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M. Jo, June-Young M. Lee, A. Assouline, P. Brasseur, [K. Watanabe](https://orcid.org/0000-0003-3701-8119), [T. Taniguchi](https://orcid.org/0000-0002-1467-3105), P. Roche, D. C. Glattli, N. Kumada, F. D. Parmentier, H. -S. Sim, P. Roulleau

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[Scaling behavior of electron decoherence in a graphene Mach-Zehnder interferometer](https://mdr.nims.go.jp/datasets/92abf023-7ec9-4083-ae45-98524706bc0c)

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Scaling behavior of electron decoherence in a graphene Mach-Zehnder interferometerArticle https://doi.org/10.1038/s41467-022-33078-2Scalingbehaviorof electrondecoherence inagraphene Mach-Zehnder interferometerM. Jo1,5, June-Young M. Lee 2,5, A. Assouline1, P. Brasseur1, K. Watanabe 3,T. Taniguchi 3, P. Roche 1, D. C. Glattli 1, N. Kumada 4, F. D. Parmentier 1,H. -S. Sim 2 & P. Roulleau 1Over the past 20 years, many efforts have been made to understand andcontrol decoherence in 2D electron systems. In particular, several types ofelectronic interferometers have been considered in GaAs heterostructures, inorder to protect the interfering electrons fromdecoherence. Nevertheless, it isnow understood that several intrinsic decoherence sources fundamentallylimit more advanced quantum manipulations. Here, we show that grapheneoffers a unique possibility to reach a regime where the decoherence is frozenand to study unexplored regimes of electron interferometry. We probe thedecoherence of electron channels in a graphene quantum Hall PN junction,forming a Mach-Zehnder interferometer1,2, and unveil a scaling behavior ofdecay of the interference visibility with the temperature scaled by the inter-ferometer length. It exhibits a remarkable crossover from an exponentialdecay at higher temperature to an algebraic decay at lower temperaturewherealmost no decoherence occurs, a regime previously unobserved in GaAsinterferometers.The field of electron quantumoptics relies on the analogy between thepropagation of electrons in a quantum conductor and that of photonsin quantumoptics experiments. This research field emerged in the latenineties with the possibility of manipulating electron beams in con-densed matter systems while preserving their wave-particle nature. Ithas proven since then to grant a fundamental understanding ofquantum electronics down to the single-particle excitation. The pro-totypical systems of electron quantum optics are two-dimensionalconductors in the quantum Hall effect regime. This regime is reachedunder strong perpendicular magnetic field and is characterized by theexistence of one-dimensional, chiral and dissipationless electronicchannels propagating along the edges of the sample. Those quantumHall edge channels canbedirectly viewed as the analogof opticalfibersfor electrons. A large majority of these experiments has been per-formed in GaAs/AlGaAs semiconductor heterostructures, where it hasbeen shown that decoherence can stem from different sources: edgereconstruction by disorder3, intra-channel Coulomb interactionswithin a single edge channel4,5, and inter-channelCoulomb interactionsbetween adjacent edge channels6–8. The lattermechanism is thought tobe the main hindrance in realizing complex quantum circuits withquantum Hall edge channels in GaAs, and has received considerabletheoretical and experimental attention. In particular, recent experi-ments have shown that it can be diminished by a somewhat cumber-some engineering of the edge channels9,10. Despite this, identifyingdecoherence sources remains an open problem, with e.g. severalpreviouslyoverlookeddissipationmechanisms that have been put intolight in the last few years11,12, and experimental observations that arestill debated after more than a decade13,14. Probing decoherence inquantum Hall edge channels realized in different 2D materials poten-tially allows tackling this issue, by providing an apparently similarsystem whose intrinsic parameters (e.g. electron velocity, capacitivecoupling and screening, geometry) are nonetheless sufficientlyReceived: 8 October 2021Accepted: 30 August 2022Check for updates1SPEC, CEA,CNRS, Université Paris-Saclay, CEA Saclay, 91191Gif sur Yvette, Cedex, France. 2Department of Physics, Korea Advanced Institute of Science andTechnology, Daejeon 34141, Korea. 