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Thomas Werkmeister, James R. Ehrets, [Yuval Ronen](https://orcid.org/0000-0002-2427-2591), Marie E. Wesson, Danial Najafabadi, [Zezhu Wei](https://orcid.org/0000-0002-2017-8530), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), D. E. Feldman, [Bertrand I. Halperin](https://orcid.org/0000-0002-6999-1039), [Amir Yacoby](https://orcid.org/0000-0002-5737-7963), [Philip Kim](https://orcid.org/0000-0002-8255-0086)

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[Strongly coupled edge states in a graphene quantum Hall interferometer](https://mdr.nims.go.jp/datasets/f04b3916-5f59-49cc-af34-da024820323c)

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Strongly coupled edge states in a graphene quantum Hall interferometerArticle https://doi.org/10.1038/s41467-024-50695-1Strongly coupled edge states in a graphenequantum Hall interferometerThomas Werkmeister1, James R. Ehrets2, Yuval Ronen 2,3, Marie E. Wesson1,Danial Najafabadi4, Zezhu Wei 5,6, Kenji Watanabe 7, Takashi Taniguchi 8,D. E. Feldman5,6, Bertrand I. Halperin 2, Amir Yacoby 1,2 & Philip Kim 1,2Electronic interferometers using the chiral, one-dimensional (1D) edge channelsof the quantum Hall effect (QHE) can demonstrate a wealth of fundamentalphenomena. The recent observation of phase jumps in a Fabry-Pérot (FP)interferometer revealed anyonic quasiparticle exchange statistics in the frac-tional QHE. When multiple integer edge channels are involved, FP inter-ferometers have exhibited anomalous Aharonov-Bohm (AB) interferencefrequency doubling, suggesting putative pairing of electrons into 2e quasi-particles. Here, we use a highly tunable graphene-based QHE FP interferometerto observe the connection between interference phase jumps and AB frequencydoubling, unveiling how strong repulsive interaction between edge channelsleads to theapparentpairingphenomena.By tuningelectrondensity in-situ fromfilling factor ν < 2 to ν > 7 , we tune the interaction strength andobserve periodicinterference phase jumps leading to AB frequency doubling. Our observationsdemonstrate that the combination of repulsive interaction between the spin-split ν = 2 edge channels and charge quantization is sufficient to explain thefrequency doubling, through a near-perfect charge screening between thelocalized and extendededge channels. Our results show that interferometers aresensitive probes of microscopic interactions and enable future experimentsstudying correlated electrons in 1D channels using density-tunable graphene.Electrons in 1D quantum systems exhibit striking phenomena, includ-ing the breakdown of Fermi liquid theory and quasiparticle formationin favor of collective modes1. Likewise, electrons confined to twodimensions and subjected to perpendicularmagnetic fields exhibit thequantumHall effects (QHEs)2. Although themicroscopic details ofQHEstates are still an active area of research3,4, their low-energy transportproperties are known to be governed by chiral, 1D edge channels5–8.These edge channels (ECs) conduct charge ballistically, allowing forphase-coherent electronic experiments9,10. In particular, electronicFabry-Pérot (FP) QHE interferometry11–13, was performed extensively inGaAs, culminating in the observation of interference phase jumps asevidence for anyonic statistics of fractional quasiparticles14–17.Recently, FPs were developed in graphene, which showed Aharonov-Bohm (AB) interference of integer ECs18–20, with oscillation periodicityset by the magnetic flux quantum for electrons Φ0 � h=e. Our pre-vious design18 utilized graphite gates encapsulating the graphenechannel, which screenedbulk charges.Without such screening layers21,however, interferometers exhibit ‘Coulomb dominated’ (CD) behaviorReceived: 23 January 2024Accepted: 18 July 2024Check for updates1John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA. 2Department of Physics, Harvard University,Cambridge, MA 02138, USA. 3Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 7610001, Israel. 4Center for NanoscaleSystems, Harvard University, Cambridge, MA 02138, USA. 5Department of Physics, Brown University, Providence, RI 02912, USA. 6Brown Theoretical PhysicsCenter, Brown University, Providence, RI 02912, USA. 7Research Center for Functional Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba305-0044, Japan. 8International Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan.e-mail: pkim@physics.harvard.eduNature Communications |         (2024) 15:6533 11234567890():,;1234567890():,;http://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/0000-0002-2017-8530http://orcid.org/0000-0002-2017-8530http://orcid.org/0000-0002-2017-8530http://orcid.org/0000-0002-2017-8530http://orcid.org/0000-0002-2017-8530http://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/0000-0002-6999-1039http://orcid.org/0000-0002-6999-1039http://orcid.org/0000-0002-6999-1039http://orcid.org/0000-0002-6999-1039http://orcid.org/0000-0002-6999-1039http://orcid.org/0000-0002-5737-7963http://orcid.org/0000-0002-5737-7963http://orcid.org/0000-0002-5737-7963http://orcid.org/0000-0002-5737-7963http://orcid.org/0000-0002-5737-7963http://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/0000-0002-8255-0086http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-50695-1&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-50695-1&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-50695-1&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-50695-1&domain=pdfmailto:pkim@physics.harvard.eduin which strong coupling of the interfering EC to localized compres-sible states in the bulk determines the oscillation periodicity andobscures the expected AB oscillations13,22–24.When bulk charges were strongly screened, GaAs FPs showedunexpected doubling of the AB oscillation frequency and shot noisecorresponding to charge 2e when interfering the outermost EC withthe bulk of the interferometer in filling 2:5 ≤ ν ≤ 4:5, suggesting apossibility of ‘pairing’ of elementary charges25. Furthermore, thecoherence and periodicity of the interfering outer EC were related tothe coherence and the enclosed flux of the adjacent inner EC26, and the‘pairing’ phenomena only occurred when the outer two modesbelonged to the same spin-split Landau level27. Independently, single-electron capacitance measurements in GaAs quantum dots revealedthat tunneling into the edge of the dot corresponded to the entranceof two electrons rather