channel short diffusive model with an effective transparency τ* = 0.86. The current is normalized to units of the critical current Ic. The schematic of the SQUID helps visualize the requirements for a \(\sin (2\varphi )\) junction: a dominant 4e supercurrent is obtained with a symmetric SQUID biased at Φ0/2.

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Figure 1b depicts a false-color electron micrograph of the device and the experimental setup. We apply a dc-current via the voltage drop over a bias resistor Rb = 1 MΩ. We damp the SQUID with a shunt resistor Rs = 10 Ω to enable a continuous transition from the superconducting to the normal conducting state. The 10 Ω-shunt increases the region of the stable voltage drop across the junction, and at the same time, it reduces both heating and hysteretic behaviors. The differential resistance is measured using standard lock-in techniques. Furthermore, the microwave setup allows probing the ac Josephson effect in two ways. On one hand, the Josephson radiation emitted from the SQUID under finite dc bias can be detected with a spectrum analyzer. Second, the reverse experiment can be performed, namely, irradiating the device with a microwave tone and measuring its dc response.

Figure 1c shows the interference between the supercurrent I1 flowing in JJ1 (blue dashed curve) and the supercurrent I2 in JJ2 (orange dashed curve) at Φext = Φ0/2. The total supercurrent flowing through the SQUID (green solid curve) is:

$$I={I}_{1}({\varphi }_{1},{\tau }_{1}^{* })+{I}_{2}({\varphi }_{2},{\tau }_{2}^{* }).$$
(2)

The phase drops over the two JJs are related by the fluxoid relation φ1 − φ2 = 2πΦext/Φ0. Here, we have assumed that the phase difference between the two JJs is solely given by the externally applied flux, neglecting loop, and mutual inductances, which is justified in our device40. When the loop is flux biased at Φext = Φ0/2 and the JJs are the same (\({\tau }_{1}^{* }={\tau }_{2}^{* }\)), Cooper-pairs are transferred with the same amplitude but opposite phase through the SQUID arms, resulting in a destructive interference of the 2e contribution with periodicity 2π and in a constructive interference of the 4e supercurrent with periodicity π. In this way, it is possible to engineer an effective \(\sin (2\varphi )\) junction.

ac and dc Josephson effect from single junction

In the following, we characterize the dc and ac Josephson effect of the individual JJs. To this end, we measure the gate dependence of the critical current and the radiation spectrum of each junction, while the neighboring one is fully depleted. Figure 2a, b show the differential resistance of JJ1 and JJ2 as a function of current bias I for different gate voltages. We identify the critical current Ic as the boundary between the superconducting regime (dark blue) and the ohmic regime (turquoise). At negative gate voltages (VG1≤ −0.9 V and VG2≤ −1.5 V) Ic is negligibly small, but it can be gradually increased with increasing VGi. \({I}_{{{{{{{{\rm{c,max}}}}}}}}}\) saturates to \({I}_{{{{{{{{\rm{c1,max}}}}}}}}}=1.1\,\mu {{{{{{{\rm{A}}}}}}}}\) for JJ1 and \({I}_{{{{{{{{\rm{c1,max}}}}}}}}}=0.8\,\mu {{{{{{{\rm{A}}}}}}}}\) for JJ2 at around VGi = 0.5 V. The slight differences in the gate dependence of the two junctions are attributed to a different junction width and gate geometry. To estimate the IcRn product of the junctions, we measure the resistance at voltage bias larger than twice the superconducting gap of the leads as obtained from multiple Andreev reflection measurements conducted on a different chip of the same wafer. Subtracting the shunt resistor, we obtain a normal state resistance of the junction Rn ~ 90 Ω, corresponding to a IcRn ~ 90 μV. We also note that potential errors in estimating Rn might have led to an underestimation of the IcRn product. Nonetheless, the significantly large IcRn product indicates a high-quality Josephson junction with a uniform current distribution.