3National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 4NTT Basic Research Laboratories, NTTCorporation, 3-1Morinosato-Wakamiya, Atsugi 243-0198, Japan. 5These authors contributed equally:M. Jo, June-YoungM. Lee. e-mail: hs_sim@kaist.ac.kr;preden.roulleau@cea.frNature Communications |         (2022) 13:5473 11234567890():,;1234567890():,;http://orcid.org/0000-0002-9547-1256http://orcid.org/0000-0002-9547-1256http://orcid.org/0000-0002-9547-1256http://orcid.org/0000-0002-9547-1256http://orcid.org/0000-0002-9547-1256http://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/0000-0001-6089-4083http://orcid.org/0000-0001-6089-4083http://orcid.org/0000-0001-6089-4083http://orcid.org/0000-0001-6089-4083http://orcid.org/0000-0001-6089-4083http://orcid.org/0000-0002-1457-0915http://orcid.org/0000-0002-1457-0915http://orcid.org/0000-0002-1457-0915http://orcid.org/0000-0002-1457-0915http://orcid.org/0000-0002-1457-0915http://orcid.org/0000-0001-7826-6894http://orcid.org/0000-0001-7826-6894http://orcid.org/0000-0001-7826-6894http://orcid.org/0000-0001-7826-6894http://orcid.org/0000-0001-7826-6894http://orcid.org/0000-0001-9319-565Xhttp://orcid.org/0000-0001-9319-565Xhttp://orcid.org/0000-0001-9319-565Xhttp://orcid.org/0000-0001-9319-565Xhttp://orcid.org/0000-0001-9319-565Xhttp://orcid.org/0000-0002-1678-874Xhttp://orcid.org/0000-0002-1678-874Xhttp://orcid.org/0000-0002-1678-874Xhttp://orcid.org/0000-0002-1678-874Xhttp://orcid.org/0000-0002-1678-874Xhttp://orcid.org/0000-0002-8397-5019http://orcid.org/0000-0002-8397-5019http://orcid.org/0000-0002-8397-5019http://orcid.org/0000-0002-8397-5019http://orcid.org/0000-0002-8397-5019http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-33078-2&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-33078-2&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-33078-2&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-33078-2&domain=pdfmailto:hs_sim@kaist.ac.krmailto:preden.roulleau@cea.frdifferent to obtain a full pictureofdecoherence. In this letter,weprobethe decoherence in an electronic Mach-Zehnder interferometer rea-lized in a graphene PN junction in the quantum Hall regime, in whichthe typical energy scales for decoherence are an order of magnitudelarger than inGaAs/AlGaAs. This allowsus toobserve for thefirst timearemarkable universal behavior in the dependence of the interferenceswith the temperature. Together with the bias voltage dependence ofthe visibility, we show that for those systems decoherence is mainlydue to intra-channel Coulomb interactions.ResultsIn our PN junction, a graphene monolayer is encapsulated by hex-agonal boron nitride layers, and the density of the left and right halvesare controlled independently using bottom and top gates, where theformer (resp. latter) is covering the whole sample (resp. only the righthalf), as shown in Fig. 1a and the “sample description” section in theSupplementaryMaterial. Under a perpendicularmagneticfield, the lefthalf becomes a P region of filling factor νp = −1. Along the boundary ofthe P region, a spin-up channel circulates clockwise. The right half is anN region of νn = 2. Along its boundary, two channels having oppositespin circulate counterclockwise. As a result, the junction interface hasthe three co-propagating channels. The two spin-up channels haveopposite valley-isospin1. A Mach-Zehnder interferometer is formed atthe PN interface by applying the top and bottom side gates (see Fig. 1).Along the top edge, the injected current I0 is carried by the two edgechannels of the N region. Half of the current, resulting from spin downcarriers, cannot flow to the P region, because of large energy cost forspin flip. The other half I0/2 with spin up carriers, on which we focushereafter, can contribute to the transmitted current IT. Therefore thetransmission probability was measured as TMZ = IT / (I0/2). The fillingfactors ν1 and ν2 below the side gates are controlled independently. Inthe “large” interferometer, νi = 1,2 = −1 (see the first panel of Fig. 1b). Inthis case, the spin-up channels from the P and N regions collide at thetop and bottom “edge” intersections of the PN interface with thegraphene edge below the side gates, leading to formation of theirbeam splitters (which are called valley splitters2). The atomic structureof the graphene edge causes sharp potential change at the edgeintersection, hence, scattering between the spin-up channels havingopposite valley-isospin15,16. The two spin-up