s41467-025-56245-7.pdf

Article https://doi.org/10.1038/s41467-025-56245-7 Gate- and ﬂux-tunable sin(2φ)J o s e p h s o n element with planar-Ge junctions Axel Leblanc 1 , Chotivut Tangchingchai1, Zahra Sadre Momtaz2, Elyjah Kiyooka1, Jean-Michel Hartmann3, Frédéric Gustavo1, Jean-Luc Thomassin1, Boris Brun 1, Vivien Schmitt 1, Simon Zihlmann 1, Romain Maurand 1, Étienne Dumur 1, Silvano De Franceschi 1 & François Leﬂoch 1 Hybrid superconductor-semiconductor Josephsonﬁeld-effect transistors (JoFETs) function as Josephson junctions with gate-tunable critical current. Additionally, they can feature a non-sinusoidal current-phase relation (CPR) containing multiple harmonics of the superconducting phase difference, a so- far underutilized property. Here we exploit this multi-harmonicity to create a Josephson circuit element with an almost perfectlyπ-periodic CPR, indicative of a largely dominant charge-4e supercurrent transport. We realize such a Josephson element, recently proposed as building block of a protected superconducting qubit, using a superconducting quantum interference device (SQUID) with low-inductance aluminum arms and two nominally identical JoFETs. The latter are fabricated from a SiGe/Ge/SiGe quantum-well hetero- structure embedding a high-mobility two-dimensional hole gas. By carefully adjusting the JoFET gate voltages andﬁnely tuning the magnetic ﬂux through t h eS Q U I Dc l o s et oh a l faﬂux quantum, we achieve a regime where the sinð2φÞ component accounts for more than 95% of the total supercurrent. This result demonstrates a new promising route towards parity-protected super- conducting qubits. Quantum information processing requires qubits with long coherence time enabling high-ﬁdelity quantum gates. Over the past two decades, superconducting circuits have led to the realization of quantum pro- cessors of ever-growing size made of qubits with steadily improving ﬁdelities

This way, superconducting qubits have become one of the most advanced physical platforms for quantum computing. Progress has been driven by material engineering and optimization, as well as by the development of new device concepts capable of providing a growing level of protection against noise sources in the environment 2– 4. Qubit protection against relaxation and dephasing processes can be granted from the symmetry properties of the qubit Hamiltonian. In this direction, a variety of possible solutions have been proposed and only partly explored 5–11. One of the leading ideas is to create superconducting qubits whose two lowest energy states are associated with odd and even numbers of Cooper pairs in a super- conducting island, respectively. Due to the different parity, these states are orthogonal to each other in both charge and phase space12– 19. This type of parity-protected qubit requires a parity-preserving Josephson element that only allows the coherent transfer of corre- lated pairs of Cooper pairs, which translates into devising a Josephson circuit with a π-periodic, sinð2φÞ current phase relation (CPR). Some proposals to engineer such a sin ð2φÞ qubits rely on con- ventional sin ðφÞ Josephson junctions, either arranged into large arrays 14 or embedded in a superconducting quantum interference device (SQUID) together with extremely large inductances 15.T h e practical realization of these ideas is technologically challenging and some signiﬁcant experimental progress was reported only recently

Another approach is to leverage the multi-harmonic CPR and the gate Received: 13 June 2024 Accepted: 10 January 2025 Check for updates 1Univ. Grenoble Alpes, CEA, Grenoble INP, IRIG-PHELIQS, 38000 Grenoble, France.2Institut Néel, CNRS/UGA, Grenoble 38042, France.3Univ. Grenoble Alpes, CEA, LETI, 38000 Grenoble, France. e-mail: [email protected]; [email protected]; francois.leﬂ[email protected] Nature Communications| (2025) 16:1010 1 1234567890():,; 1234567890():,; tunability of superconductor(S)– semiconductor(Sm) Josephsonﬁeld- effect transistors (JoFETs)20– 29. Various signatures of sinð2φÞ Josephson elements were recently reported 30– 35 and harnessed to demonstrate some ﬁrst experimental evidence of parity protection 16.H o w e v e r ,a direct measurement of a sinð2φÞ CPR, a precise quantitative