than one for ν ≥ 2, and that near ν ≈ 2:5 thecharging peaks followdoubledmagneticflux frequency28.Mechanismsof electron pairing are important questions in emergent phenomena,e.g. high-temperature superconductivity29 and the even-denominatorfractional QHE states in GaAs30 and bilayer graphene31,32. However,theoreticalwork concerning FP interferometerswas able to explain thedoubled AB oscillation frequency based on a microscopic modelwithout explicit introduction of electron pairing, though explainingother related phenomena in GaAs remains challenging33.In this work, we experimentally address the microscopic mechan-ism of coupling betweenQHE edges by elucidating the relation betweenAB oscillation phase jumps and frequency doubling, employing a highlytunable QHE FP interferometer with strongly screened bulk charge ingraphene. We observe periodically modulating interference phasejumps on the outer EC leading to nearly doubled AB oscillation fre-quency as we increase the electron density in-situ, unveiling a density-induced transition which was not explored in GaAs. We find that strongrepulsive interactions between the outermost pair of spin-split ECs canexplain both the observation of interference phase jumps and theapproximately doubled interference frequency.ResultsInterferometer design and tuningWedesigned a graphene-based FP interferometer tuned by a local gatearray (Fig. 1a). The FP cavity is defined electrostatically using separatedgraphite top-gates (Methods and Supplementary Fig. 1), which ensuresFig. 1 | Highly tunable Fabry-Pérot interferometer in graphene. a False-colorscanning electron microscopy image of a Fabry-Pérot (FP) device identical to thedevice measured here. The graphite top-gate layer is selectively etched to form 8separated top-gates (purple). Metal bridges (blue) connect to each graphite top-gate region and two additional bridges (yellow) suspend over the quantum pointcontacts (QPCs). The lithographic area of the interferometer cavity (areaA= 1:16μm2) is defined by the central hexagonal top-gate. Scale bar: 1μm.b Simplified schematic of the FP tuned so that filling factors νLG = νMG = νRG = 2 andνSG1 = νPG = νSG2 =0 illustrating interference of the partitioned outer edge channel(EC) (red) while the inner EC (blue) forms a closed annulus inside the FP. Voltage V iapplied to the top-gate labeled ‘i’ tunes the local filling factor νi . Voltages VQPC1 andVQPC2 applied to the suspended metal bridges selectively gate the QPC constric-tions through the etched graphite gaps, tuning theQPC transmissions.Wemeasurethe diagonal conductance GD = Id=ðV +D � V�D Þ, where V ±D and Id are measured vol-tages in ð± Þ probes and drained current, respectively. See Supplementary Fig. 1 forthe full device details. In addition tomagnetic field, we tune the interference phaseusing voltage VMG on the ‘middle gate’ or VPG on the ‘plunger gate’. c Conductanceas a functionofVQPC1 withVQPC2 = 7V (i.e.TQPC2 = 2) demonstratingQPC1 tunings tointerfere outer EC (red dot) and inner EC (blue dot) in ν = 2. See SupplementaryFig. 3 for QPC tuning details and voltages set on the other gates to form thenecessary QPC saddle-points to acquire this data. d Same type of plot as (c), butdemonstratingQPC2operation insteadofQPC1. e, fCharacteristic FPoscillations asa function of VPG for the inner EC and outer EC, respectively, at the QPC tuningsindicated in (c) and (d). Vertical dashed lines indicate edges of plateaus of fillingfactor νPG. All data is at fixed magnetic field B=6T.Article https://doi.org/10.1038/s41467-024-50695-1Nature Communications |         (2024) 15:6533 2a high channel quality and allows a high degree of density tunability in-situ. Metal bridges contact each top-gate, and we additionally suspendmetal bridges over the two quantum point contacts (QPCs), illustratedin Fig. 1b. By applying voltages VQPC1 and VQPC2 to these suspendedbridge gates, we can tune the transmission of eachQPC independentlywhile keeping the filling factor of the surrounding regions fixed(Supplementary Note 1).In our experiments, we measure the diagonal conductance GD,as defined in Fig. 1b and Supplementary Fig. 1. In the regime that westudy, GD = e2h νQPC where νQPC counts the number of edge channelstransmitted through the device, with a partially transmitted channelcounted as fraction34,35. To characterize the QPC transmissions, wemeasure GD as a function of the bottom-gate voltage and split-gatevoltage for each QPC with the bulk of the interferometer tuned toν =2 at B=6T (Supplementary Fig. 3). At ν =2, there are two spin-splitLandau levels, of which the lower energy spin species hosts an ECcloser to the effective boundary of the sample. Hence, we refer tothe EC belonging to the lower (higher) energy spin species as the‘outer’ (‘inner’) EC. Once appropriate bottom-gate and split-gatevoltages are set, we tune VQPC1 and VQPC2, voltages applied on thesuspended bridges to control the individual QPC transmissions.Figure 1c, d shows themeasured GD as a function of VQPC1 and VQPC2,respectively, with the other QPC fully open (i.e. νQPC2 = 2 andνQPC1 = 2, respectively). GD exhibits plateaus at ð0,1,2Þ e2h , corre-sponding to (neither, outer, both) ECs transmitted. In this regime,we define TQPC � GDhe2 as the transmission of the QPC34, where0<TQPC < 1 corresponds to a partially transmitted outer EC and1 <TQPC < 2 corresponds to a partially transmitted inner EC.Tuning to partial transmission of the inner EC for both QPCs,TQPC1 =TQPC2 = 1:5, we observe high-visibility conductance oscillationsas a function of plunger gate voltage VPG, which tunes the filling factorνPG under the plunger gate, in Fig. 1e. Similarly, we tune toTQPC1 =TQPC2 =0:5 andmeasure conductance oscillations on the outerEC in Fig. 1f. In both cases, oscillations are largest for νPG <0, whichcorresponds to a fully gate-defined interference path since electronsare depleted under the plunger gate. Increasing νPG brings the inter-fering edge closer to the etched graphene boundary, inducingdephasing18. Notably, the inner EC oscillations survive until νPG = 2,when it flows close to the etched boundary of the graphene, while theouter EC reaches the boundary by νPG = 1: Another difference is theapparent irregularity of the oscillations on the outer EC compared tothe inner EC, which we will understand in this work.Phase jumps and AB oscillation frequency transitionHigh-visibility oscillations allow us to probe the dependence of