Fig. 2: dc and ac Josephson effect of the individual junctions.
figure 2

a, b Differential resistance dV/dI of the individual Josephson junctions JJ1 and JJ2 as a function of gate voltage VG1, VG2 and current bias I. c, d Integrated voltage Vint as a function of current bias I at VG1 = −0.75 V and VG2 = −0.7 V obtained by integrating the corresponding dV/dI along the white dashed lines shown in a, b. e Illustration of the expected peak evolution in the emission spectrum of voltage-biased JJ as a function of detection frequency \({f}_{\det }\). A junction with finite transparency emits photons at the fundamental Josephson frequency (red dashed line) and integer multiples of it (orange and pink dashed lines), here corresponding to the coherent transport of pairs of Cooper-pairs. The dashed gray lines indicate processes associated with the up- and down-conversion of environmental photons at frequency fenv. f, g Normalized radiation power Pdet,norm. as a function of \({f}_{\det }\) and Vint for the same configuration in c and d. The orange arrow points to the 4e emission peak.

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In Fig. 2c, d, the IV-curves at VG1 = −0.75 V and VG2 = −0.7 V, respectively are obtained by integrating the measured dV/dI curves along the white dashed lines in Fig. 2a, b. Both junctions show an ohmic behavior down to 2 μV, which allows stable voltage biasing in the microwave regime of the Josephson emission.

According to the ac Josephson effect, the phase difference of a voltage-biased Josephson junction will evolve linearly in time following

$$\varphi (t)=\frac{2\pi }{{\Phi }_{0}}Vt,$$
(3)

with V being the voltage drop across the junction. Consequently, an applied dc voltage causes an oscillating supercurrent at the Josephson frequency fJ = 2eV/h. This transforms into the emission of microwave photons at fJ. If higher harmonics are present, photon emission at higher frequencies fJ,m = m × 2eV/h also occurs41,42. In Fig. 2e we show the expected peak evolution in the emission spectrum of voltage-biased JJ as a function of detection frequency \({f}_{\det }\) and V. For every voltage bias position, peaks emerge in the emission spectrum, if the detection frequency matches an integer multiple of the Josephson frequency \({f}_{\det }={f}_{{{{{{{{\rm{J}}}}}}}},m}\). These peaks induce a fan-like pattern, capturing the linear relation between voltage and the emission frequency with slope h/(m2e). Emission lines evolving as \(h{f}_{\det }/(m2\,{{\mbox{e}}}\,)\) correspond to the coherent transport of m Cooper-pairs across the junction (red, orange, and pink dashed lines for m = 1, 2, and 3). In addition to the fan-like pattern, replicas of the Josephson emission lines can appear at a constant frequency offset on the right and on the left of the predicted peak position due to photon-assisted emission through environmental modes43. A photon from a spurious environmental mode can be upconverted to a detector photon by taking up the energy 2eV provided by the inelastic tunneling of a Cooper-pair (right shift in frequency). The energy balance in this case reads \(h{f}_{\det }=h{f}_{{{{{{{{\rm{env}}}}}}}}}+2\,{{\mbox{e}}}\,V\), where fenv corresponds to the resonant frequency of an environmental cavity. Such resonance can be caused for example by a standing wave pattern along the microwave lines. The complementary process is also possible, meaning that a photon coming from a Cooper-pair tunneling can be downconverted to a detector photon by giving up the energy hfenv to the environment (left shift in frequency). The energy balance in this case reads \(h{f}_{\det }=2\,{{\mbox{e}}}\,V-h{f}_{{{{{{{{\rm{env}}}}}}}}}\).

In Fig. 2f, g we plot the normalized radiation power Pdet,norm. as a function of \({f}_{\det }\) and Vint for JJ1 and JJ2 respectively. The power is normalized at each detection frequency to compensate for the frequency-dependent background. A pronounced emission peak at frequency fJ,1 (red dashed line) corresponding to the 2e single Cooper-pair transport is measured over the entire frequency range from 5 GHz to 8 GHz. The signal due to the 4e double Cooper-pair transport at frequency fJ,2 (orange dashed line) is weaker but becomes clearly visible in the emission spectrum around 7 GHz (orange arrow). Emission peaks at frequencies corresponding to higher harmonics, m > 2, are below our detection limit. In addition to the fan-like pattern, there is a strong replica of the fundamental Josephson emission appearing at a constant frequency offset (fenv ~ 1.95 GHz) on the right of the predicted peak position. Its contribution diminishes for \({f}_{\det } \, > \, 6\) GHz. Changes in power spectrum as a function of detection frequency arise from a frequency-dependent probability of photon emission due to inelastic Cooper pair tunneling. The emission probability depends on the impedance of the environment surrounding the Josephson junction44, which in turn has a complex behavior as a function of frequency caused, for example, by standing wave patterns in the rf lines due to spurious impedance mismatch conditions. By setting \({f}_{\det }=7.1\) GHz, we can disregard the contribution of this environmental mode in the following investigation.