channels co-propagatingalong the PN interface and their beam splitters at the top and bottomedge intersections constitute the large interferometer that exhibitstransmission oscillations of period ΔB = 20mT as a function of themagnetic field (note that this period is slightly different from the onereported in ref. 2 although it is the same device. Between the twomeasurements, the experimental setup has been deeply modified. Italso corresponds to two different cooldowns with different gate vol-tages and edge electrostatics). The arm length of the large inter-ferometer is estimated as L = 1.5μmfrom the samplegeometry (see the“sample geometry” section in the Supplementary Material). When aside gate is further tuned to have ν1 = 0 or ν2 = 0, the Aharonov-Bohmoscillations disappear as reported in refs. 1, 2, but they reappear incertain ranges of the side gate voltage (see the “tuning length of theinterferometer” section in the Supplementary Material). Two differentperiods ΔB = 34.5mT and 81mT of the reappeared Aharonov-Bohmoscillations are observed (see the second and third panels of Fig. 1b),and they are much larger than the period ΔB = 20 mT of the largeFig. 1 | Experimental setup and Aharonov-Bohm oscillations. a Schematicrepresentation of the PN junction. The N region is depicted in blue, the P one inpink. Electrons are injected from the upper right ohmic contact by applying a biasvoltage VDC, and electron transmission TMZ is measured at the lower left contact.b Left panel: TMZ with respect to change ΔB of the magnetic field. Its oscillationperiod decreases as the interferometer length increases. Right panel: Schematicview of the Mach-Zehnder interferometer. At the beam splitters (dotted curves),mixing between the spin-up channels having opposite valley-isospin occurs. In thelarge interferometer, the beam splitters are formed at the intersections of the PNinterface with the graphene edge. In the intermediate one, the top beam splitter isformed at the intersection with the edge, while the bottom splitter is at the inter-sectionwith the ν =0 region in the bulk. In the small one, the two beam splitters areformed in the bulk. c TMZ oscillations of the large interferometer plotted at dif-ferent temperature. The absence of temperature dependence of the average TMZunambiguously ruled out the effect of the temperature on the intervalleyscattering rate.Article https://doi.org/10.1038/s41467-022-33078-2Nature Communications |         (2022) 13:5473 2interferometer. The disappearance is due to the fact that an edgeintersection, hence a beam splitter there, cannot be formed below theside gate of ν =0, because of spatial separation of the channel of the Pregion from those of the N region by the ν = 0 region. The interestingreappearance of the Aharonov-Bohmoscillations, accompanied by theperiods much larger than that of the large interferometer, implies theformation of a beamsplitter not at the edge intersection but at anotherposition below the side gate. The only possible position, where thechannels fromtheP andN regions cancollide, is the “bulk” intersectionof the PN interface with the ν = 0 region inside the graphene bulk (seethe schematic views in Fig. 1b). The resulting interferometers aresmaller than the large one; in the “intermediate” interferometer, ν1 = −1and ν2 = 0, while νi = 1,2 = 0 in the “small” interferometer. Their inter-ferometer length is estimated as L = 1.05 μm and 0.62μm from thelocation of the bulk intersections in the sample geometry (see Sup-plementary Fig. S2 in the Supplementary Material). The ratios of theestimated lengths of the three interferometers match with those oftheir Aharonov-Bohmperiods; small mismatch can be interpreted thatthe spacing between the two interferometer arms (the two spin-upinterface channels) slightly differs between the interferometers,depending on the filling factor of the side-gate regions. So it is con-cluded that a beam splitter is formed inside the bulk rather than at theedgewhen νi = 1,2 = 0. The originof the beam splitter (namely, the valleymixing) at the bulk intersectionsmay be atomic defects, as reported inrecent STM experiments17 on the same source of graphene as ours(NGS graphenium flakes), or many-body states at ν =0.The arms of the three interferometers are expected to be similar.The gate configurations of the three interferometers are the same inthe regionbetween the side gates, anddifferent