inter-ference phase θ on magnetic field variation δB and gate voltage var-iations, which distinguishes the AB from the CD regimes13,15,18,19,22. Forsmall variations in field and gate voltages in the AB regime, we expectδθ=2π ≈AδB=Φ0 +CPGδVPG=e+CMGδVMG=e, where A, CPG, and CMG,are the (approximately constant) area enclosed by the interfering EC,interfering EC—plunger gate capacitance, and interfering EC – middlegate capacitance, respectively. Importantly,VMG alsodirectly tunes theelectron density in the interferometer, so sweeping VMG over a largerangewill change the FP cavity filling factor νMG. To calibrate the fillingthat we expect in the cavity, we first measure standard Hall con-ductance in the region gated by V LG (see Supplementary Fig. 1e) andobserve conductance plateaus (Fig. 2a). Since the top gates are iden-tically coupled to the channel directly beneath them, an identicalsweep ofVMG will tune νMG through the same filling factors. Data in theremaining panels of Fig. 2 were taken with the QPCs set toTQPC1 =TQPC2 =0:5 i.e. partially transmitting the outer EC. Near thelowest density of the νMG =2 plateau (Fig. 2b), we observe a typical ABinterference pattern. Constant phase stripes ðδθ=0Þ trace out anegative slope δVPG=δB with magnetic field period ΔB yieldingΦ0=ΔB= 1:13μm2, matching the designed area A= 1:16μm2. Plungergate period ΔVPG yields 1=ΔVPG = 19:2V�1. Increasing νMG using VMGreveals more complicated interference patterns in Fig. 2c, d. Periodicshifts in the interference pattern persist and modulate until near thecenter of νMG =4, as seen in Fig. 2e, when a simple stripe patternreturns. However, now Φ0=ΔB= 2:32μm2 and 1=ΔVPG = 36:3V�1, bothapproximately doubled from Fig. 2b. Since A is fixed, a doubling ofΦ0=ΔB indicates oscillations with Φ0=2 =h=2e periodicity instead ofΦ0 so that Φ0=2ΔB= 1:16μm2. Similarly, assuming a fixed CPG, then1=ΔVPG doubling corresponds to adding twice asmany electrons to thesystem per flux quanta. Both could be interpreted as an effectivecharge e* = 2e for the interfering particle, as argued inGaAs25–27, but ourobservations indicate a different interpretation in our graphene-basedinterferometer.Importantly, we can observe the entire density-tuned transition tothe AB frequency-doubled regime at fixed B by sweeping VMG andobserving oscillations with VPG, as shown in Fig. 2f. Remarkably, thefrequency transition occurs continuously. From the top panel, Φ0interference is apparent. AsVMG increases, periodic phase jumps beginto appear. Both the VMG spacing and magnitude of the phase jumpsincrease, until eventually the most apparent periodicity correspondsto Φ0=2 oscillations (i.e., doubled frequency 2Φ�10 ).To better understand the phase jumps, we use a general relationbetween charge and phase in FP interferometers36. When a single ECpasses through the two constrictions with weak backscattering, theinterference phase seen by the device at zero temperature isθ=2πQ+θ0,mod2π, whereQ is the total electron charge (inunits e) inthe region between the two scattering points and θ0 is a constant forsmall variations in B, VPG, and VMG. In our experimental regime, ν ≥ 2,we expect this relation to hold with Q=Q1 +Q2, where Q1 is the totalcharge residing in the lowest spin-split Landau level and Q2 is thecharge in the higher energy spin state (and also higher Landau levels).Q1 can vary continuously since the outer EC is connected to thesource and drain charge reservoirs. In contrast, Q2 is required to beinteger, as the corresponding energy levels are isolated throughthe incompressible QHE bulk. An integral change in Q2 has no obser-vable effect on the interference signal unless it produces a non-integralchange in Q1 due to Coulomb coupling between the two typesof charge. Hence, we can redefine θ to include only the charge Q1 inthe lowest spin-split Landau level, and the values Q1 in the groundstate of the interferometer determine θ. Following similar modelsused to understand the CD regime15,24,37 and considering small changesin Q1 and Q2, we expand the change in ground state energyE =K1δQ12 +K2δQ22 + 2K12δQ1δQ2, where Ki is the charging energy ofthe charge species i and K12 describes the mutual capacitive couplingbetween them. Energetic stability requires that K12�� ��2 ≤K1K2. Withinthis capacitive coupling model, when Q2 increases by 1, the charge Q1correspondingly decreases by a discrete (generically non-integral)amount ΔQ1 to screen the added charge, leading to a phaseshift Δθ=2π =ΔQ1 =�K12=K1.By taking 1D fast Fourier transforms (FFTs) along lines parallel tothe phase jumps14,15, we extract several values ofΔθ=2π near the centerof the periodicity transition in Fig. 3a. We observe that the locationswhere the phase jumps occur (marked in Fig. 3b) follow a steeper slopethan the slope δVPG/δVMG of constant phase lines of the main inter-ference oscillation in the VMG-VPG planes. A steeper slope also occursin the B-VPG plane (Fig. 2c, d). Moreover, these phase jump lines havenegative slopes δVPG=δB<0, like the constant phase lines of ABoscillations. This observation is in sharp contrast to the phase jumpsreported in the FP interferometer operated in the fractional QHEregime14,15 or in the FP interferometer operated in the integer CDregime37, where phase jump lines follow positive slope δVPG=δB>0.The different slope suggests a different structure to the energy levelsthat are being populated in our sample. Considering that the outer ECis partitioned at the QPCs, while the inner ECs are well isolated, wehypothesize that the charging events seen as phase jumps representArticle https://doi.org/10.1038/s41467-024-50695-1Nature Communications |         (2024) 15:6533 3charge added to the annular, closed inner EC, illustrated in Fig. 3. Thedominant coupling K12 is directly between the outer and inner ν =2ECs. Any charges added to higher Landau levels or to localized states inthe bulk are not measurably coupled to the outer EC, presumablybecause of effective screening by the gates.AB frequency doubling from strongly coupled QHE edge statesWe provide further evidence for capacitively coupled QHE edgestuning the AB frequency in Fig. 4. At fixed VMG in the transitionregime, we compare interference in the B-VPG plane for the inner EC,Fig. 4a, to the outer EC, Fig. 4c. This direct comparison is onlypossible because we can control QPC transmissions independentlyof bulk filling. We observe that the slope of the oscillation maximaon the inner EC (dotted lines in Fig. 4a) matches the slope of thephase jump lines on the outer EC (dotted lines in Fig. 4c). Reducingthe transmission for the inner EC, the interference maxima in Fig. 4abecome sharper charging resonances, corresponding to chargeQ2 ! Q2 + 1 through the inner EC. When the transmission of theinner EC vanishes, the inner EC is fully disconnected from the sourceand drain charge reservoirs, and the outer EC is now partitioned atthe QPCs to form a new interference path (shown in the left inset inFig. 