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Ko-Fan Huang, Yuval Ronen, Régis Mélin, Denis Feinberg, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Philip Kim

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[Evidence for 4e charge of Cooper quartets in a biased multi-terminal graphene-based Josephson junction](https://mdr.nims.go.jp/datasets/79e598d0-1686-46ec-a131-31be2067267e)

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Evidence for 4e charge of Cooper quartets in a biased multi-terminal graphene-based Josephson junctionARTICLEEvidence for 4e charge of Cooper quartets in abiased multi-terminal graphene-based JosephsonjunctionKo-Fan Huang1,6, Yuval Ronen 1,6, Régis Mélin2, Denis Feinberg2, Kenji Watanabe 3, Takashi Taniguchi 4 &Philip Kim 1,5✉In a Josephson junction (JJ) at zero bias, Cooper pairs are transported between two super-conducting contacts via the Andreev bound states (ABSs) formed in the Josephson channel.Extending JJs to multiple superconducting contacts, the ABSs in the Josephson channel cancoherently hybridize Cooper pairs among different superconducting electrodes. Biasing three-terminal JJs with antisymmetric voltages, for example, results in a direct current (DC) ofCooper quartet (CQ), which involves a four-fermion entanglement. Here, we report half a fluxperiodicity in the interference of CQ formed in graphene based multi-terminal (MT) JJs with amagnetic flux loop. We observe that the quartet differential conductance associated withsupercurrent exhibits magneto-oscillations associated with a charge of 4e, thereby presentingevidence for interference between different CQ processes. The CQ critical current showsnon-monotonic bias dependent behavior, which can be modeled by transitions betweenFloquet-ABSs. Our experimental observation for voltage-tunable non-equilibrium CQ-ABS influx-loop-JJs significantly extends our understanding of MT-JJs, enabling future design oftopologically unique ABS spectrum.https://doi.org/10.1038/s41467-022-30732-7 OPEN1 Department of Physics, Harvard University, Cambridge, MA 02138, USA. 2 Université Grenoble—Alpes, CNRS, Grenoble INP, Institut NEEL, 38000Grenoble, France. 3 Research Center for Functional Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 4 InternationalCenter for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 5 John A. Paulson School ofEngineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA. 6These authors contributed equally: Ko-Fan Huang, Yuval Ronen.✉email: pkim@physics.harvard.eduNATURE COMMUNICATIONS |         (2022) 13:3032 | https://doi.org/10.1038/s41467-022-30732-7 | www.nature.com/naturecommunications 11234567890():,;http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-30732-7&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-30732-7&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-30732-7&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-30732-7&domain=pdfhttp://orcid.org/0000-0002-2427-2591http://orcid.org/0000-0002-2427-2591http://orcid.org/0000-0002-2427-2591http://orcid.org/0000-0002-2427-2591http://orcid.org/0000-0002-2427-2591http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-8255-0086http://orcid.org/0000-0002-8255-0086http://orcid.org/0000-0002-8255-0086http://orcid.org/0000-0002-8255-0086http://orcid.org/0000-0002-8255-0086mailto:pkim@physics.harvard.eduwww.nature.com/naturecommunicationswww.nature.com/naturecommunicationsAt a normal (N)-superconductor (S) boundary, below thesuperconducting gap Δ, current is induced via Andreevreflection (AR)1, i.e., an electron impinging on S binds toanother electron near the interface, transmitting a Cooper pairinto the S region while a hole is reflected. By constructing twosuch boundaries one creates an SNS Josephson junction (JJ),which can be viewed as an electronic analog of the optical Fabry-Perot interferometer: Each boundary acts as an AR mirror andresonances are formed in the junction. In this case, these reso-nances coherently superimpose electron and hole waves, formingthe so-called Andreev bound states (ABSs)2–4. Each AR picks upthe phase of the corresponding superconductor; therefore, theABS wave-functions and energies depend on the phase differenceφ between the two superconductors. Each populated ABS αcontributes a current, derived from its ABS energy EαðφÞ withrespect to φ, to the total Josephson current. Recently, muchattention has been paid both in theoretical predictions5–7 and inexperiments8–12 to multi-terminal Josephson junctions (MT-JJs),where a single metallic region bridges three or more super-conductors. With ARs taking place at each SN interface, the SNSphysics is generalized in several aspects. First, the equilibriumABS spectrum of a multi-terminal JJ depends on multiple phasedifferences φi, where φi is the phase of ith S electrodes connectedto the junction13,14. The ABS energy appears as a contour in amulti-dimensional voltage space11,12. The high dimension phasespace spanned by φi’s offers the prospect of engineering artificialhigh-dimensional crystal band structures with topologicalproperties15–19. Second, multi-dimensional current-voltagecharacteristics may present a complex subgap structure due tolocal (between two terminals) or non-local (among multipleterminals) multiple Andreev reflections (MARs)9–12,20. Third,MT-JJs