below the side gates. Arecent calculation18 studying edge-channel reconstruction along a PNinterface, based on the Chklovski-Shklovskii-Glazman model, esti-mates that for the gate configuration of our experiment, the spacingbetween the two arms is 102 nm / 90 nm in the presence/absence ofthe side gates. Note that this estimation is comparable with 110 nm/83 nm obtained from the experimental Aharonov-Bohm period ofthe large/small interferometer2. Therefore, the side gates change theproperties of the arm not largely, and it is reasonable to compare theinterferometers based on their arm length difference as the firstapproximation.We study the interference visibility Vis=(TMZ,max – TMZ,min)/(TMZ,max + TMZ,min), where TMZ,max (min) is the maximum (minimum)value of the oscillation of TMZ (see Fig. 1c). We first discuss thermaldecoherence. Figure 2a shows the interference visibility normalized byV0, that is the visibility at base temperature, as a function ofFig. 2 | Scaling behavior of thermal decoherence. a Thermal decay of the inter-ference visibility Vis in log scale for the three interferometers. V0 is the visibilitymeasured at the fridge base temperature. The visibility is defined as: Vis=(TMZ,max-TMZ,min)/(TMZ,max+ TMZ,min)=Td/Ts with Td = TMZ,max- TMZ,min and Ts = TMZ,max+TMZ,min. Td and Ts and their associated errors δTd and δTs are obtained from the fit.Visibility’s error bar is given by δVis=(Ts*δ Td – Td*δ Ts)/ (Ts)2. b Thermal decay ofthe interference visibility Vis in log scale for the two intermediate interferometerconfigurations. c The decay is redrawn with scaled temperature LT/L0 where T istemperature, L is the interferometer length, and L0 is the length of the largeinterferometer. The decayof the three interferometer lies on the same curve, whichis in good agreement with an intra-channel interaction model. In the theory, wechoose the parameters of v =4.4 × 104m/s and g = 3.3.dVoltage spectral density asa function of the frequency for different refrigerator temperature. The red curvesare the theoretical fits given by the Johnson-Nyquist noise of the circuit which iscomposed of the sample resistance in parallel with an RLC resonator. The gain ofthe amplification chain is extracted from the fit. e Average current noise as afunction of temperature.Article https://doi.org/10.1038/s41467-022-33078-2Nature Communications |         (2022) 13:5473 3temperature. Notably, the interference persists above 1.5 K, a tem-perature much higher than the usual operating temperature of theGaAs edge channels. Below 1 K, the visibility decays not exponentially,but algebraically, which means that thermal decoherence is sup-pressed. The crossover temperature from the algebraic to exponentialregime becomes higher for the smaller interferometers. We have alsomeasured the interference visibility as a function of the temperaturefor the two intermediate configurations (ν1 = −1, ν2 = 0 and ΔB = 34.5mT or ν1 = 0, ν2 = −1 and ΔB = 39 mT) and observe the same behavior(see in Fig. 2b). The graphene interface channels are quite robustagainst thermal decoherence even around 1 K and the coherencelength, extracted from the exponential decay regime, is 1.24μm at 1 K.We note that the algebraic decay does not originate from heatingeffects. Heating effects are excluded in our case by measurement ofelectron temperature through the Johnson-Nyquist noise (see the“noise setup” section in the Supplementary Material). We use home-made cryogenic amplifiers combined to a LC tank circuit at 2.2MHz toavoid the 1/f noise generated by the amplifiers’ HEMT and the parasiticnoise induced by the dry dilution refrigerator vibrations. After a secondstage of amplification at room temperature, the voltage fluctuations aredigitized with an acquisition card and the noise spectral density iscomputed for different refrigerator temperatures (see Fig. 2d). Thecurrent fluctuations of the Hall resistance are given by SI =4kBTeRHwithkB the Boltzmann constant, Te the electronic temperature and RH = h2e2 :The average value of SI is linear down to a refrigerator temperature of25mK (Fig. 2e), confirming that electrons are perfectly thermalized.Remarkably, the visibility curves lie on a single curvewhenplottedwith temperature scaled by the interferometer length L with a clearcrossover from an algebraic to an exponential decay of the visibilitythat has never been observed in conventional semiconductors (seeFig. 