4c). Since the bulk density and electrostatic configurations forFig. 4a and Fig. 4c are identical, the regions in between the phasejump lines in Fig. 4c correspond to fixed Q2, and we see that theinterference phase on the outer EC shifts when the charge on theinner EC discretely changes.Taking Fourier transform of the interference signal provides fur-ther understanding of interactions between the two ECs involved infD (%)�GPG(V)VMG (V)V-1.2-1.00.7 0.82.51-2.483.61-3.340.8 0.98.84-7.770.9 1.013.4-11.71.0 1.116.3-12.81.1 1.220.5-13.251.2 1.318.6-11.91.3 1.411.6-9.141.4 1.513.6-11.11.5 1.69.41-7.571.6 1.7aLG (V)V0.5 2.01.0 1.5xy(e2 /h)G1234bMG = 0.73 VV(T)B-1.2PG(V)V-0.85.98 6.00D (e2/h)G0.640.340.83cMG = 1.10 VV(T)B-1.2PG(V)V-0.86.00D (e2/h)G0.525.98= 1.30 Vd0.84MGV(T)B-1.2PG(V)V-0.86.00D (e2/h)G0.465.98= 1.90 Ve0.86MGV(T)B-1.2PG(V)V-0.86.00(e2/h)G0.72(T)5.98DFig. 2 | Density-tuned Aharonov-Bohm frequency doubling transition ofouter EC. a Hall conductance Gxy in the region gated by V LG, demonstrating thatV LG (equivalently, any of the top gates) tunes the filling ν underneath it at a fixedmagnetic field B=6T. Colored dots indicate the filling (set by equivalent VMG vol-tages) at which interference data are shown in (b–e); vertical dashed lines show therange of VMG swept for (f). Top inset: schematic of compressible regions expectedin the FP cavity when VMG is swept. b–e Conductance GD oscillations on the outerEC with VPG and B, for each of the indicated VMG values. f GD oscillations on theouter EC with VPG and VMG, for VMG swept continuously over the transition fromapparent h=e to h=2e oscillations periodicity, at B=6T. GD is plotted as a percen-tage of e2h deviation from the average value calculated for eachfixedVMG linecut andsubtracted off. Further phase jumps or periodicity changes are not observed pastVMG ≈ 1:7V (checked up to ν = 7). QPCs are retuned to maintain transmissionsTQPC1 =TQPC2 =0:5 over the dataset while νLG = νRG = 2 and νSG1 = νSG2 =0 are fixed.Article https://doi.org/10.1038/s41467-024-50695-1Nature Communications |         (2024) 15:6533 4the interference. Figure 4b, d shows the 2D FFT of the conductanceoscillations in Fig. 4a, c. For interference of the inner EC (Fig. 4b), weobserve a simple FFT pattern of peaks corresponding to the funda-mental frequency of the inner EC f i, a vector containing the peakposition in the 2D FFT, and its harmonics (nf i, where n is an integer).The FFT pattern of the outer EC interference (Fig. 4d) exhibits a morecomplicated lattice of Fourier peaks. If we label one of the dominantpeaks as the fundamental frequency of the outer EC, f o, we can thenidentify the rest of the peaks by addition or subtraction of the samevector f i evident in the inner EC data. The lowest order peaks corre-spond to the sum f o+ i = f o + f i and the difference f o�i = f o � f i. Weshow a similar Fourier lattice construction in Supplementary Fig. 6 forinterference in the B-VMG plane.By tuning VMG, we modulate the filling factor of the inter-ferometer cavity in a wide range and observe the evolution of theinterferencepatterns and correspondingpeaks for the outer (inner) ECin Supplementary Fig. 7 (8). As in Fig. 2, phase jumps appear onlywithin the periodicity transition. Figure 4e shows the average magni-tude of individual phase jumps as a function of VMG. We find that thephase jump continuously evolves from Δθ=2π ≈0 (VMG <0:6V)through the periodicity transition to Δθ=2π ≈�1(VMG > 1:6V), corre-sponding to the strongly coupled limit K12=K1 ≈ 1. The transitionregime marked by non-trivial phase jumps spans from the appearanceof the inner EC (VMG ≈0:6V) to the strongly coupled outer two EClimit (VMG ≈ 1:6V).The Fourier peaks’ evolution tuned by VMG provides furtherinsight into the interaction between ECs. Figure 4f displays the nor-malized Fourier peak intensity as a function of VMG.The amplitude ofthe Fourier peak f o decays through the transition regime(0:6V<VMG < 1:6V), replaced by f o+ i as the dominant peak. We plotthe magnetic field frequency multiplied by Φ0 (Fig. 4g) and the plun-ger gate frequency (Fig. 4h), respectively, for each of the lowest-orderpeaks f o, f i, f o+ i, and f o�i as a function ofVMG. At the beginning of thetransition regime where the ECs are not interacting, both f o and f o+ iapproach the corresponding AB frequency Φ�10 = e=h through thedesigned area. As VMG increases, however, f o stays nearly unchanged,while f o+ i increases to reach the doubled value 2Φ�10 . The experi-mental observation that the dominant peak in the frequency-doubledregime corresponds to f o+ i precludes the possibility of 2e chargepairing within the outer EC alone.Instead, our frequency-doubled regime arises from Coulombinteraction between the spin-split ECs combined with charge quanti-zation on the inner EC (Methods). Electrons would naturally tend toenter the inner EC at frequency f i, but, due to charge quantization,cannot enter continuously. Hence, as the magnetic flux increasescontinuously, the area enclosed by the inner EC must shrink to main-tain fixed charge. During this shrinking process, electron charge istransferred continuously into the interior, leaving missing electroncharge between the outer and inner ECs. In the strongly coupled EClimit, thismissing charge attracts anequal charge onto the outer EC for= -0.45�� �� -0.48 -0.47-0.47-0.49-0.45a1.00MG (V)V0.870.52D (e2/h)G1.02bPG(V)VMG (V)V�����0-1-2-312�1.00 1.02-1.4-1.0Fig. 3 | Phase jump extraction in the transition regime. a Phase θ of the 1D FFTextracted along linecuts parallel to the phase jumps in (b). The phase is evaluated atthe dominant frequency in the FFT amplitude spectrum for the linecuts in betweenphase jumps. A linear increase in phase extracted from regions without phasejumps is subtracted off to make the phase jump magnitude evident as the verticalshift between plateaus in panel (a). From this datawe extract an averageΔθ=2π ≈ �0:47, reflecting