ac and dc Josephson effect from a SQUID

Next, we exploit the interference between the two junctions when both carry a finite supercurrent in order to realize an effective Josephson element with the negligible first harmonic component. We require two conditions: (i) the flux is to be set to Φext = Φ0/2, and (ii), the JJs are gate-tuned into balance, such that \({c}_{1,{{{{{{{{\rm{JJ}}}}}}}}}_{1}}={c}_{1,{{{{{{{{\rm{JJ}}}}}}}}}_{2}}\). The key challenge in the experiment is the balancing of the junctions. As a solution, we adopt an approach proposed in45 that is based on the observation that Ic for the forward and reverse current-bias directions, Ic,+ and Ic,−, is mismatched unless both junctions are balanced and Φext = nΦ0/2 with n being an integer. To balance the SQUID, we look at regions in gate voltage without diode effect, meaning Ic,+ and Ic,− are equal (symmetric junctions).

In Fig. 3 we measure the SQUID in three different configurations. Firstly, we fix the gate voltages such that the junctions are symmetric and sweep Φext. Secondly, we fix VG2 and sweep VG1 at Φext = Φ0/2. Finally, we fix the gate voltages and sweep Φext in the case of asymmetric junctions.

Fig. 3: Tuning a superconducting quantum interference device into a symmetric configuration.
figure 3

a Differential resistance dV/dI of the superconducting quantum interference device (SQUID) as a function of external flux Φext and current bias I for symmetric junctions. Here, the gate voltage on one junction is VG1 = −0.865 V and the gate voltage on the other junction is VG2 = −0.9 V. In this balanced configuration, there is no diode effect. b Normalized radiation power Pdet,norm. at a detection frequency \({f}_{\det }=7.1\) GHz plotted vs external flux Φext and normalized voltage drop over the SQUID Vint. The map is measured at the same time as in a. At half flux quantum, the 2e radiation signal is suppressed, and the 4e peak becomes the dominant feature. The plots in blue and orange are line cuts in the power map taken at Φext = 0.22 Φ0 and Φext = Φ0/2 respectively, as indicated by the arrows. In c, we bias the SQUID at Φext = Φ0/2, and fix VG2 = −0.875 V. We measure the SQUID differential resistance as a function of current bias and VG1. Moving from left to right, we go from Ic2 > Ic1 to Ic1 > Ic2, crossing a balanced configuration. d Same as in b but for the gate and flux configuration as in c). For specific values of VG1, we see a clear increase in the visibility of the 4e peak. The plots in blue and orange are line cuts in the power map taken at VG1 = −0.89 V and VG1 = −0.8 V, respectively. e Same as in a), but for VG1 = −0.9 V and VG2 = −1 V. In this unbalanced configuration, there is a diode effect. f Same as in b but for the gate configuration as in e. Here, throughout the flux bias range, the 2e peak remains the dominant feature.