evaluation of its harmonic purity including the ﬂux and gate tunability and the inﬂuence of the SQUID arm inductance, have been missing so far. These important aspects are addressed in the present work. Our experimental study takes advantage of a recently developed S–Sm platform based on SiGe/Ge/SiGe quantum-well heterostructures36. Notably, these heterostructures embed a high-mobility two-dimensional hole gas enabling the reproducible top-down fabrication of multi-har- monic, gate-tunable Josephson junctions. We investigate the CPR of a SQUID incorporating two of such junctions, in short a G-SQUID. We demonstrate ample gate and magnetic-ﬂux control of the Josephson harmonic content. By a quantitative analysis based on a fully compre- hensive model of our circuit, we show that half aﬂux quantum through the SQUID results in an unprecedented suppression of theﬁrst junction harmonic, which is reduced by two orders of magnitude. Under these conditions, the desired sin ð2φÞ (i.e., charge-4e) contribution to the supercurrentﬂow reaches up to 95.2% of the total supercurrent. This achievement is a signiﬁcant step forward in the development and opti- mization of a semiconductor-based parity-protected qubit. Results Device and measurement setup The G-SQUID (shown in Fig. 1a) consists of an aluminum super- conducting loop with nearly symmetric arms embedding two nominally identical JoFETs fabricated out of a SiGe/Ge/SiGe quantum-well hetero- structure. The compressively strained Ge quantum well lies 22 nm beneath the semiconductor surface and hosts a two-dimensional hole gas exhibiting mobility of 10 5 cm2/Vs measured at a carrier density of 6×1 011 cm−2. The two JoFETs,JN1 and JN2,h a v ea1μm-wide and a 300 nm- long Ge channel, short enough to be considered as quasi-ballistic junc- tions (for more details on the JoFETs see Supplementary Note 1). The G-SQUID is embedded in a second la rger loop together with a wider reference JoFET,J W,( 1 0μm-wide and 300 nm-long Ge channel), enabling a direct CPR measurement20,22,37–42. To this purpose, JW is designed to have a critical current much larger than those ofJN1 and JN2.T h es m a l l and large superconducting loops are locallyﬂux-biased by means of two 10 μm-wide and 50 nm-thick Al lines whose cross-talks have been cali- brated in situ and then implicitly compensated throughout the rest of the paper (see Supplementary Note 6). Furthermore, we stress that the Al arms have small but non- negligible inductances, mostly of kinetic origin, that we label asL 1, L2, and LW. As our later analysis will reveal, properly extracting the intrinsic harmonic content of the JoFET CPRs from the measurements requires taking these inductances into account. In the rest of the paper, all of the calculated curves are obtained using the circuit model shown in Fig. 1a (see Supplementary Note 2 for more details). All measurements were performed in a dilution refrigerator at a base temperature of 38 mK, see Supplementary Note 8. A detailed description of the fabrication process and the measurement methods was provided in our previous publications 32,43. Characterization of the individual JOFETs To access the individual DC transport characteristic of a given JoFET in such a parallel con ﬁguration, we purposely apply large positive gate voltages (≈1.5 V) to the other JoFETs, thereby suppressing currentﬂow through their respective arms. The resulting individual characteristics of J W, JN1,a n d JN2 as a function of their respective gate voltages are shown on the color scale in Fig. 1b– d, with the corresponding circuit measurement schematics displayed in the insets. For all the JoFETs, the current at which the device switches from superconducting to normal state, close to the critical currentIC, is clearly visible as an abrupt change of the measured source-drain voltage drop from 0 to a ﬁnite value. The three JoFETs exhibit a similar behavior denoting consistent properties of the Ge channel and the super- conducting contacts. In particular, we note that the two narrow JoFETs, designed to be identical, have very similarI Ni C ðV Ni G Þ characteristics. By varying the magnetic ﬂux, Φ1, through the larger loop we can sequentially measure the CPR of each JoFET in the G-SQUID. To