approximately half of an electron repelled from the outer EC foreach charge added to the inner ECwithin this range ofVMG. Inset: illustration of thecoupling K12 between the outer and inner ECs contributing to the phase jumps.bConductanceGD oscillations on the outer ECwith VPG and VMG near the center ofthe transition regime showing periodic phase jumps along the dashed black lines.Note that increasing VMG adds electrons to the system or equivalently increasesphase, so the phase jumps correspond to negative shifts in phase i.e., repulsion ofelectrons from the FP cavity. Similar interference patterns are observed in both thestrong and weak QPC backscattering regimes (Supplementary Fig. 4) as well as atelevated temperatures (Supplementary Fig. 5).Article https://doi.org/10.1038/s41467-024-50695-1Nature Communications |         (2024) 15:6533 5screening. In the absence of this screening effect, charge is con-tinuously added to the outer EC with frequency f o according to theincreased AB phase. In the coupled ECs, the combination of thescreening-induced charge and the natural AB effect results in the outerEC charging at a frequency f o+ i. Therefore, the interference phasefollows f o+ i. In addition to this continuous charging effect, electronscan tunnel into the inner EC from the external reservoirs. As previouslydiscussed, each electron addition repels some electron charge fromthe outer EC, causing the negative interference phase shifts that weobserved. For larger values of VMG, as the bulk density increases, theinner and outer EC move closer together, and the system approachesthe strong coupling limit, where the phase jumps are close to�2π andunobservable, reflecting a full electron charge screening. Moreover, asthe inner and outer ECs asymptotically enclose the same area, set bythe confining potential of the device, the frequency f o+ i approa-ches 2Φ�10 .Note: a concurrent work also observed apparent AB frequencytripling, corresponding to the sum of the three ν =3 edge channelfrequencies38. The framework that wedeveloped here can be expectedto naturally explain this observation, since in devices utilizing thegraphene crystal edge, the sharp confining potential can lead to mul-tiple ECs developing within a few magnetic lengths of the edge8.Fig. 4 | Comparison of inner and outer EC interference and couplings acrosstransition. aConductanceGD oscillations on the inner EC (TQPC1 =TQPC2 = 1:5) withVPG and B, for VMG = 1:2V. Dotted black lines highlight conductance maxima. Leftinset: illustration of inner EC interference configuration. b 2D FFT of the GD oscil-lations in (a) showing peak f i (vector corresponding to blue arrows) and its har-monics, whereΦ0 � h=e. c GD oscillations on the outer EC (TQPC1 =TQPC2 =0:5) atthe same density set by VMG. Dotted black lines with identical slope to (a) highlightphase jumps. Left inset: illustrationofouter EC interferenceconfiguration.d 2DFFTof oscillations in (c) showing the peaks f o (red arrows), f o+ i, and f o�i and theirharmonics. e Magnitude of the phase jump on the outer EC as a function of VMG.Each data point is averaged over a ∼0:25V range in VMG; error bars indicate ± 1standard deviation in this range. Unfilled data points represent zero observablephase jumps over the range, hence we infer a magnitude of 0 or �1. Gxy of thedevice taken in an identical measurement to Fig. 2a, reflecting the expected fillingνMG, is plotted for reference. Top inset: cartoon of the outer and inner EC evolutionwith increasing VMG. fMagnitudes Io, Io+ i, and Io�i of the respective peaks f o, f o+ i,and f o�i as a function of VMG. Io, Io+ i, and Io�i are normalized by the sumIo + Io+ i + Io�i to show their relative contributions. Eachdata point is extracted froma 2D FFT (Supplementary Fig. 7). g Magnetic field frequency multiplied byΦ0 forpeaks f o, f i , f o+ i, and f o�i tracked through the transition. Note that f i is measuredfrom a separate measurement of interference on the inner EC (SupplementaryFig. 8). h Same as (g) but for plunger gate frequency. Horizontal dashed lines in(g, h) indicate the corresponding f o and 2f o values before the transition. Black(red) dots show calculated f o ± f i from outer and inner EC data, which match thepeaks identified as f o+ i and f o�i , respectively.Article https://doi.org/10.1038/s41467-024-50695-1Nature Communications |         (2024) 15:6533 6The combination of reduced spatial separation and reduced screeningby nearby graphite gates may account for the observation of apparenttripling, arising from the outer EC screening both internallocalized ECs.DiscussionWe have investigated phase jumps and AB frequency modulation in ahighly tunable graphene QHE FP interferometer with coupled co-propagating edge modes. We identify that interference phase jumpsare related to the single electron charging events in the inner EC, andthe transition of the AB frequency can be connected to the corre-sponding screening effect of the outer EC. As VMG increases, the ECcoupling becomes strong and the AB frequency doubles, indicating anear-perfect screening between the ECs. Thus, our experimentalobservation supports the proposal that AB frequency doubling can beexplained without explicitly introducing electron pairing within theouter two ECs33. In other words, a half flux quantum introduced in thetwo strongly coupled ECs can bring a full charge from the externalreservoir and a 2π evolution of the observed interferometer phase.Our observations do not exclude the possibility of further corre-lation effects in the strongly coupled ECs; instead, the tunably coupledECs discovered here provide a system to test the emergence of elec-tron correlations in 1D systems39. However, AB frequency multi-plication, which we explained within a single particle picture, cannotsubstantiate the correlation effect. Further experiments probing thetransition from the weakly to strongly coupled limit, such as shotnoise25,40,41, finite-bias dependence15,42, energy relaxation43, and high-frequency transport44–46 will provide further insight into the groundstate andexcitations.Moregenerally, inter-edge screening could affectinterferometry in fractional fillings containing multiple ECs42,47–49,though fractional QHE experiments so far appear to be in the weakcoupling regime42,50. The recent observations of interference in frac-tional quantum Hall states using similar bilayer and monolayer gra-phene devices51–53 will enable further experiments