2c). Namely, the visibility does not dependon temperature and thelength independently, but only on the product of the two. The cross-over temperature, which is inversely proportional to L, is 350mK in thelarge interferometer. Note that the scaling behavior is satisfied overthe large length variation by 300%. This confirms the formation of theinterferometers and the beam splitters at the bulk intersections whenνi = 1,2 = 0, and also validates that the algebraic decay does not originatefrom electron heating. In the following we discuss the differentdecoherence mechanisms that lead to such a scaling behavior.The algebraic decay implies suppression of thermal decoherence.The algebraic decay has not been reported in GaAs interferometers;there has been a report on a non-exponential decay that may originatefrom heating effects14. By contrast, in our graphene interferometersbeing ten times smaller than the GaAs ones13,14, the universal crossoveris clearly observed. The scaling behavior requires a decoherencemechanism to follow the scaling with the interferometer length orhave a length scale much longer or shorter than the interferometerlength. This excludes disorders or small charge puddles of the bulkfrom the mechanism. As possible mechanisms, one can cite inter-channel interactions between adjacent interface channels and intra-channel interactions. The short-range inter-channel interactions cancause decoherence through fractionalization of electron flow into slowand fast modes as in the GaAs edge channels7,8. We will show in thefollowing that this latter mechanism is negligible and that the inter-channel interactions are dominated by the intra-channel interactions.To compare an intra-channel interaction model with our experi-mental observations, we consider a simple capacitive Hamiltonian5,Hint = ECPα = l,rQ̂α=e� Ng� �2, ð1Þwhere Q̂α = l,r is the charge inside the left and right interferometer armsrespectively, Ng is a reference charge number determined by the gatevoltages, EC = gv_= 2Lð Þ is the charging energy, v is the drift velocity,and g is a dimensionless interaction parameter. Physically, when anelectron enters an interferometer arm, charge density fluctuations inthe arm provide which-path information through the capacitiveinteraction, reducing the interference. The interference visibility,computed with the intra-channel interaction model, satisfies thescaling behavior (Methods), exhibits a universal crossover from thealgebraic to the exponential regime, andfits verywell the experimentalthermal decay (Fig. 2c). The crossover happens at the temperatureT ~ _v=ðkBLÞ comparable with the single-particle level spacing. In thealgebraic regime below the crossover temperature, the electronthermal length is longer than the interferometer arms, hence thethermal charge fluctuations and the resulting decoherence by theinteraction are suppressed.By contrast, inter-channel interactions are negligible. We observethe evolution of the visibility with the temperature as we add additionalPN interface channels by changing their respective filling factors. InFig. 3, the visibility decay for (νn, νp) = (4,−1) ismoreor less similar to the(2,−1) case, and the decay for (2,−2) is stronger only slightly. TheoverallFig. 3 | Thermaldecoherence at variousfilling factors. a Interference visibility Vis versus temperature, as in Fig. 2a, for the large interferometer at (νn, νp) = (2, −1), (2, −2),(4, −1). b Schematic representation of the different configurations.Article https://doi.org/10.1038/s41467-022-33078-2Nature Communications |         (2022) 13:5473 4similarities among the decay curves indicate that the inter-channelinteractions are not the dominant source of the dephasing, althoughthey may not be completely suppressed. This observation is supportedby the geometry of the sample and the configuration of the edgechannels. First, in the sample geometry, the vertical distance (30 −50 nm) between the gates and the graphene layer is shorter than thespacing (50 − 60nm) between two adjacent edge channels leading toscreening of the inter-channel interactions; the spacing is indicated,with assuming a symmetric PN junction, by the 110 nm spacing betweenthe two arms estimated2 from the Aharonov-Bohm period. This is insharp contrast with the GaAs edge channels, where the distancebetween the edge channels and the top gates is typically 90 − 100nm.Note that in our geometry, theN region