allow DC transport of multiple entangled Cooper pairsfor commensurate combinations of applied voltages. For instance,in a three-terminal junction with two leads biased with anti-symmetric bias scheme at (V, −V), a Cooper pair from thegrounded terminal is split into two quasiparticles via crossedAR21–23. The two quasiparticles propagate toward two distinctterminals and then recombine with the ones originated from theother pair splitting, forming two entangled Cooper pairs—theCooper quartet (CQ) within the junction6,7 (see Fig. 1a). We notethat the crossed AR is a local AR at one of the SC electrodes asopposed to CAR across a SC metal.Since a CQ minimally requires four coherent ARs, its under-lying mechanism is distinct from a simple extrinsic lockingbetween two separated JJs biased at opposite voltages. In theexternally coupled JJs with the antisymmetric bias condition, thiscan produce an alternating current (AC) Josephson oscillationswith the same frequency. Synchronization of these oscillationscan occur by photon exchange between the JJs via a classicalimpedance24. This view of mode-locked JJs, however, only con-siders external coupling between local AR processes. For MT-JJswith low energy ABS in the weak link, new possibility arises foran intrinsic synchronization of asymmetrically biased JJs via non-local AR processes6,7, leading to the entangled CQ spreading overmultiple JJs.MT-JJs with conducting weak links have been fabricated in 2Dmetallic8 and 1D semiconducting9 channels as well as ingraphene10. While non-local supercurrent was probed in MT-JJsby measuring cross-correlated current noise9, a direct experi-mental observation of the presence of phase coherent entangledCQs has yet to be realized. In this work, we employ a magneticflux loop coupled to the MT-JJs to modulate the junction prop-erties. Using both bias voltage and threaded magnetic flux, wecontrol the CQ dynamics, including coherent CQ-ABS andinterference between different CQ processes. As the bias increa-ses, we find non-monotonic behavior of the CQ critical current asa function of bias, which can be interpreted within a simplifiedmodel by transitions between Floquet CQ-ABSs generated byintrinsic synchronizing of the entangled CQs25–27.Results and discussionsCharacterization of MT-JJ in the Josephson regime. Along withthe ability of controlling the number of conducting channels, lowsuperconducting contact resistance and weak back-scattering28–30make graphene an ideal choice for exploring MT-ABS physics.Utilizing the tunability of graphene chemical potential, one canmodulate the coupling strength at each contact, thereby engi-neering the ABS spectrum. Our graphene-based MT-JJs use Ti/Alas the superconducting contacts, where Al is chosen owing to itslarge superconducting coherence length (~1 μm). A three-terminal JJ (four contacts including a superconducting loop) isfabricated on the graphene-hBN-SiO2 structure as shown inFig. 1a (additional fabrication information can be found in the“Method” section).All measurements were performed at 300 mK. Before weconduct the MT-JJ measurement, we first characterized ourdevice with a two-terminal measurement as the S-loop imple-ments a superconducting quantum interference device (SQUID)geometry. For this measurement, we applied a bias voltage V tothe loop via two series connected RC filters. The output current Iis measured at S2 while S1 is floating. Figure 1c shows an I-Vmeasurement curve of the junction. In the smaller bias regime,supercurrent flows in the junction and the bias voltage drops areonly on the series connected resistors RRC (200Ω each) in thefilters. As the current exceeds the critical current Ic of the SQUID,the slope of I-V curve changes suddenly at the correspondingapplied voltage Vc. Since the bias voltage is distributed amongtwo filter resistors and the normal junction resistance, the criticalcurrent can be obtained from Ic ¼ Vc=2RRC . Upon applying themagnetic field B, Ic is modulated and exhibits SQUID-like patternas a result of the two interfering superconducting paths in theloop (blue and red dashed lines in Fig. 1b). Figure 1d shows thedifferential conductance (G ¼ dI=dV) as a function of biasvoltage and magnetic field. The higher conductance area near thezero-bias regime (central part) is the supercurrent region and itsedges mark the value of Ic. As the magnetic field is swept, Ic ismodulated with a periodicity of δB ¼ 145 μT, corresponding tothe unit flux quantum Φ0 ¼ h=2e for an enclosed area ofA ¼ 14:2 μm2, matching our device loop size (including the areaincrease due to London penetration depth). An additional lowerfrequency (δBF ¼ 3 mT) originated from the Fraunhofer oscilla-tions is observed, corresponding to an area of 0.69 μm2, whichagrees with the junction dimensions. We find that the strength ofthe critical current can also be tuned according to the graphenecarrier density via a back-gate voltage Vbg . As shown in Fig. 1e, Icdecreases monotonically as Vbg approaches the charge neutralitypoint of graphene located at Vbg � �32 V. Reduction of Ic closeto the Dirac point is expected due to the decreasing number ofABS carrying current in the graphene channels31.Cooper quartet. With reconfiguration of the external circuitry,our device can serve as a MT-JJ where the common N-regiongraphene