to probe theinteracting co-propagating fractional QHE edge modes, where itremains unclear whether the strong coupling regime can be realizedbetween fractional edge channels.MethodsSample preparationThe monolayer graphene stacks with hBN and graphite encapsulationused in this study were fabricated using the standard polycarbonate(PC) polymer dry transfer method (ref. 18). The graphite top-gates andbottom-gate, which encapsulate the graphene channel after stacking,are crucial to screen charge disorder from the graphene channel, sta-bilizing robust integer and fractional QHE states at lowmagnetic fields(Supplementary Fig. 2). The stack used for all data shown here had atop (bottom) hBN thickness of 49 (27) nm.After adhering the stack to asubstrate and annealing in vacuum at 300 °C, the top gate was firstetched into a simplified shape by reactive ion etching in an inductivelycoupled plasma etching chamber with 30W O2 plasma using a poly-methyl methacrylate (PMMA) resist patterned with electron-beamlithography as the etchmask. Next, a full etch through the entire stackwas performed to define all outer boundaries. This etching was inseveral steps: first a pure 30W O2 etch remnants of the top graphite;then a 30W process O2/CHF3 to etch through the underlying hBN,graphene, and hBN; and finally another 30W O2 etch to remove thebottom graphite. Next, edge contacts to the exposed graphene weremade by a 30WCHF3 etch on the exposed hBN/graphene/hBN contactregions and thermal evaporating 2/7/150nm of Cr/Pd/Au at an anglewith rotation. Then, air bridge contacts weremade to the top-graphitein various locations using a bilayer PMMA process followed by a short20–25 s 30WO2 plasma PMMA residue clean and thermal evaporationof 2/7/350 nm Cr/Pd/Au. Then, to etch the ~100 nm width trenches toseparate the top graphite regions, a thinner PMMA resist was used andagain a reactive ion etchwith gentle 30WO2 plasma alonewas done in~1minute steps. In between etches, the two-probe resistance betweeneach bridge-contacted gate was checked until they were all separated.Finally, bridge contacts to the separated central hexagon gate andsuspended bridges over the QPC regions were deposited. See Sup-plementary Fig. 1 for more details on the fabrication process and thefinal device.MeasurementsThe 8 top graphite gates in the devicewere separately controlled to setfilling factors in each region at perpendicular magnetic field B, sinceLandau level filling factor (also simply called ‘filling’) ν � ne=nϕ, wherenϕ = eB=h and ne is the areal electron density. At the region in themiddle of the top-gate split-gates, where the graphite is etched awayfor a separation of ~150nm, the electrostatics are tuned to create asaddle-point potential at the QPC. See Supplementary Note 1 fordetails of this tuning process. Once an approximate saddle-point isformed at the QPCs using the graphite top-gates and bottom-gate, thesuspended metal bridges over the QPCs are tuned to precisely settransmissions TQPC1 and TQPC2. The neighboring top-gates screen outstray fields generated by the suspended bridges such that VQPC1 andVQPC2 are primarily coupled to the graphene at the saddle-point of theQPCs. We interpret non-integer values 0<TQPC<1 as a transmissionprobability for electrons in the outer EC, which is partially transmitted,while for 1 <TQPC < 2, TQPC � 1 gives the transmission probability forthe inner EC.Experiments were performed in an Oxford wet dilution systemwith base temperature ~20mK and estimated ~20–25mK electrontemperature. The 24DCmeasurement lines of the fridgewerecarefullythermalized through Thermocoax cables and 3 Sapphire platesbetween room temperature and the mixing chamber. A series oflumped element Pi and RC filters at the mixing chamber reducedelectronic noise and ensured low electron temperature. Unlessotherwise noted, a constant 6 T perpendicular magnetic field wasapplied.Measurementswere taken using standard low-frequency lock-in amplifier techniques with a typical AC excitation current of 1 nA at17.77 Hz applied to the sample and simultaneously measured AC vol-tage drops and drained current. Graphite and suspended bridge gateswere controlled with a house-made, low-noise 16-bit D/A voltagesource. Bias dependence (see Supplementary Note 4) was taken byvoltage biasing instead and adding in a DC bias at the source. Simul-taneously, the DC voltage drop VD was measured on the same probesmeasuring the AC conductance so that the accurate voltage dropacross the FP cavity was known. All data collected and analysis pro-grams have been made available.Estimation of the coupling strengthAlthough we have not attempted a detailed calculation of the cou-pling constants important for our analysis, we can at least advancesome qualitative arguments for the trend that emerges from ouranalysis. The edge of the sample consists of alternating compres-sible and incompressible stripes whose width is set byelectrostatics6. ECs are located in compressible stripes. It may beexpected that the outermost EC is located along an electron densitycontour where the local Landau-level filling factor is ~ 0.5, while thesecond EC is located along a contour with filling ~1.5. Due to residualdisorder and electron-electron interactions, the Hall plateau at ν =2will set in when the bulk filling is smaller than 2, though larger than1.5. The density profile produced by charges on confining gatesshould be relatively smooth, so that the spatial separation betweenthe outer most EC and the second EC should be relatively large atthis point, and the Coulomb coupling between the channels,screened by the gates, should be relatively weak. As the electrondensity is increased, the inner EC should move closer to the outeredge, and the coupling should become stronger, and it is plausibleArticle https://doi.org/10.1038/s41467-024-50695-1Nature Communications |         (2024) 15:6533 7that by the time the device enters the ν =3 plateau, the value ofK12=K1 is close to 1.Further increases in the density should produce additional ECs,which are totally reflected at the QPCs and do not contribute directlyto the transport. The number of electrons on any additional closedECs, as on other localized states, will be restricted to integer values,and in principle, due to Coulomb interactions, there should be a jumpin the interference phase of outer edge states each time this integerchanges by one. However, Coulomb interactions in our system arestrongly