hasmore screening of the inter-channel interactions than the P region, since it is more affected by thetop gate. Second, in the (νn, νp)= (2, −1) case, the additional channel issandwiched between the interferometer arms along the PN interface sothat its interaction with the left armwill be similar to its interaction withthe right arm if the interactions are present. When it interacts with thetwo arms equally, our theoretical calculation (see the “theoreticalmodels” section in the Supplementary Material) shows that no deco-herence is induced by the interaction. Instead, to fit the thermal visi-bility decay in Fig. 2 by the calculation based on the inter-channelinteraction, it is required that one arm interacts with the additionalchannel about 10 times more strongly than the other arm. This asym-metry is unlikely in our setup. In the (2, −2) case, on the other hand,another additional channel is formed outside of the interferometer inthe P region, resulting in asymmetry in its interactionwith the two arms.In this case, the inter-channel interaction can cause, yet weak, deco-herence. In the (4, −1) case, the two more channels are added in the Nregion in comparison with the (2, −1) case, and they are far apart fromthe interferometer arms, so decoherence by them will be negligible.These suggest that the inter-channel interactions do not provide thedominant decoherence mechanism.Next, we examine non-equilibrium decoherence by a finite biasvoltage applied to the large interferometer at (νn, νp) = (2, −1). Figure 4shows the voltage dependence of the visibility, which is called the lobepattern14,19–21. In the pattern, a dip occurs near 220 μV and a single sidelobe appears at larger voltages. Interestingly, Fig. 4 shows that thepattern depends onlyweakly on the beamsource, i.e., whether the biasvoltage is applied to the upper right ohmic contact (biasing the twoedge channels of the N region) or the upper left contact (biasing thesingle edge channel of the P region). This feature is in contrast with theGaAs interferometers at ν = 2, whose lobe pattern depends largely onwhether a bias voltage is applied to only one14,19,20 or the two edgeFig. 4 | Dependenceof lobe pattern on the beamsource.TransmissionTMZof thelarge interferometer at (νn, νp) = (2, −1) as a function of the magnetic field and theDC bias VDC applied to a the upper right ohmic contact or b the upper left contact.In a (resp. (b)), electron beam is injected from the N region of νn = 2 (resp. P regionof νp = −1) to the interferometer, biasing the two (resp. single) edge channels of theregion. Note that the DC bias VDC is inverted in b for comparison. c Interferencevisibility as a function ofVDC in the cases a, b. It is measured at 9 T. The similarity ofthe lobe pattern between a, b implies that inter-edge interactions between theinterface channels are weak.Article https://doi.org/10.1038/s41467-022-33078-2Nature Communications |         (2022) 13:5473 5channels21 because of inter-channel interactions. Note that our intra-channel interactions model qualitatively reproduces the lobe pattern(see the “theoretical models” section in the Supplementary Material).This confirms that contrary to conventional GaAs interferometers,decoherence in graphene in the quantum Hall regime effect is domi-nated by intra-channel interactions.Before concluding, wemention recent related works on grapheneFabry-Pérot interferometers22,23. In22 the thermal decay of the inter-ference visibility of the Fabry-Pérot interferometer is discussed fordifferent lengths of the interferometer. The decay is only exponential;no algebraic decay is found probably due to the presence of etchdefined edges24. However, in a gate defined interferometer23, nosaturation of the visibility at low temperature is observed. Theextracted coherence lengthof 8.1 µmat 32mk is 4.78 times smaller thanin the presentMZI. A smaller coherence length associatedwith a largerarea may explain the absence of visibility saturation at low tempera-ture in a graphene Fabry-Pérot.DiscussionQuantum Hall systems provide a promising platform for the imple-mentation of quantum information processing. Indeed, the one-dimensional dissipationless quantum Hall edge channels form an idealand tunable propagation medium for quantum coherent single-electron wave-packets, the spatial trajectories of which encode theinformation to process. The basic building blocks of this so-calledflying