this aim, the reference JoFET, J W, is biased at full accumulation (V W G = /C0 1:5 V) such that the necessary condition IW C ≫ IN1 C , IN2 C is fulﬁlled37,44. To measure the JN1 ( JN2 )C P R ,w ea p p l y V N1 G = /C0 0:6V (V N2 G = /C0 1:5 V) while JN2 ( JN1 ) is pinched off by setting V N2 G =1 :5V (V N1 G =1 :5V ) .T h eo n - s t a t ev o l t a g e sV N1 G = /C0 0:6 V and V N2 G = /C0 1:5Va r e chosen to obtain equal amplitudes of the ﬁrst CPR harmonic. As we Fig. 1 | Double SQUID device and single-JoFET characteristics. a Schematic and scanning electron micrograph of the device. A Ge-based SQUID (G-SQUID), embedding JoFETsJ N1 and JN2, is connected in parallel to a wider JoFET (JW)u s e da sa reference Josephson junction for current-phase-relation (CPR) measurements. The aluminum arms are modeled by three inductances: L1, L2 and LW. b– d Current- biased measurements of the JoFET characteristics forJW, JN1,a n dJN2,r e s p e c t i v e l y .I n each panel, the measured source-drain voltage is plotted as a function of gate voltage,V i G (i = W, N1, N2), and source-drain current biasIDC. e (resp. f), Left: critical current, IC, as a function of magnetic ﬂux Φ1 through the large loop in (a). The reference JoFETJW is biased to strong accumulation (V W G = /C0 1:5 V), JN2 (resp. JN1)i s pinched off and V N1 G = /C0 0:6 V (resp. V N2 G = /C0 1:5 V). The IC oscillations are a direct measurement of JN2 (resp. JN1) CPR. Right: Fast Fourier transform (FFT) of the CPR on the left, calculated over 15Φ0. Black dashed line: calculated CPR and FFT based on the circuit model in a with parameters obtained from aﬁt of the data in Fig. 2a. Article https://doi.org/10.1038/s41467-025-56245-7 Nature Communications| (2025) 16:1010 2 shall see below, operating the G-SQUID at these gate voltages enables the suppression of the ﬁrst harmonic by ﬂux-induced destructive interference, hence leaving a dominant sin(2φ)c o m p o n e n t . T h em e a s u r e dC P R sa r es h o w ni nF i g .1e, f together with the respective Fourier transforms. A total of 15ﬂux periods were measured in order to ensure suf ﬁcient resolution of the harmonics in Fourier space. Each reportedIC data point represents the median value obtained from 10 measurements, with the light green area indicating ±1 standard deviation. Both CPRs are clearly skewed and we distinguish up to ﬁve harmonics. This multi-harmonicity indicates a high transparency of the superconducting contacts, which is consistent with earlier observations with similar devices 30,32. Yet, as we shall discuss below, the higher har- monics, especially the fourth and ﬁfth ones, have largely enhanced amplitudes due to theﬁnite inductance of the aluminum arms. sin(2φ) Josephson element With the two JoFETs JN1 and JN2 independently characterized, we now turn to the study of the G-SQUID CPR, once again using JW as a refer- ence ( V W G = /C0 1:5 V). With the ultimate goal to engineer a sin(2 φ) Josephson element, we symmetrize the G-SQUID by applyingV N1 G = /C0 0:6Va n d V N2 G = /C0 1:5 V. These two values have been chosen such that the critical current amplitudes of the two junctions are equal. As it will be shown later, it also implies very similar harmonic contents, corro- borating the reproducible electronic properties of our junctions. Figure 2a shows a measurement of the G-SQUID critical current as a function ofΦ 1 and Φ2, the latter being the magneticﬂux through the G-SQUID loop. This is the most important data set. We ﬁt the entire two-dimensional plot to the circuit model shown in Fig.1a, withﬁxed inductances of the Al arms (see Supplementary Note 1) and 12 free parameters accounting for the amplitudes of theﬁrst four harmonics of the three JoFETs (see Supplementary Note 2 for details). Interestingly, theﬁt yields negligible amplitudes for all the fourth-order harmonics, implying that only three harmonics per JoFET are sufﬁcient to reproduce the data. This outcome apparently contrasts with the experimental data in Fig.1e, f, where up to ﬁve harmonics can be dis- tinguished for bothJ N1 and JN2. The presence of higher harmonics arises from theﬁnite inductances of the Al arms42,45.W ee s t i m a t eL1 ≈ L2 ≈ 50 pH (i.e., 20 times smaller than the inductances of JN1 and