screened by the nearby gates, so if the additional channels arenot close to the outer two ECs, the jumps would be too small to beobservable. In monolayer graphene, the energy gap at ν =2, which isdue to the cyclotron energy, is much larger than the gaps at ν = 1, 3, 4,and 5, which arise from electron-electron interactions. Consequently,we expect that the spatial separation between the outermost EC andthe second EC will tend to be small compared to the separationbetween the second EC and any additional ECs.Another issue is the stability criterion embodied in the require-ment K12�� ��2 ≤ K1K2. This requirement is automatically satisfied if weassume that when the two outer ECs are close together, the energy foradding an electron to either one of them is dominated by an electro-static energy that depends primarily on the total charge on the edges,and only weakly on the difference between them, so thatE =aδQ12 + bδQ22 + 2K12ðδQ1 + δQ2Þ2, with a and b small compared toK12. Then, K1 and K2 will be approximately equal to each other andslightly larger than K12.This analysis is compatible with experiments in GaAs inter-ferometers where the ECs occur at the boundary between two QHEstates of different integer filling fractions (ref. 27). There it was foundthat the h=2e periodicity occurred only if the outer EC and second ECbelong to the same orbital Landau level, and not if they belong todifferent levels. In the first case, the energy gap for the QHE statebetween the two ECs will arise from electron-electron interactions,while the energy gap in the second case will be dominated by thegenerally larger cyclotron energy. Therefore, in the first case, when thedensity is increased enough to populate a thirdQHE state in the bulkofthe sample, the two outer ECs might be pushed so close to each otherthat they are strongly coupled, while this might not be expected tohappen in the second case.Physics of AB frequency doubling at strong couplingThe meaning of the charge fluctuations δQ1 and δQ2 can be mademore precise as follows. As stated in the main text, we defineQ1 as thenumber of electrons in the lowest spin-split Landau level enclosed bythe outer edge mode and Q2 as the number of electrons in the higherspin state enclosedby the innermode. These charges are related to theenclosed areas A1 and A2 by Qi =AiB=Φ0, where i= 1 or 2. These areasare allowed to deviate slightly from the ideal areas �Ai, which areassumed to be smooth functions of VPG and, at most weakly varyingfunctions ofB and VMG. Then δQi =Qi � B�Ai=Φ0, and the energy E maybe expanded to quadratic order in δQi as stated above.When the inner mode is completely reflected at the QPC, thecharge Q2 is constrained to be an integer, while the charge Q1 canchange continuously, assuming that the outer edge is mostly trans-mitted through the QPCs. At low temperatures the charges will bedetermined so as to minimize E, subject to the integer constraint.If Q2 is held fixed while the magnetic field is increased by a smallamount dB, the inner edge charge δQ2 will change by an amount�dB�A2=Φ0. This happens because, as the area shrinks, charge istransferred from the edge region to the interior, where it is effectivelyscreenedby the gates, leaving a charge deficit at the edge. In the strongcoupling limit, this will cause δQ1 to increase by an equal amount.Thus, the total charge Q1 in the lowest spin-split Landau level willincrease by dQ1 =dB �A1 + �A2� �=Φ0, and the interferometer phase θ willincrease by 2πdQ1.If B is increased by a large amount, the value ofQ2 will not be fixedbut will undergo periodic integer jumps. In the strong coupling limit,the jump in Q1 caused by a jump in Q2 will also be an integer. This willcause θ to jump by a multiple of 2π, which will be invisible in aninterferometer experiment. Thus, the observed rate of change of thephasewill bedθ=dB=2π �A1 + �A2� �=Φ0, which is equal to 4π�A1=Φ0, ifweneglect the difference between �A1 and �A2. This rate of change is twiceas fast as would have been observed in the absence of couplingbetween the inner and outer edge modes.We remark that in the course of adding one flux quantum to thearea �A1, onewould expect on average to have a jumpbyone electron ineach spin state. So, in general, one will have one positive jump in Q2and one negative jump in Q1. Thus, while the observed interferencephase will change by an amount equivalent to a change of two elec-trons, the actual change in Q1 will only be one electron.Influence of bulk-edge coupling and screening by nearby gatesScreening of long-range Coulomb interactions by nearby gates in ourdevice is essential to considerwhen determining the influenceof edge-edge and bulk-edge coupling on the interference signal. In the dataanalysis and preceding discussion, we have neglected bulk-edge cou-pling since the radius of the interference loop extracted from the outerEC interference (~600 nm) is always much larger than the averagedistance to the graphite screening gates (~38 nm), as shown in Sup-plementary Fig. 9. Therefore, bulk-edge interaction should be wellscreened by the gates. In contrast, the distance between the inner andouter EC approaches ~85 nm in the strong-coupling limit, whichenables strong Coulomb interaction that will not be screened out bythe gates. We also observe that the contribution of edge-edge cou-pling, as evidenced by the presence of the f o+ i component in thesignal, becomes significant only when the edge-edge separationbecomes about 200nm or less, within about a factor of 5 of the gatedistance. The behavior of the inner EC interference over the range ofdensity we explored is also consistent with these rough estimates, asthe interference signal continues to show Aharonov-Bohm inter-ference (Supplementary Fig. 8) through a shrinking area down to anenclosed radius of 200nm. Hence, we find that bulk-edge coupling isnegligible in all experimental regimes that we explored.According to the model by Frigeri et al. (ref. 33), for negligiblebulk-edge coupling, the transition to frequency doubling occurs whenthe inter-channel interaction drops to roughly half of the chargingenergy for a bare outer EC. This would correspond to the point wherethe inter-channel distance is comparable to the distance to thescreening gates. We find that indeed this agrees with our data, sincethe system enters the strongly coupled limit when the average inter-channel separation approaches ~85 nm, approximately twice theaverage distance to