channel is proximitized. MT-JJ with magnetic flux loopswas studied theoretically and experimentally in bi-SQUIDdevices32–34, where the equilibrium (i.e., no potential differencebetween the junctions) ABS spectrum was investigated. Our four-contact device geometry with gate-tunable graphene weak linkallows us to study biased MT-JJs in the non-equilibrium regime,where the non-local CQ formation can be investigated35. More-over, by threading a flux through the device loop we aim tomodulate the CQ-ABS spectrum. Figure 2a shows theARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-30732-72 NATURE COMMUNICATIONS |         (2022) 13:3032 | https://doi.org/10.1038/s41467-022-30732-7 | www.nature.com/naturecommunicationswww.nature.com/naturecommunicationsmeasurement scheme adopted in this study for phase sensitivequartet detection. We apply DC bias voltages V1 and V2 to S1 andS2, respectively, and a small AC bias voltage δV to the loopelectrodes S0a and S0b. At given bias voltages, we measure the ACcurrent contributions δI1 and δI2 flowing to S1 and S2, respec-tively. Voltages V1 and V2 are applied to the total circuitincluding RRC , which is about 100 times larger than the actualvoltage applied at the junction (see circuit in Fig. S1).Figure 2b shows the differential conductance measured at S1(G1 ¼ δI1=δV) as a function of the two DC bias voltages V1 andV2 with Vbg ¼ 40 V. We identify four high conductance regions(marked by four white dashed lines crossing at the origin), whichcorrespond to four different supercurrents. For instance, when S2and S0 are equipotential along V2 = 0, a Josephson supercurrentflows between these two contacts carried by a Cooper pair-ABS.Subfigures (i), (ii) and (iii) illustrate these local supercurrentsbetween different pairs of S-contacts. The critical values of thesupercurrents can be extracted from the widths of the signals,which are 0.47, 0.42, 0.38 μA, respectively. Similar data can beobtained for differential conductance G2 ¼ δI2=δV measured atS2 (see Section 2 in the Supplementary Information).In addition to the two-terminal Josephson currents (i)–(iii), weobserve another supercurrent signal along the V1 ¼ �V2 line, asshown in Fig. 2b. This line originates from the sharp black lines,which define the 2-terminal critical current contour (CCC). Toascertain the intrinsic nature of this signal and that it is due toquartets, let us first remark that no clear MARs are observed inthis sample in the bias voltage range where we observe a quartetsignal. Indeed, given the low value of the junction voltage, thoseMARs, whether local or non-local, would have very high orders.In a non-ballistic graphene with interface scattering, such high-order MARs are unlikely to take place, in contrast to clean InAs2DEG samples such as those in ref. 11. In the work of ref. 11,where the critical currents are high, the situation is very different:several bright local MAR lines were observed, but no supercurrentwas observed along the V1 ¼ �V2 line beyond the CCC. This isnot surprising because quartets require four ARs, two local andtwo non-local processes, and are easily masked by bright localMARs. Notice that the same conditions (low voltage compared tothe gap, no MARs or very weak ones) were met in refs. 8,9 and aquartet line was indeed observed.We labeled the V1 ¼ �V2 line as (iv) Quartet and it signals theexistence of non-equilibrium CQ-ABSs within the junction,despite the fact that all contacts are at different chemicalpotentials. In this regime all 2e Josephson currents taking placebetween each pair of terminals are AC. On the contrary, in thisconfiguration, as depicted in Fig. 1a, two Cooper pairs from twoS-contacts (S1 and S2) are entangled into a four-electron state viatwo local ARs and two crossed ARs at the middle S-contact ðS0Þ toform CQ-ABS5–9. The shape and width of the anomaly is verysimilar to that of an ordinary Josephson current, and it allows todefine a “quartet critical current” Iqc. Remarkably, in this regimewhere the local DC-Josephson currents disappear, the CQ-ABSsform only when the junction is biased antisymmetrically and theycarry non-local DC supercurrent flowing among all terminalssimultaneously. The corresponding bias condition V1 þ V2 ¼ 0satisfies the energy conservation for the CQ DC current, wherecorrelated Cooper pairs originating from S1 and S2 aresimultaneously transmitted into S0. Notice that this necessarycondition does not tell anything about the microscopic mechan-ism for quartets. Our experiment precisely helps elucidating thismechanism, by using the tool of a magnetic flux and byinvestigating the periodicity and the voltage dependence of thefield modulation.Two types of Cooper quartet processes. Similar quartet super-current signatures were inferred previously in three-terminal JJmade from diffusive metal8 and 1D nanowires9. The novelty inab-40-30-20-100102030400.70.91.11.31.50V bg (V)0.1-0.5 1V (mV)0.5G/GN1 μmS0aS0bS2S121VRRCCRRCCI -1.5-1-0.500.511.50.911.021.141.25G/GNB (mT)0-0.5 0.5 1V (mV)Vbg= 0V+V-Vhe heDOS DOSEnergy EnergyS1 S2S0 GG2e2e-1 -0.5 1V (mV)-0.6-0.300.30.6 Ic-Ic0 0.5I (μA)Vc-VccdeVbg= 30Vn=1 n=2n=4n=3Fig. 1 Illustration of quartet formation and a single source voltage bias characterization of four-terminal Josephson junction including a loop.a Schematic illustration