JN2)a n d LW ≈ 210 pH. Even such relatively small inductances can signiﬁcantly enhance the harmonic amplitudes. The enhancement becomes proportionally larger as the harmonic order increases 42,45.T h i sa s p e c ti sf u l l yc a p t u r e d by our circuit model. Indeed, t he black dashed lines in Fig. 1e, f are calculated using the parameters extracted from theﬁto fF i g .2a. The experimental data are accurately reproduced despite the fact that only three harmonics effectively contribute to the CPR of each JoFET. In Fig. 1e, f the arm inductances amplify the 1st, 2nd, and 3rd harmonics of J N1 (JN2) by 7(6)%, 22(30)%, and 250(244)%, respectively, and lead to the emergence of a 4th and a 5th harmonic 46.H e n c eo u r analysis reveals the importance of including even small contributions of arm inductances in order to avoid a crude overestimation of the harmonic amplitudes. Finally, we remark that arm inductances can also induce a phase shift in the CPR 45. Moreover, we notice that the harmonic content of the two junc- tions obtained from our analysis is very similar (Supplementary Tab. 1) despite the different gate-voltage dependence of their critical current (Supplementary Fig. 10). This high level of symmetry indicates a good reproducibility of the fabrication process. In addition, we argue that the very large number of channels in our junctions should make the CPR less sensitive to the distribution of transmission coefﬁcients. This can be a clear advantage over junctions based on one-dimensional semiconductor nanowires. The magnetic-ﬂux dependence of the G-SQUID CPR for three values of Φ 2 is shown in Fig. 2 together with the Fourier decomposi- tion. AtΦ2 =0( F i g .2b, e), we expect the G-SQUID CPR to be the sum of the JN1 and JN2 CPRs shown in Fig. 1e, f. Instead, we observe a CPR containing about ten harmonics. Moreover, all harmonics beyond the ﬁrst one exhibit amplitudes clearly larger than expected from a simple addition. Fully captured by our circuit model (see dashed lines in Fig. 2b, e), this ﬁnding is mostly a consequence of the moderate ratio between the critical current of the reference junctionJ W and the one of the G-SQUID44, i.e., IW C =IG/C0 SQUID C /C25 5a t Φ2 =0 . At Φ2 = −Φ0/4 (Fig.2c, f), JN1 and JN2 are dephased byπ/2, resulting in a destructive interference between even harmonics. The 2nd and 4th harmonics are consequently suppressed while the 1st and 3rd preserve 2.5 3.0 3.5IC(μA) 2.8 3.2IC(μA) −2 −1 0 1 Φ/ Φ10 2.9 3.0IC(μA) −2 −1 0 1 Φ/ Φ10 −0.50 −0.25 0.00 0.25 0.50 Φ/ Φ20 IC(μA) 2.5 3.0 3.5ae 10 0 10 2 IC FFT(nA) 10 0 10 2 IC FFT(nA) 02468 1 0 1 2 (Φ /Φ )10 −1 10 0 10 2 IC FFT(nA) f g b c d

Fig. 2 | Frequency doubling of the G-SQUID CPR.The G-SQUID is symmetrized by setting V N1 G = /C0 1:5 V and V N2 G = /C0 0:6 V, which equalizes the amplitudes of theﬁrst harmonics in JN1 and JN2. JW is kept in strong accumulation (V W G = /C0 1:5 V). a Critical current as a function of the two compensated magneticﬂuxes, Φ1 and Φ2, threading the large superconducting loop and the G-SQUID, respectively.b–d G-SQUID CPR (IC vs Φ1)f o rΦ2 =0 , −Φ0/4, and −Φ0/2, i.e., at the line cuts denoted by green lines in (a). e–g The FFTs obtained from the CPRs in ( b–d), respectively, are calculated over 15Φ0. Black dashed lines inb–g are calculated CPRs (FFTs) based on the circuit model in Fig.1a with parameters obtained from aﬁto ft h ed a t ai na.I n d, following a suppression of the odd harmonics (clearly shown ind), we observe the doubling of the CPR frequency as expected for a sinð2φÞ Josephson element. Article https://doi.org/10.1038/s41467-025-56245-7 Nature Communications| (2025) 16:1010 3 the same amplitude. The resulting CPR is clearly less skewed than at Φ2 = 0. From our model, we conclude that the residual 2nd and 4th harmonics are again due to a moderate ratio IW C =ISQUID C . Increasing the JW critical current by a factor of ten would further suppress the 2nd harmonic by the same factor. At Φ2 = −Φ0/2 (Fig. 2d, g), a π phase shift induces a destructive interference between the odd harmonics ofJN1 and JN2 with a reduction of the G-SQUID critical current. Following the suppression of the 1st and 3rd harmonics, the 2nd harmonic becomes the dominant one resulting in the emergence