qubit approach were demonstrated in GaAs/AlGaAs hetero-structures. However, the limited phase coherence length in thismaterial (∼20μm at 20mK) critically hampers the development ofcomplex multi-qubit architectures needed for quantum informationprocessing. To circumvent thismajor issue, a newparadigm in termsofmaterial is necessary.In this work we perform a detailed study of the decoherenceprocesses in a graphene Mach-Zehnder interferometer. While weobserve the usual exponential thermal decay of the visibility at hightemperature, below a crossover temperature (∼350mK) the deco-herence is suppressed. We expect that the presence of the top andbottomgates close to the graphene layer screens interactions betweenco-propagating edge states, a large source of decoherence. We finallyreach a regimewhere intra channel interactions are themain source ofdecoherence. Thepossibility towork in a regimewhere decoherence issuppressed makes graphene a very promising platform for applica-tions to flying qubits25–27, orbital entanglement generation28,29 andvalleytronics30.MethodsVisibilityThe visibility noted “Vis” is defined as: Vis = (TMZ,max- TMZ,min)/(TMZ,max+TMZ,min) with TMZ,max the maximum value of the MZI transmission andTMZ,min the min value.The normalized visibility is defined as Vis/ V0 with V0 the visibility atbase temperature.In ref. 2, a precise study of the transmission (T) dependence of V0 isdone that clearly demonstrates theffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTð1� TÞpdependence of thevisibility. Nevertheless, it is also shown that the maximum visibility is60%. It remains unclear why we could not reach 100%.We note I0 the injected current. The definition of the transmission TMZdepends on the injected current and the pn junction configuration.In the νL = −1 / νR = +2 configuration, there is a spectator edge state thatdoes not participate to the interference. The interfering edge statescarry the same spin while the spectator edge state carries an oppositespin. The energy cost to flip the spin at 9 T is given by the Zeemanenergy gµB= 1mV that is much larger than the electronic temperature:T = 25mK corresponds to V = kBT/e = 1.2µV. Therefore, the spectatoredge channel is fully reflected and half of the current cannot betransmitted leading to a definition of theMZI transmission TMZ = IT/(I0/2). In the νL = −1 / νR = +4 configuration, three edge channels from theright region will be reflected andTMZ = IT/(I0/4). Note that for the voltage-dependent lobe pattern,the visibility is also defined with differential conductance.Theoretical model and scaling behaviorWe explain the theoretical model based on the intra-edge capacitiveinteraction model, and show how the scaling behavior appears in themodel. The full Hamiltonian is described byH =H0 +Hint +HTwhere H0 = � _vPα= l,rRdxψyα xð Þi∂xψαðxÞ is the Hamiltonian forelectrons in the left (α = l) and right (α = r) arms of the MZI. TheHamiltonian Hint for the intra-channel interaction is given in Eq. (1).The Hamiltonian HT =TU +TD +h:c: describes the MZI beam splitter.Here, TU tð Þ= _vtUeieVDC t_ eiϕAB2 ψyr 0,tð Þψl 0,tð Þ and TD tð Þ= _vtDeieVDC t_ eiϕAB2ψyr L,tð Þψl L,tð Þ describe electron tunneling from the right arm to theleft arm at the first and second beam splitters, respectively. ϕAB is theAB phase enclosed by the MZI loop.To show the universal thermal crossover of the MZI visibility, weconsider the regime of small tunneling amplitudes tU and tD. Thecurrent through the MZI is computed,IT = ∣tD∣2 + ∣tU ∣2� � e2hVDC � e_2Zdt TU 0ð Þ,TyD tð Þh iD E+ c:c:� �:The maximum and minimum values of the MZI transmission followsTMZ,max +TMZ,min = 2ð∣tD∣2 + ∣tU ∣2Þ and TMZ,max � TMZ,min =4π_2∣Rdt ith½TU ð0Þ,TyDðtÞ�i∣. Hence, the visibility of the differential conductancethrough the MZI at the zero bias limit is given byVis =2∣tD∣ � ∣tU ∣∣tD∣2 + ∣tU ∣2 ∣2πv2Xη= ±ηZdt itGη tð Þ∣: ð2ÞGη (t) is an electronic Green’s function at finite temperature under theHamiltonianH0 +Hint:G+ ðtÞ= hψyl ðL, tÞψl ð0, 0ÞihψrðL, tÞψyr ð0, 0Þi andG�ðtÞ= hψlð0, 0Þψyl ðL, tÞihψyr ð0, 0Þψr ðL, tÞi: It is calculated by using the bosonizationmethod,Gη tð Þ= exp δGηðtÞð Þ2vkBTsin½πkBTv ða� iηðL� vtÞÞ�� �2 ,δGη tð Þ= i2ηLZdqg2πsinqL=2qL=2� �21 + g2πsinqL=2qL=2 e�iηqL=2eiηqðL�vtÞ1� e�qv=kBTwhere a (> 0) is an infinitesimal length cutoff. δGηðtÞ comes from theintra-channel interaction.The scaling behavior Vis(L, T) = Vis(LT) of