of the three-terminal quartet process with the Andreev reflection picture. The middle superconductor S0 is grounded while theother two superconductors are biased at +V, −V, respectively. The two entangled Cooper pairs (with red and blue electrons) are formed in S0 through twolocal Andreev reflections and two crossed Andreev reflections. b False color scanning electron microscopy (SEM) image of the device with measurementconfigurations. Graphene (purple) is top-contacted by Ti/Al superconducting electrodes (blue) and the electrode separations typically are 80–100 nm.Here we split S0 in a into to two contacts S0a; S0b connected by a loop, c I� V curve of the device from the measurement configuration in b Ic is the criticalcurrent and the corresponding voltage value is labeled as Vc (the blue dots). d Magnetic field dependence, dI/dV as a function of the bias voltage andmagnetic field. Bright region (high conductance) is the supercurrent and the edge corresponds to the value of critical current, which is modulated by themagnetic field. The SQUID-like pattern indicates the interference between two supercurrent paths (red and blue dashed lines in b). The periodicity of thefast oscillation (white dashed curve) corresponds to the loop area and the slow oscillation (yellow dashed curve) is the first lobe of Fraunhofer pattern.e Gate dependence of the supercurrent, dI/dV as a function of the bias voltage and global back-gate voltage Vbg. The critical current reaches the minimumas graphene is tuned to the Dirac point near Vbg = −32 V.NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-30732-7 ARTICLENATURE COMMUNICATIONS |         (2022) 13:3032 | https://doi.org/10.1038/s41467-022-30732-7 | www.nature.com/naturecommunications 3www.nature.com/naturecommunicationswww.nature.com/naturecommunicationsour four-terminal JJ device is the study of the nontrivial biasvoltage dependence, and the presence of a magnetic flux loop,enabling direct probing of the CQ-ABS coherence via the periodicdependence of the critical current with magnetic field. The leftpanels of Fig. 3a, b show the quartet differential conductancemeasured at S2 (i.e., G2 along the quartet line V1 ¼ �V2) as afunction of the magnetic flux Φ ¼ B � A measured at differentback-gate voltage Vbg . The quartet differential conductance Gi¼1;2probes the quartet critical current Iqc (see Section 5 in the Sup-plementary Information). As a function of Φ, clear oscillations ofGi are observed, demonstrating periodic modulation of IqcðΦÞ dueto phase coherence of the CQ-ABS. By taking the Fouriertransform of GðΦÞ (the right panel of Fig. 3a, b), we find twomajor periodicities Φ0=2 and Φ0, where Φ0 ¼ h=2e. The relativestrength of the periodicities is tuned non-monotonically, sinceVbg modifies the number of channels in graphene as well as thecoupling of S-electrodes, which modifies the ABS spectrum.In particular, at Vbg ¼ 25 V (Fig. 3a), Iqc exhibits a prominentcontribution from Φ0=2-periodicity, which, as we show below,provides direct evidence for the charge 4e associated with theCQ-ABS.At first sight, the observation of the two periodicities tuned bythe gate voltage resembles the SQUID oscillation in Fig. 1d, wherethe Φ0=2 oscillation would be viewed as the second harmonic ofthe fundamental quantum flux periodicity. However, themagneto-oscillation here in Fig. 3a, b cannot be related to usualDC-SQUID harmonics since, as stressed above, we do not operatein the Josephson regime but well beyond, i.e., in a range whereAC Cooper pair Josephson currents flow between each pair ofterminals, rather than DC ones. Only the junction between S0aand S0b is equipotential but the current is not measured throughthis junction. Furthermore, as opposed to the 2-terminal case, in aMT-JJ, quasi-particle current and quartet DC current flowsimultaneously due to the inequivalent chemical potential of theaV1 (mV)(iv) Quartetb0.42 0.72V2 (mV)1.01.5(i) S0-S1V0=0V1=0(ii) S0-S2V0=0V2=0(iii) S1-S2VV(iv) Quartet:      S0-S1S2V-V(iii)S 1-S 2(i)S0 -S1(ii)S0-S2S1S2 δVV0=0S0aS0bV2RRCCδI2V1RRCCδI1 0.50.91.0G1/GNG1/GNFig. 2 Dual source voltage bias characterization for quartet detection. a Configuration for the quartet measurement. The loop S0 is grounded whilepotential of the first (S1) and second (S2) electrodes are controlled via DC voltages V1 and V2, respectively. An additional AC excitation δV is applied to theloop, and the AC current δI1 (δI2) through S1 (S2) is measured. b top panel: differential conductance G1 (= δI1=δV) measured at S1 as a function of the DCbias voltage V1 when V2 is tuned from 0.42 to 0.72mV. Bottom panel: color plot of G1 as a function of V1 and V2 (excluding the circuit resistance RRC), withVbg ¼ 40V. In total, there are four different supercurrents in the device. Inset (i)–(iii) show the local Josephson supercurrent between any pair of leads atthe same potential. Inset (iv) Quartet shows the non-local quartet supercurrent flowing among all three superconducting leads and the quartet signal in(V1, V2)-plane is the narrow yellow region along the −45 degree direction (where the red arrow points at).ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-30732-74 NATURE COMMUNICATIONS |         (2022) 13:3032 | https://doi.org/10.1038/s41467-022-30732-7 | www.nature.com/naturecommunicationswww.nature.com/naturecommunicationsSC contacts under the quartet condition. We delineate our signalfrom the quasi-particle current contribution by measuring anoscillatory differential conductance, following the Iqc Φð Þ variationalong the quartet bias condition (V1 ¼ �V2) on top of the quasi-particle current background (see Fig. S3c).The observed periodicities are therefore intrinsic to the quartetprocess itself. In a perturbative model approach expanded towardthe finite bias regime (see Section 5 in the SupplementaryInformation for detailed discussion), we consider the minimalnumber of four ARs required for quartet processes, taking placebetween four terminals instead of three. We find that themodulation of the periodicity is indeed associated withinterference of three different contributions to the CQ-ABS:two conventional quartets (3-terminal), denoted as Qa and Qb,and a novel process, the split-quartet (specific to four terminals),denoted as SQab. As shown in Fig. 3c, the two conventionalquartets, Qa and Qb, take place among S1, S2 and only oneelectrode of the S-loop. In these processes, the entangled Cooperpairs enter the loop either through S0a or S0b. Since every ARpicks up the phase of the superconducting contact, theseconventional quartet processes acquire phase factors eiðφ1þφ2Þ atS0a and eiðφ1þφ2þ4πΦ=Φ0Þ at S0b, where φ1 (φ2) is the phasedifference between S1 (S2) and S0. Note that the factor 4 in theexponent reflects that two Cooper pairs depart from the sameelectrode of the grounded loop. If there were only this type of3-terminal quartet process in the system, the phase factor atΦ=Φ0 ¼ 0 would become equivalent to that at Φ=Φ0 ¼ 1=2,leading to Φ0=2-periodicity in Iqc Φð Þ.While the conventional quartet process Qa and Qb describedabove is common with simple three-terminal JJs, the three-terminal JJ with a loop enables a new type of quartet, the split-quartet process SQab. As shown in Fig. 3d, two entangled Cooperpairs are spatially separated into the two electrodes of the loop,yielding a phase factor eiðφ1þφ2þ2πΦ=Φ0Þ. Interference of split andconventional quartet processes leads to Φ0-periodicity. Weobserve that the Fourier component associated with Φ periodicitystays constant while the Φ/2 component varies sensitively withthe gate voltage. Although a full understanding of thisdependence is beyond the scope of this work, it indicates thatthe strengths of the two different (i.e., conventional and split)quartet processes are determined by the relative contactcouplings, which are tunable via gating (see Section 5 in theSupplementary Information).Bias voltage dependence of quartet supercurrent. Mostimportantly, the quartet supercurrent can be modulated by thequartet voltage Vq, which is the actual voltage applied on thejunction along the V1 ¼ �V2 line. The variation of GðΦ;VqÞwith Φ or Vq is proportional to that of IqcðΦ;VqÞ along thequartet line since it is an increasing function of the critical current(see Section 5 in the Supplementary Information). Therefore, thisdifferential conductance measurement serves as a good indicatorto investigate the behavior of quartets as a function of magneticfield and the quartet voltage Vq. Figure 4a shows a 2D color plotof G1 (the quartet conductance measured at S1) as a function ofVq and the normalized magnetic flux Φ=Φ0 at a fixed gate voltageVbg =−5 V, where the quartet current is strong (see Fig. Sup-plementary 4 in the Supplementary Information). Both voltagescales (at the junction and at the circuit resistance) are presented,Fig. 3 Different types of quartet process. a Left panel shows the quartet differential conductance G2 (= δI2=δV) measured at S2 as a function of the fluxΦ ¼ B � A=Φ0, where Vbg= 25 V. In the right panel, discrete Fourier transform (DFT) analysis of the data shows prominent harmonics, a consequence of 8oscillations in the left panel. The quartet is biased at V1=−V2= 0.4 mV, where the DC 2e process and MARs are not effective. b for Vbg = 26–30 V. Theperiodicity evolves from half-flux quantum to one flux quantum as Vbg increases. c (Qa) (Qb) shows the conventional three-terminal quartet process withonly one out of the two loop contacts involved. Electron-hole conversion happens twice at the same contact of the loop (either S0a or S0b), resulting inperiodicity of half-flux quantum. d (SQab) shows the split-quartet process involving both contacts of the loop. With the odd parity of Cooper pairstransferred, the periodicity is one flux quantum.NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-30732-7 ARTICLENATURE COMMUNICATIONS |         (2022) 13:3032 | https://doi.org/10.1038/s41467-022-30732-7 | www.nature.com/naturecommunications 5www.nature.com/naturecommunicationswww.nature.com/naturecommunicationswith the latter equal to the quartet line in the zero-field map ofFig. 