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In Fig. 3a we plot the SQUID differential resistance dV/dI as a function of current bias I and Φext in a gate configuration where Ic1 ≈ Ic2. No diode effect is observed over the entire flux bias range. Differences between the gate values at which symmetry is achieved and those expected from Fig. 2a, b are caused by the fact that the critical current of each junction depends on whether the junction is measured individually or embedded in SQUID46. Simultaneously, we measure the SQUID ac emission at fixed detection frequency \({f}_{\det }=7.1\) GHz. Figure 3b shows the normalized radiation power Pdet,norm. as a function of Φext and integrated voltage drop over the SQUID Vint. Because the signal peaks at \({V}_{m}=h{f}_{\det }/(m2\,{{\mbox{e}}}\,)\), we scale the voltage axis by \(h{f}_{\det }/2\,{{\mbox{e}}}\,\). The emission pattern changes in a striking manner around Φext = Φ0/2. The fundamental Josephson signal at a scaled Vint = 1, corresponding to the 2e supercurrent, vanishes almost completely, while a sharp bright peak at a scaled Vint = 1/2 appears, which corresponds to the radiation signal coming from the simultaneous inelastic transport of pairs of Cooper-pairs. An additional horizontal line is visible in the map due to the spurious environmental mode, as addressed before. On the right panels, we plot cuts along Vint at Φext = 0.22 Φ0 (blue) and Φext = Φ0/2 (orange). The radiation power is here presented on a linear scale. At Φext = Φ0/2, the 4e peak emerges as the dominant feature, yet its amplitude is approximately ~ 25 times smaller compared to the amplitude of the 2e peak measured at Φext = 0.22 Φ0. This is expected, since the amplitude of the power emission peak is proportional to the square of Ic, which at Φext = Φ0/2 is only determined by the second harmonic of the CPR, and is ~5 times smaller than Ic at Φext = 0.22 Φ0. A detailed analysis of the ratio between the 4e and 2e peaks can be found in Supplementary Discussion 1.

We investigate the dependence of the emission spectrum as a function of VG1, when the magnetic flux is set to Φext = Φ0/2 and VG2 = −0.875 V, shown in Fig. 3c, d. Away from the balanced configuration, the more distinct peak in the emission spectrum is the one corresponding to the 2e transport. However, once we approach the balanced situation at VG1 ~ −0.8 V the signal at V1 = hfdet/2e is suppressed, and instead, the dominant peak in the emission spectrum becomes the one at V2 = hfdet/4e.

Lastly, in Fig. 3e we plot the dV/dI of the SQUID as a function of I and Φext in a gate configuration where Ic1 ≠ Ic2. Apart from Φext = Φ0/2, there is a clearly visible diode effect. Figure 3f shows Pdet,norm. as a function of Φext in the same gate configuration. The 2e emission peak remains the dominant feature throughout the whole flux bias range. Its amplitude decreases asymmetrically on the left- and right-hand side of Φext = Φ0/2, following the asymmetry of the SQUID critical current. Even though the junctions are not balanced, one can still see that the emission signal slightly increases at voltages \({V}_{2}=h{f}_{\det }/4\,{{\mbox{e}}}\,\), in the vicinity of Φext = Φ0/2. A study of the evolution of the 4e peak emission amplitude at different gate voltage configurations is presented in Supplementary Discussion 2.

These findings show that a continuous transition between a 2e and a 4e supercurrent can be achieved by tuning both gate voltages and the magnetic flux. Importantly, the 4e supercurrent dominates over a finite window in parameter space and is not limited to exactly matching boundary conditions.

Shapiro steps

So far, we have used the Josephson radiation measurements to identify the emergence of a 4e supercurrent in the SQUID. In the last part of this work, we discuss Shapiro step measurements that complement the radiation experiment. When a microwave drive tone is sent to a JJ, distinct voltage plateaus in the V(I) characteristic appear, known as Shapiro steps47,48,49,50,51. For a conventional \(\sin (\varphi )\) junction, each plateau corresponds to a Cooper-pair absorbing n photons with frequency fd to overcome the Shapiro step voltage Vn, and the energy relation reads 2eVn = nhfd. The presence of higher harmonics in the CPR of the junction changes the energy relation to 2meVn = nhfd, corresponding to m Copper-pairs absorbing n photons to overcome the voltage step.

We apply a microwave tone of fixed frequency fd = 7.5 GHz to the SQUID with different output power Pd values. The signal is applied to the microwave input line, connecting the device to the amplification chain through a directional coupler (see Supplementary Note 1). In Fig. 4a, we plot the SQUID differential resistance dV/dI at Φext = 0 as a function of current bias I and Pd in a symmetric gate configuration. In the left panel, we plot dV/dI versus I, and on the right, we plot the data as a function of the integrated voltage Vint scaled by hfd/2e. Shapiro steps occur at integer values of the scaled voltage as dips in differential resistance.