of a Φ0/2 ﬂux periodicity in the CPR. In conclusion, at half ﬂux quantum, the G-SQUID behaves as a sin ð2φÞ Josephson element. Harmonic tuning and sin(2φ) purity Magnetic ﬂux tunability.F i g u r e2a shows how, in a symmetrized conﬁguration with balanced Josephson junctions, the magneticﬂux Φ2 can profoundly change the harmonic composition of the G-SQUID CPR with singularities atΦ2 =± Φ0/4 and Φ2 =± Φ0/2. In order to gain more insight into this magnetic- ﬂux control and to quantify the level of harmonic “distillation” at the singularity points, we show in Fig.3at h e complete Φ2 dependence of the ﬁrst four harmonics. The plotted amplitudes of these harmonics (colored dots) are extracted from Fig. 2a by performing a Fourier transform of the measured I C(Φ1)a t every Φ2 value. The corresponding uncertainties are represented by ±1σ-wide colored bands. These uncertainties are signiﬁcant and visible only when the harmonic amplitudes are below ~1 nA, which is always the case for the fourth harmonic. The overlaid black dashed lines represent the amplitudes of the ﬁrst four harmonics calculated using the circuit model of Fig. 1aw i t h model parameters obtained from the ﬁtting of the data in Fig. 2aa s previously discussed. The remarkable quantitative agreement over four orders of magnitude conﬁrms the validity of our circuit model. At Φ 2 =± Φ0/2, the ﬁrst and third harmonics exhibit cusp-like dips where their amplitude is suppressed by two orders of magnitude, while the second and fourth harmonics simultaneously attain local maxima. In particular, the amplitude of the second harmonic atΦ 2 =± Φ0/2 is almost identical to the one atΦ2 =0 .W ec a nd eﬁne a“purity”level of the sinð2φÞ CPR as the ratio between the second harmonic amplitude,A2φ,a n dt h e sum of the four harmonic amplitudes, ΣnAnφ.I t s ﬂux dependence is displayed in Fig.3b. At Φ2 =± Φ0/2, we reach a second-harmonic purity of 95.2(2.4)%, largely exceeding the state-of-the-art33,35.W en o t et h a t such a high purity level was reached owing to the very close electronic properties of the two S–Sm SiGe/Ge junctions in the G-SQUID. We also argue that, with these very same junctions, further circuit optimization should have resulted in an even higher value of the measured purity level. In fact, based on the circuit model of Fig. 1a, our measurement underestimates the purity value due to the relatively smallI W C =IG/C0 SQUID C ratio. With a wider reference junction yielding a ten times larger IW C =IG/C0 SQUID C ratio, the same measurement should have given a purity level of 96.3%, much closer to the intrinsic value expected for exactly the same G-SQUID. Additionally, the intrinsic purity level could be increased by acting on other circuit parameters. To illustrate that, we begin by noting that ourﬁto fF i g .2ar e v e a l sa slightly imperfect symmetry of the G-SQUID, quantiﬁed by a 0.7% dis- crepancy between the amplitudes of theﬁrst harmonics ofJ N1 andJN2 (see Supplementary Note 2). Reducing this discrepancy to less than 0.14% is, in principle, possible through aﬁne adjustment of the gate voltages. This would increase the weight of the even harmonics to more than 99%, with the fourth harmonic accounting for a few percent of the total weight. We note that both the second and the fourth harmonics contribute to parity protection since they reﬂect the simultaneous transport of even num- bers of Cooper pairs. As discusse d before, the fourth harmonic is essentially absent in the CPR of the individual JoFETs, and it originates from the non-negligible inductance for G-SQUID arms. Based on our circuit model, we estimate that in a properly symmetrized G-SQUID, reducing the arm inductances L 1 and L2 from 51 to 46 would largely suppress the fourth harmonic resulting in sinð2φÞ purity above 99%. Finally, at Φ2 =± Φ0/4 the contribution of the second harmonic goes down to 2.6(0.1)%. This suppression of the second harmonic could be further pushed below 1% by changing the gate conﬁguration (see Supplementary Note 5). Gate voltage tunability . So far, we have demonstrated the ﬂux dependence of the harmonic content in the CPR of the symmetrized G-SQUID. We will now address its gate voltage tunability while keeping the ﬂux Φ 1 ﬁxed at Φ0/2. In