2b. This shows that this voltage region lies beyond theJosephson regime (black line in the bottom of Fig. 4a). At aconstant Vq, G1ðΦÞ exhibits oscillations corresponding to Iqc Φð Þwith periodicity Φ0=2 and Φ0 components as discussed in Fig. 3.Interestingly, we find that the oscillation period and phase ofIqc Φð Þ are also tunable as Vq varies. As shown in Fig. 4b, in thelow bias regime (Vq < 7:2 μV), Iqc Φð Þ shows predominantly theΦ0-periodic oscillation, in phase with the SQUID phase of equi-librium supercurrent. However, as Vq increases, Iqc Φð Þ oscillationbecomes predominantly Φ0=2-periodic near Vq � Vin � 7:2 μV.Above this critical bias voltage Vin, Iqc Φð Þ oscillation resumes theΦ0-period, but the phase is shifted by π compared toIqcðΦ;Vq < VinÞ: For this high bias quartet regime ðVq > VinÞ, theflux dependence of the quartet critical current is «inverted», i.e.,Iqc Φ ¼ 0ð Þ < Iqc Φ ¼ Φ0=2� �, suggesting that an unusual quartetbehavior occurs as we approach the high bias limit. Indeed, a naiveexpectation is that destructive interference would instead decreasethe quartet critical current for half-flux in the loop. The quartetbias condition (V1=−V2) is essential for observing this 0-π phaseππ π π πFig. 4 Quartet conductance and the Floquet spectrum as a function of Vq and magnetic flux. a Quartet conductance G1 measured along the quartet line in(Vq;Φ)-plane. Left y-axis is quartet voltage, taken across the junction. Right y-axis is the external applied voltage as shown in Fig. 2 along the yellow dottedline. The red dashed line traces the minimum conductance for �2:5 <� Φ=Φ0 < 0 and the red sphere represents the local minimum at Φ=Φ0 ¼ �2.b waterfall plot of G1ðΦÞ for Vq = 6.9–7.6 μV. It shows clear evolution of G1 from maxima to minima at integer values of flux. At the critical quartetVq � Vin � 7.2 μV, periodicity is Φ0=2 and for Vq> Vin, the quartet critical current is «inverted». c Zoom-in surface plot of G1(Vq;Φ) for �2:5 < Φ=Φ0 < 0.The winding of the red sphere (local minimum) is marked with the red dashed line, matching that in a. The gray spheres represent quartet conductance atdifferent values of Vq. d waterfall plot of G1(Vq) for Φ=Φ0 =−2~ 0. The local minimum V� presents a zig-zag pattern as flux is tuned. e When the quartetvoltage Vq is in the adiabatic limit, the adiabatic Andreev levels <EABS> depend only on one phase variable, the quartet phase φq. The minimum differencebetween the two levels is the Andreev gap Δ0 and a finite Vq creates resonant coupling between the two levels. f upper panel shows the energy of theFloquet states as a function of the quartet phase φq at different values of Vq. The corresponding quartet current Iq carried by these Floquet states is shownin the lower panel. The gray and red spheres mark the critical values of the quartet current Iqc ¼ Iqðφ�qÞ, matching the ones in c. In (III), the red spheredenotes Vq = V� when Iqc reaches a local minimum, reflecting an avoided crossing in the Floquet spectrum.ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-30732-76 NATURE COMMUNICATIONS |         (2022) 13:3032 | https://doi.org/10.1038/s41467-022-30732-7 | www.nature.com/naturecommunicationswww.nature.com/naturecommunicationschange as such phase change is absent at incommensurate biascondition (V1 ≠�V2) (see Fig. Supplementary 7 in the Supple-mentary Information).Furthermore, for a fixed Φ, G1ðVqÞ displays distinct non-monotonic behavior as Vq varies near Vin. As shown in Fig. 4c(zoom-in 3D map) and Fig. 4d (line-cuts for flux in the range[−2.5Φ0, 0]), G1ðVqÞ first decreases to yield a local minimum atVq � V� (represented by the red sphere) and then increases forVq > V�. We note that V� shifts in a zig-zag pattern in the (Vq,Φ)-plane centered at Vin. Particularly, V�ðΦÞ is the largest atinteger Φ0 and the smallest at half-integer Φ0, similar to theinverted quartet current in previous discussion. The non-monotonic variation of Iqc Vq� �and the inversion of the quartetcurrent flux dependence provide a clue to the dynamic behaviorof quartets in the non-equilibrium condition at a finite Vq. Wenote that, in order to reproduce such a behavior with aphenomenological extrinsic locking MT-JJ model24, one wouldneed to introduce an ad hoc anomaly of the circuit impedanceZ(ω) at the Josephson frequency corresponding to V, andmoreover assume that Z(ω) is modulated by the magnetic flux,which is unlikely. Another possible cause of a voltage-dependentanomaly are Fiske steps36, realized in a tunnel junction that iscoupled to a cavity resonance. To confine these modes, asufficiently large magnetic field is required, corresponding to thearea of the JJ. In our study, the anomaly appears in a much lowerfield range, corresponding to the larger SC loop area. Therefore,our observation cannot be associated with the Fiske steps. Finally,loop impedance effects may also be neglected at the measuredvoltage ranges (~µV, ~GHz) as it is estimated at 2.5 mΩ.Emergence of Floquet energy bands. To better understand theVq dependence of quartet current, we suggest considering thesuperconducting phase modulation due to the AC Josephsoneffect at finite bias. In the presence of a voltage bias V, theJosephson relation _φ tð Þ ¼ 2eV_ implies a periodic sweeping in timeof the ABS energies, defined at equilibrium as functions of φ1 andφ2. We set the phase of the grounded loop S0 to be zero and theother two superconducting leads S1 and S2 have phases φ1 and φ2,respectively. When voltages are applied to S1 and S2, the phasesacquire time (t)-dependence following the Josephson relation:_φ1 tð Þ ¼ 2eV1_ , _φ2 tð Þ ¼ 2eV2_ . Under the quartet condition(V1 ¼ �V2) and by choosing a new set of phase variablesφq � φ1 tð Þ þ φ2 tð Þ;φr � φ1 tð Þ � φ2 tð Þ ¼ 4eVq_ t, we obtain a sta-tionary quartet phase φq and a running