Fig. 4: Shapiro steps measurements at zero and a half flux quantum for symmetric junctions.
figure 4

a On the left, differential resistance dV/dI as a function of drive power Pd and bias current I at constant drive frequency fd = 7.5 GHz and zero external flux Φext = 0 for a gate voltage on the first junction of VG1 = −0.73 V and a gate voltage on the second junction of VG2 = −0.5 V. The drops in differential resistance correspond to the emergence of Shapiro steps. On the right, differential resistance as a function of Pd plotted vs normalized voltage drop Vint over the device. At zero flux, mostly integer Shapiro steps are visible. b Same as in a, but at Φext = Φ0/2. The destructive interference of the first harmonics produces a current-phase relation with double the periodicity of the individual junctions, inducing the emergence of half-integer Shapiro steps.

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The data in Fig. 4b is measured for the same gate values as in Fig. 4a, but at Φext = Φ0/2. In this configuration, the SQUID resembles an effective \(\sin (2\varphi )\) junction because the 2e supercurrent is suppressed. The energy relation for the appearance of Shapiro steps is given in this case by 4eVn = nhfd, resulting in a doubling of the number of observed steps. In line with the theoretical expectations45, both integer and half-integer Shapiro steps are equally visible in the data. Differences between this measurement and Shapiro steps measurements performed on Josephson junction with a high-quality factor52 are attributed to the 10 Ω-shunt in our device.

Conclusion

We have demonstrated the realization of an effective \(\sin (2\varphi )\) Josephson junction using a dc SQUID consisting of two planar Josephson junctions formed in a proximitized InAs two-dimensional electron gas. We probe the emergence of a dominant second harmonic in the CPR of the SQUID by measuring the ac Josephson effect as a function of gate voltages and magnetic flux. Photon emission at the fundamental Josephson frequency is suppressed when the SQUID is in a symmetric configuration and biased at half flux and instead, photons are only emitted at fJ,2. We provide evidence on how to continuously tune from the 2e to the 4e supercurrent regime by adjusting the junction gate voltages and the external magnetic flux. The results are further substantiated through complementary Shapiro step measurements in a symmetric SQUID configuration at half flux, revealing additional half-integer steps with the same visibility as the integer steps.

Our results indicate, that a robust \(\sin (2\varphi )\) JJ can be engineered and could be used to realize parity-protected qubits with this material system. Such parity-protected qubit provides an alternative route to the protection of quantum information in superconducting devices and may complement alternative approaches based on fluxonium qubits53,54,55,56 and qubits based on topological wavefunctions57,58,59,60,61,62,63. Looking ahead, the 2D platform would make it easier to further protect the qubit from noise and offsets by concatenating several SQUIDs in parallel21.

Methods

The proximitized InAs two-dimensional electron gas used in this project is grown starting from a semi-insulating InP (100) substrate. A 1 μm thick InxAl1−xAs buffer layer is used to match the lattice constant of InP to the one of InAs. The quantum well consists of a 7 nm InAs layer sandwiched between a 10 nm (top barrier) and a 4 nm (bottom barrier) In0.75Ga0.25As layer. The 10 nm Al layer is epitaxially grown on top of a capping GaAs thin film without breaking the vacuum, ensuring a pristine interface between the semiconductor and the superconductor. Here we show results obtained from a wafer stack with mobility μ = 11,000 cm2 V−1 s−1 at electron densities of 2.0 × 1012cm−2, measured on a different chip coming from the same wafer.

The device is fabricated using standard electron beam lithography techniques. The SQUID is electrically isolated by etching the Al layer and 300 nm of buffer around it. First, the Al film is removed with Al etchant Transene D, followed by a deep III-V chemical wet etch H2O:C6H8O7:H3PO4:H2O2 (220:55:3:3). Next, JJs are formed by selectively removing the Al over 150nm-long stripes on each branch of the loop. A 15 nm-thick layer of insulating HfO2 is grown by atomic layer deposition at a temperature of 90 ∘C over the entire sample. The set of gates is realized in two steps. A thin Ti/Au (5/25 nm) layer is evaporated on top of the mesa to define the gate geometry, and then leads and bonding pads are defined by evaporating a Ti/Au (5/85 nm) layer at an angle of ±17∘ to overcome the mesa step. More information about the full wafer stack and the fabrication procedure can be found in refs. 31,33,34,64.