this experiment, the gate voltage ofJN2 is ﬁxed at V N2 G = /C0 1:5V , while the gate voltage of JN1 is swept. The amplitude of the ﬁrst har- monic exhibits a strong suppression aroundV N1 G = /C0 0:7V( s e eF i g .4a), which corresponds to the symmetric condition. (In comparison to Fig. 3, here, this condition is achieved for a slightly different gate- voltage setting, due to a small electrostatic charge reconﬁguration that occurred between the two measurement runs, which were taken sev- eral weeks apart.) Since the second-harmonic shows only a weak dependence on V N1 G , the sin ð2φÞ purity attains its maximum at V N1 G = /C0 0:7V , a s s h o w n i n F i g .4b. The maximal purity level is robust against V N1 G deviations of up to 50 mV from the symmetry point. This robustness is largely reduced when operating the JoFETsJN1 and JN2 in a regime of moderate accumulation where their critical current is more sensitive to gate voltage variations (see Supplementary Note 4). 10 0 10 1 10 2 10 3 Harmonic Amplitude (nA) sin( )φ sin(2 )φ sin(3 )φ sin(4 )φa −0.5 0.0 0.5 Φ/ Φ20 0.0 0.2 0.4 0.6 0.8 1.0 −0.52−0.48 0.8 0.9 model b Fig. 3 | Flux modulation of the G-SQUID harmonics. aAmplitudes of theﬁrst four harmonics in the G-SQUID CPR as a function of Φ2.T h eﬁrst harmonic vanishes at Φ0/2, while the second one vanishes atΦ0/4. b Flux dependence of sinð2φÞ purity, which is deﬁned as the ratio between the amplitude of the second harmonic, A2φ, and the sum of all four harmonic amplitudes,ΣnAnφ. The sinð2φÞ purity has a sharp maximum at Φ0/2 where it reaches 95.2(2.4)%. Inset: close-up around the max- i m u m .C o l o r e db a n d si na and b represent the ± σ standard deviation originating from the experimental uncertainty on the CPR data points. Black dashed lines in a and b represent the harmonic amplitudes calculated from our circuit model using the ﬁt parameters from Fig. 2a. Article https://doi.org/10.1038/s41467-025-56245-7 Nature Communications| (2025) 16:1010 4 Discussion Our study provides important insight to devise practical realizations of parity-protected superconducting qubits based on the demonstrated sinð2φÞ Josephson element 16,17. By leveraging a recently developed S–Sm platform based on SiGe/Ge/SiGe heterostructures, we obtained highly symmetric, multi-harmonic Josephson junctions leading to an unprecedented second-harmonic purity as high as 95%. Such a high purity should allow for enhanced protection against charge relaxation protection. We expect an improvement of two orders of magnitudes with respect to the nanowire-based cosð2φÞ qubit reported by Larsen et al. 16 or to typical Transmon 7 and Fluxonium8 qubits (see Supple- mentary Note 9). In principle, parity protection enhances the qubit lifetime at the cost of rendering the qubit resilient to external control pulses. Therefore, qubit operation would require temporarily exiting the protected regime, e.g., by controlling the ratio between the second and ﬁrst harmonics. We show here that this ratio can be tuned by a magnetic ﬂux or a gate voltage. A ﬂux shift δΦ 2 = 0.025Φ0 from Φ2 = Φ0/2 lowers the second-harmonic content from 95% to 60%. In Fig. 4, we show that starting from a symmetric biasing of the G-SQUID, a gate voltage shift δVG = 100 mV in one of the two JoFETs lowers the sinð2φÞ purity by about 20%. An even smaller swing, δVG =2 0m V , would result in a similar purity reduction in a regime of moderate hole accumulation (see Supplementary Fig. 6e). Suchﬂux and gate-voltage shifts are experimentally accessible with electrical control pulses on the typical time scale (~10 ns) of single-qubit operations. Building on the S – Sm platform based on the SiGe/Ge/SiGe het- erostructure presented in this report, future experimental efforts should integrate the sin ð2φÞ element into a circuit quantum electro- dynamics (cQED) architecture to investigate its potential parity pro- tection. To achieve this, the challenge posed by the relatively large dielectric losses in the SiGe substrate 47,48 could be addressed by fab- ricating the superconducting microwave circuitry on a separate low- loss chip and subsequently employingﬂip-chip assembly to connect it with the semiconductor-based Josephson elements 49. In conclusion, we have reported