the top and bottom gates (Supplementary Fig. 9).Robustness of the theoretical predictionsAs discussed in ref. 36, when a single EC passes through the two con-strictions, with weak backscattering at the constrictions, the inter-ference phase seen at low temperatures and low source-drain voltageis given by θ=2πQ+θ0,mod2π, where Q is the total electron charge(in units e) in the region between the two scattering points (theexpectation value of the charge on the interferometer in its groundstate) and θ0 is a constant for small variations in B, VPG, and VMG. Theargument is essentially the same if the backscattering is not weak. Theprincipal effect of stronger backscattering at the QPCs is to add a termto the energy E that favors integer values of the charge Q1 and henceinteger values of the total charge on the interferometer. This meansthat as the control parameters are varied continuously, the phasedifference θ� θ0 will undergo an additional modulation pulling ittowards the nearest integer multiple of 2π. If we define θb as the valueof the interferometer phase that would occur in the limit of weakbackscattering, for the given value of the control parameters, then theArticle https://doi.org/10.1038/s41467-024-50695-1Nature Communications |         (2024) 15:6533 8actual value of θ should have the form θ=θb + δθ, where δθ is a peri-odic function of θb � θ0. In addition, in the presence of finite backscattering, interference contribution to the measured resistivity mayno longer be a simple sinusoidal function of θ but can contain higherharmonics. The combination of these effects means that the inter-ference current will remain a periodic function of θb, with period 2π,but the relative amplitudes of various harmonics may be modified. Inthe main text, it was argued that cos θb should be a two-dimensionalperiodic function of B and the gate voltages, with frequenciesexpressed in terms of two non-colinear fundamental vectors in reci-procal parameter space. The effect of finite backscattering at the QPCswill be to modify the amplitudes of the various Fourier components,but not to change their positions.Using similar arguments, wemayargue thatmeasurement at finitetemperature should not change the locations of the fundamental fre-quency vectors, but thermal fluctuations will reduce the Fourieramplitudes. In general, at high temperatures T , the amplitude of agiven Fourier component will fall off, proportional to e�T=ε, where εwill be different for each Fourier component. At sufficiently hightemperatures, therefore, only the one or two components with thelargest values of ε will remain visible. The values of ε will depend ondetails of the system, but typically the Fourier components that aremost prominent at T =0 will be the ones that persist to highesttemperatures.For our system, in the case where there is only a single EC, as wefind for bulk filling less than 2, the value of ε for the lowest Fouriermode is predicted to be ε=hv=ð2π2PÞ, where v is the EC velocity and Pis the perimeter of the interferometer path. For the case of twostrongly coupled edge channels, the prediction is ε=hv=ð4π2PÞ, wherev is now the velocity of the fast charge mode. In both cases, thedominant effects come from thermalfluctuations eδQ of the charge onthe edge, whose energy cost is given by ðeδQÞ2=ð2γPÞ, where γ is thecapacitance per unit length of the edge. The velocity v is given byv= δσxy=γ, where δσxy is the change in Hall conductance across theedge. 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P.K., T.W., and Y.R.acknowledge support from DOE (DE-SC0012260) for sample prepara-tion, measurement, characterization, and analysis. J.R.E. acknowledgessupport from ARO MURI (N00014-21-1-2537) for sample preparation,measurement, characterization, and analysis. K.W. and T.T. acknowl-edge support from the Elemental Strategy Initiative conducted by theMEXT, Japan, Grant Number JPMXP0112101001, JSPS KAKENHI GrantNumber JP20H00354 and the CREST(JPMJCR15F3), JST. D.E.F. and Z.W.acknowledge support by the National Science Foundation under GrantNo. DMR-2204635. B.I.H. acknowledges support from NSF grant DMR-1231319. M.E.W. and A.Y. acknowledge support from Quantum ScienceCenter (QSC), aNationalQuantum InformationScienceResearchCenterof the U.S. Department of Energy. Nanofabricationwas performed at theCenter for Nanoscale Systems at Harvard, supported in part by an NSFNNIN award ECS-00335765.Author contributionsT.W. and D.N. stacked the graphite-encapsulated heterostructures. T.W.performed the nanofabrication, measurements, and data analysis. J.R.E.and Y.R. assisted in the measurement and analysis. D.E.F., B.I.H., andZ.W. contributed the theoretical analysis. M.E.W. and A.Y. provided themeasurement cryostat andcollaboratedon thediscussions andanalysis.K.W. andT.T. provided thehBNcrystals. T.W., B.I.H., J.R.E., andP.K.wrotethe paper with input from all authors.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-024-50695-1.Correspondence and requests for materials should be addressed toPhilip Kim.Peer review information Nature Communications thanks the anon-ymous reviewers for their contribution to the peer review of this work. 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To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.© The Author(s) 2024Article https://doi.org/10.1038/s41467-024-50695-1Nature Communications |         (2024) 15:6533 10http://arxiv.org/abs/2312.14767http://arxiv.org/abs/2312.14767https://arxiv.org/abs/2405.05486v1https://arxiv.org/abs/2405.05486v1https://arxiv.org/abs/2402.12432v1https://arxiv.org/abs/2402.12432v1https://arxiv.org/abs/2403.18983v2https://arxiv.org/abs/2403.18983v2https://arxiv.org/abs/2403.19628v1https://doi.org/10.1038/s41467-024-50695-1http://www.nature.com/reprintshttp://creativecommons.org/licenses/by-nc-nd/4.0/http://creativecommons.org/licenses/by-nc-nd/4.0/ Strongly coupled edge states in a graphene quantum Hall interferometer Results Interferometer design and tuning Phase jumps and AB oscillation frequency transition AB frequency doubling from strongly coupled QHE edge states Discussion Methods Sample preparation Measurements Estimation of the coupling strength Physics of AB frequency doubling at strong coupling Influence of bulk-edge coupling and screening by nearby gates Robustness of the theoretical predictions Data availability Code availability References Acknowledgements Author contributions Competing interests Additional information