the visibility can be seenby the fact that Eq. (2) is invariant under the rescaling of variables witha scaling parameter b>0: arm length L→ bL, temperature T→ b−1 T,length cutoff a→ ba, momentum q→ b−1 q, and time t→ bt.Supplementary Materials include the dependence of the thermaldecay of the visibility on the interaction parameter g, and also thecalculation of the dependence of the visibility on the bias voltage forarbitrary tunneling amplitudes tU and tD.MeasurementsWe used a Cryoconcept dry dilution refrigerator with a base tem-perature of 13mK. Measurements of transmitted currents and RHallvalues were performed using multiple Lock-in amplifiers with lownoise preamplifiers. AC excitations 1nA-5nA with different frequenciesArticle https://doi.org/10.1038/s41467-022-33078-2Nature Communications |         (2022) 13:5473 6(70Hz-300Hz)wereused. Buried ohmic contacts underneath top gatesenabled us the direct determination of filling factors from regions ofinterest.Data availabilityAll data, code, and materials used in the analysis are available in someform to any researcher for purposes of reproducing or extending theanalysis.References1. Wei, D. S. et al. Mach-Zehnder interferometry using spin-and valley-polarized quantum Hall edge states in graphene. Sci. Adv. 3,e1700600 (2017).2. Jo, M. et al. Quantum Hall Valley Splitters and a Tunable Mach-Zehnder interferometer in Graphene. Phys. Rev. Lett. 126,146803 (2021).3. Chamon, C., de, C. & Wen, X. G. Sharp and smooth boundaries ofquantum Hall liquids. Phys. Rev. B 49, 8227 (1994).4. Seeling, G. & Buttiker, M. Charge-fluctuation-induced dephasingin a gated mesoscopic interferometer. Phys. Rev. B 64,245313 (2001).5. Youn, S.-C., Lee, H.-W. & Sim, H.-S. NonequilibriumDephasing in anElectronic Mach-Zehnder Interferometer. Phys. Rev. Lett. 100,196807 (2008).6. Roulleau, P. et al. 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It is also supported by Korea NRF via theSRC Center for Quantum Coherence in Condensed Matter (Gran-tNo.2016R1A5A1008184) and NRF-2019-Global Ph.D. fellowship.Author contributionsM.J., P.B., A.A. & P.Rou performed the experiment with help fromF.D.P. and P.R.; M.J., P.B., A.A., F.D.P, H.S.S & P.Rou analysed anddiscussed the data with help from P.R.; J.Y.L, H.S.S, developed thetheoretical model and discussed it with P. Rou; T.T., K.W. provided theBN layers; M.J. fabricated the device with inputs fromW.D., P.B., A.A.,F.D.P & P.Rou; J.Y.L, H.S.S and P.Rouwrote themanuscript with inputsfrom all coauthors; H.S.S and P.Rou designed and supervised theproject.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-022-33078-2.Correspondence and requests for materials should be addressed toH. -S. Sim or P. Roulleau.Peer review informationNatureCommunications thanks Boris Brun, andthe other, anonymous, reviewer for their contribution to the peer reviewof this work. Peer reviewer reports are available.Reprints and permission information is available athttp://www.nature.com/reprintsPublisher’s note Springer Nature remains neutral with regard to jur-isdictional claims in published maps and institutional affiliations.Article https://doi.org/10.1038/s41467-022-33078-2Nature Communications |         (2022) 13:5473 7https://doi.org/10.1038/s41467-022-33078-2http://www.nature.com/reprintsOpen Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing,adaptation, distribution and reproduction in any medium or format, aslong as you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons license, and indicate ifchanges were made. The images or other third party material in thisarticle are included in the article’s Creative Commons license, unlessindicated otherwise in a credit line to the material. If material is notincluded in the article’s Creative Commons license and your intendeduse is not permitted by statutory regulation or exceeds the permitteduse, you will need to obtain permission directly from the copyrightholder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2022Article https://doi.org/10.1038/s41467-022-33078-2Nature Communications |         (2022) 13:5473 8http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Scaling behavior of electron decoherence in a graphene Mach-Zehnder interferometer Results Discussion Methods Visibility Theoretical model and scaling behavior Measurements Data availability References Acknowledgements Author contributions Competing interests Additional information