phase φr2 that is periodi-cally driving the system with a frequency2eVq_ , enabling intrinsicsynchronization of CQs. In the adiabatic limit, i.e., Vq beingmuch smaller than the Andreev minigaps Δ0 between the ABSpairs, one can take a time average of the equilibrium ABS spec-trum over φr and obtain an adiabatic ABS energy <EABS>, whichnow only depends on the quartet phase φq (see Section 5 in theSupplementary Information). For simplicity, let us illustrate theeffect of a running phase on the ABS spectrum by consideringonly a single pair of ABSs at a bias small enough for the adiabaticapproximation to work. Iq , the supercurrent carried by quartets,can then be derived from the usual JJ current-phase relation:Iq ¼ 2e_ ∂ <EABS>=∂φq. However, as Vq increases, we eventuallyenter the non-adiabatic regime: the running phase φr2 creates aninternal effective RF-field, which triggers non-adiabatic transi-tions between adiabatic CQ-ABS (Fig. 4e and SupplementaryInformation), and thus favors the occupation of higher levelABSs. This eventually creates resonances, in a way reminiscent ofShapiro steps5 or microwave resonances in transparent metalliccontacts37–39, and this manifests as a quartet current minimum.This demonstrates, by analogy to other microwave resonancephenomena in metallic junctions, that a non-monotonic depen-dence of the quartet current on applied voltages is expected. In a setof non-equilibrium ABSs, its effective separation depends on thebias voltage in analogy to the Floquet bands40,41 separated by 2eVq;emerging from the periodically driven Bloch bands35,42.This picture is corroborated by detailed non-equilibriumcalculations within a single-level quantum dot model (seeSupplementary Information Section 5Ba). While this simplemodel is not intended to be quantitative in a multi-channeljunction as the one in our experiment, similar physics can beapplied: when the Josephson frequency due to Vq matches thespacing between the ABS levels, resonance would occur, resultingin an oscillation of Iqc with Vq. Numerical studies on multi-channel models have confirmed this picture (see SupplementaryInformation Section 5Be).Employing the Floquet energy levels EFloquet that are derivedfrom a pair of <EABS> biased by the quartet bias Vq (Fig. 4e), wecan now explain the experimentally observed non-monotonicbehavior of IqcðVqÞ. Figure 4f illustrates in the simplest case of asingle-channel junction the evolution of two first-order EFloquet asa function of quartet phase φq. The corresponding quartet currentIqðφqÞ, shown in the bottom panels, is obtained with the Floquet-Landau-Zener43 consideration (see Section 5 in SupplementaryInformation). The critical quartet current Iqc ¼ max Iq φq� �n o� �takes place at φ�q . As Vq increases, four different regimes appear:(I) for 2eVq < Δ0; no resonant coupling exists between the two<EABS> and the quartet current is the same as near equilibrium.(II) 2eVq � Δ0, i.e., the Landau-Zener (LZ)-like transitionsbetween the two <EABS> bands become appreciable, openinggaps between different Floquet bands. Hybridization between twolevels and mixing of states that carry opposite directions ofcurrents reduce the net quartet current, resulting in a drop inIqc ¼ max Iq φq� �n o�and the shifting of φ�q . (III) At even largerquartet voltage Vq ¼ V�, the resonances occur at the φ�q in regime(I), denting the peak in Iq and thereby Iqc reaches a minimumvalue. (IV) When 2eVq is increased to be greater than the largestgap between the two levels, there is no more hybridization. Boththe energy levels and the quartet current resume the nearlyadiabatic situation, similar to regime (I). For a more accurateconsideration, the non-equilibrium Keldysh formalism is alsoapplied to multi-level ABSs25–27. It reveals that the inversion ofIqc Φð Þ can be associated with the avoided crossings due to LZtransition in the Floquet bands (see Section 5 of SI). As shown inthe Supplementary Information, a minimal model considers twoquantum dots, each coupled principally to one of the loopelectrodes. The two quantum dots are connected by matrixelements mimicking the underlying graphene layer. This modelshows «inversion» in a very low Vq range and in a wide range ofjunction parameters, consistent with the experimentalobservations.Summary. In conclusion, we experimentally demonstrate theexistence of CQ using MT-JJ with gate-tunable graphene channel.With a magnetic flux threaded through the loop in the unbiasedbranch of our three-terminal junction, the CQ critical oscillationexhibits two distinct Φ0 and Φ0=2 periodicities, indicatinginterferences between different CQ-ABS processes. At a largebias, we observe non-monotonic variation of the quartet criticalNATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-30732-7 ARTICLENATURE COMMUNICATIONS |         (2022) 13:3032 | https://doi.org/10.1038/s41467-022-30732-7 | www.nature.com/naturecommunications 7www.nature.com/naturecommunicationswww.nature.com/naturecommunicationscurrent, which can be associated to Landau-Zener tunnelingbetween Floquet CQ-ABS levels, driven by the intrinsic effectiveRF-field due to the running phase of CQ.During the publication period of our work a complementarytransport signature of quartet physics on 2D InAs quantum wellheterostructure have been demonstrated44, showing the