an experimental realization of a sinð2φÞ Josephson element leveraging the intrinsic multi-harmonicity and gate tunability of SiGe-based JoFETs. The CPRs of these JoFETs can be accurately described by the sum of only three harmonics albeit higher harmonics are measured due to the relatively modest ratio between I ref C and IW C . The almost complete suppression of the odd harmonics at half a ﬂux quantum leaves a sinð2φÞ CPR with a remark- ably high purity level exceeding 95%. We argued that even higher values beyond 99% could be reached through further circuit optimi- zation. All data analysis was based on a relatively simple but accurate circuit model taking into account the non-sinusoidal CPRs of the JoFETs as well as the inductances of the superconducting arms. Data availability All of the data used to produce theﬁgures in this paper and to support our analysis and conclusions are available at https://zenodo.org/ records/14169434. This repository includes the original data, jupyter notebooks for data analysis and ﬁgure plotting. Additional data are available upon request to the corresponding author. References 1 . B r a v y i ,S . ,D i a l ,O . ,G a m b e t t a ,J .M . ,G i l ,D .&N a z a r i o ,Z .T h ef u t u r eo f quantum computing with superconducting qubits.J. Appl. Phys. 132, 160902 (2022). 2. 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Lett. 112, 167001 (2014). 10 −1 10 0 10 1 10 2 Harmonic Amplitude (nA) sin( )φ sin(2 )φ sin(3 )φ sin(4 )φ −1.0 −0.5 0.0 VG N1(V) 0.2 0.4 0.6 0.8 1.0 −0.8 −0.6 0.8 0.9 b a Fig. 4 | Gate modulation of the G-SQUID harmonics. aAmplitudes of theﬁrst four harmonics in the G-SQUID CPR as a function ofV N1 G ,w h i l eV N2 G is ﬁxed at −1.5 V and Φ1 is set to Φ0/2. The symmetric SQUID regime is reached when V N2 G approaches −0.7 V, coinciding with the maximum suppression of the ﬁrst harmonic. b Gate dependence of the sinð2φÞ purity, deﬁned as the ratio of the second harmonic amplitude, A2φ, to the sum of the amplitudes of all four harmonics, ΣnAnφ.T h e sinð2φÞ purity reaches its maximum near V N2 G = /C0 0:7 V, as shown in the inset. Colored bands ina and b represent the ±σ standard deviation originating from the experimental uncertainty on the CPR data points. Article https://doi.org/10.1038/s41467-025-56245-7 Nature Communications| (2025) 16:1010 5 1 5 . 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From our model we conclude that the 4 th harmonic is completely due to branch inductances. 47. Sagi, O. et al. A gate tunable transmon qubit in planar Ge. Nat. Commun. 15, 6400 (2024). 48. Kiyooka, E. et al. Gatemon qubit on a germanium quantum-well heterostructure.Nano Letters 25,5 6 2– 568 (2025). 49. Hinderling, M. et al. Direct microwave spectroscopy of andreev bound states in planar Ge josephson junctions.PRX Quantum 5, 030357 (2024). Acknowledgements This work has been supported by the ANR project SUNISIDEuP (ANR-19- CE47-0010), the PEPR projects ROBUSTSUPERQ (ANR-22-PETQ-0003), and PRESQUILE (ANR-22-PETQ-0002), the ERC Grant e-See (Grant No. 758385), and the Grenoble LaBEX LANEF. We thank the PTA (CEA-Gre- noble) for the nanofabrication. We thank J. Renard and S. Messelot for discussions. Author contributions A.L. performed the experimental measurements. C.T., Z.S.M., E.K., F.G., and J.-L.T. developed the fabrication recipe. F.G. and J.-L.T. fabricated the device. J.-M.H. designed and grew the semiconductor hetero- structure. E.K. characterized the heterostructure. A.L. and E.D. per- formed the data analysis. A.L., B.B., V.S., S.Z., R.M., E.D., S.D.F., and F.L. wrote the paper with inputs from all co-authors. F.L. and S.D.F. super- vised the project. Competing interests The authors declare no competing interests. Additional information Supplementary informationThe online version contains supplementary material available at https://doi.org/10.1038/s41467-025-56245-7. Correspondenceand requests for materials should be addressed to Axel Leblanc, Silvano De Franceschi or François Leﬂoch. Peer review informationNature Communicationsthanks Federico Pao- lucci, and the other, anonymous, reviewers for their contribution to the peer review of this work. A peer review ﬁle is available. 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