univers-ality of the quartet ABS physics.MethodsFabrication. The van der Waals heterostructure—monolayer graphene on top of40–60 nm thick hBN—is assembled via the inverted stacking technique, where hBNserves as the dielectric substrate to minimize disorder45. The flakes are picked upthrough procedures similar to the dry transfer technique46 except the order is reversed,where the bottom hBN is picked up first. Via this method the top surface of graphene isguaranteed to be clean without any polymer contact in the assembling process. Thegraphene layer we use in our device is larger in size compared to the MT-JJ, and havenot been etched in any step of the fabrication; thereby eliminating natural/etched edgeeffects from interfering with the JJ transport characteristics. The superconductingcontacts are made of 80 nm thick aluminum with 5 nm thick sticking layer of titanium,directly deposited on graphene through electron-beam evaporation at a pressure of low10−7 torr. Each channel is designed to be 80–90 nm to ensure the existence ofsupercurrents among all of the superconductors.Measurement setup. The measurements are performed in He-3 fridge with thebase temperature 300 mK, well below the superconducting critical temperature ofaluminum (Tc ~ 1.1 K) and the dual voltage source measurement scheme allows thedetection of quartet signal (see Supplementary Information for additional details).Data availabilityThe data generated in this study have been deposited in the online depository Zenodo(https://doi.org/10.5281/zenodo.6549095).Received: 22 July 2021; Accepted: 16 May 2022;References1. Andreev, A. F. The thermal conductivity of the intermediate state insuperconductors. J. Exp. Theor. Phys. 19, 1228–1232 (1964).2. Kulik, I. O. Macroscopic quantization and the proximity effect in S-N-Sjunctions. J. Exp. Theor. Phys. 30, 944 (1970).3. Furusaki, A. & Tsukada, M. Current-carrying states in Josephson junctions.Phys. Rev. B 43, 10164–10169 (1991).4. Beenakker, C. W. J. & van Houten, H. 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Landau-Zener-Stückelberginterferometry. Phys. Rep. 492, 1–30 (2010).44. Graziano, G. V. et al. Selective control of conductance modes in multi-terminalJosephson junctions. arXiv https://doi.org/10.48550/arXiv.2201.01373 (2022).45. Dean, C. R. et al. Boron nitride substrates for high-quality grapheneelectronics. Nat. Nanotechnol. 5, 722–726 (2010).46. Wang, L. et al. One-dimensional electrical contact to a two-dimensionalmaterial. Science 342, 614–617 (2013).AcknowledgementsWe thank B. Douçot for his collaboration on the Floquet theory. K.-F.H. acknowledgessupport from DOE (DE-SC0019300) for sample preparation and fabrication. Y.R.acknowledges support from NSF (QII-TAQS MPS 1936263) for device characterization. P.K.ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-30732-78 NATURE COMMUNICATIONS |         (2022) 13:3032 | https://doi.org/10.1038/s41467-022-30732-7 | www.nature.com/naturecommunicationshttps://doi.org/10.5281/zenodo.6549095https://doi.org/10.48550/arXiv.2201.01373www.nature.com/naturecommunicationsacknowledges NSF (DMR1809188) for data analysis. R.M. acknowledges the use of theresources of the Mésocentre de Calcul Intensif de l’Université Grenoble-Alpes (CIMENT)and the French National Research Agency (ANR) in the framework of the Graphmon project(Grant No. ANR-19-CE47-0007). K.W. and T.T. acknowledge support from the ElementalStrategy Initiative conducted by the MEXT, Japan, Grant Number JPMXP0112101001, JSPSKAKENHI Grant Number JP20H00354 and the CREST(JPMJCR15F3), JST.Author contributionsY.R., K.-F.H. and P.K. designed the experiment. P.K. supervised the project. K.-F.H. and Y.R.fabricated the devices. T.T. and K.W. provided single crystals of hBN. K.-F.H. and Y.R.performed the measurements. K.-F.H., Y.R., and P.K. analyzed the data. R.M. and D.F.carried out the theoretical modeling. K.-F.H., Y.R., and P.K. prepared the manuscript andSupplementary Information with input from all the authors.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version contains supplementary materialavailable at https://doi.org/10.1038/s41467-022-30732-7.Correspondence and requests for materials should be addressed to Philip Kim.Peer review information Nature Communications thanks the anonymous reviewer(s) fortheir contribution to the peer review of this work. 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To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2022NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-022-30732-7 ARTICLENATURE COMMUNICATIONS |         (2022) 13:3032 | https://doi.org/10.1038/s41467-022-30732-7 | www.nature.com/naturecommunications 9https://doi.org/10.1038/s41467-022-30732-7http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/www.nature.com/naturecommunicationswww.nature.com/naturecommunications Evidence for 4e charge of Cooper quartets in a biased multi-terminal graphene-based Josephson junction Results and discussions Characterization of MT-JJ in the Josephson regime Cooper quartet Two types of Cooper quartet processes Bias voltage dependence of quartet supercurrent Emergence of Floquet energy bands Summary Methods Fabrication Measurement setup Data availability References References Acknowledgements Author contributions Competing interests Additional information