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N. L. Samuelson, L. A. Cohen, W. Wang, S. Blanch, [T. Taniguchi](https://orcid.org/0000-0002-1467-3105), [K. Watanabe](https://orcid.org/0000-0003-3701-8119), M. P. Zaletel, A. F. Young

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[Hard and Soft Phase Slips in a Fabry-Pérot Quantum Hall Interferometer](https://mdr.nims.go.jp/datasets/40ac0110-2c7d-4020-a21d-9a87a80d9481)

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Hard and soft phase slips in a Fabry-P\'erot quantum Hall interferometerHard and soft phase slips in a Fabry-Pérot quantum Hall interferometerN. L. Samuelson,1, ∗ L. A. Cohen,1, ∗ W. Wang,1 S. Blanch,1 T. Taniguchi,2 K. Watanabe,3 M. P. Zaletel,4, 5 and A. F. Young1, †1Department of Physics, University of California at Santa Barbara, Santa Barbara CA 93106, USA2International Center for Materials Nanoarchitectonics,National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan3Research Center for Functional Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan4Department of Physics, University of California, Berkeley, California 94720, USA5Material Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA(Dated: September 15, 2025)Quantum Hall Fabry-Pérot interferometers are sensitive to the properties of bulk quasiparticles enclosed bythe interferometer loop, with the interference phase containing information about both the quasiparticle statis-tics and the Coulomb-mediated bulk-edge coupling. Previous studies have explored the role of the bulk-edgecoupling in an equilibrium picture where quasiparticles enter and exit the interferometer rapidly compared to thetimescale over which the interferometer phase is measured. Here, we present data from a monolayer graphenequantum Hall interferometer in the integer quantum Hall regime at ν = −1 and ν = −2. Quantum interfer-ence shows phase slips associated with the entrance of quasiparticles to the interferometer bulk. Tracing thedependence of these phase slips on the magnetic field, we show that the equilibration time can become as longas several minutes. We further use our multi-gated geometry to identify two classes of phase slips. The firstis associated with the addition of a quasiparticle to a bulk ‘puddle’ of quasiparticles uniformly coupled to theentire chiral edge state, while the second is associated with the addition of a quasiparticle trapped by a defectsite that couples predominantly to the closest portion of the edge.In the quantum Hall edge-state Fabry-Pérot interferometer,the interference signal measures the phase accumulated bythe current-carrying excitations on the edge of an interferenceloop. In the integer quantum Hall (IQH) regime, the measuredphase arises due to the Aharonov-Bohm effect, making it pro-portional simply to the magnetic flux enclosed by the edgestate[1]:θ2π=AIBΦ0(1)where AI is the interferometer area, B is the applied mag-netic field, and Φ0 = h/e is the magnetic flux quantum. Inthe simplest picture, then, one expects a linear variation of theinterference phase as the magnetic field and area (tuned by,for instance, a plunger gate voltage) are changed. Howeverthis picture is complicated by the fact that small changes inthe total electron density are accommodated by the creationof localized quasiparticles in the bulk of the interference loop.These quasiparticles exert a Coulomb force on the compress-ible edge state, causing it to move when the number of bulkquasiparticles, Nqp, changes. Consequently, AI can have acomplicated, geometry-dependent relationship to Nqp as wellas B, leading to discrete step-like shifts in θ as the number ofenclosed quasiparticles changes. This effect is known in theliterature as bulk-edge coupling. In this work we use the pres-ence of these small “slips” in the interference phase to deter-mine when individual quasiparticles are added to or removedfrom the bulk of an IQH interferometer.Prior works have measured and characterized bulk-edgecoupling phenomena both experimentally [2–6] and theoret-ically [7–9], and the role of the competing electrostatic ener-gies in typical device geometries is now well-understood. Of-ten the goal is to suppress these effects as much as possible,to which end significant progress was made by engineering(a)0.60.0G2T (e2/h)31 2 4VP (V)VSD  (mV)2.00.0ν = -1, B = 4.0T(b)(c) (d)1μmVCM (mV)dI/dVSD (e2/h)0 2 40.0-0.40.4 535 μVFIG. 1. Monolayer Graphene Fabry-Pérot interferometer. (a)Device schematic showing patterned graphite top gates, hBN dielec-tric spacers, monolayer graphene, and graphite bottom gate. (b)AFM topograph of the graphite top gate. (c) High-visibility Fabry-Pérot oscillations in the cross-device conductance as a function ofthe plunger gate voltage VP . (d) dI/dVSD|VCMas a function of thedc common mode voltage VCM = 12(VS + VD) and source-draindifference VSD = VS − VD across the sample.devices with nearby conducting layers to screen the interac-tion between bulk and edge; in many such devices, Aharonov-Bohm (rather than Coulomb-) dominated interference was ob-served in both the integer and fractional quantum Hall regimes[10–20]. These efforts have culminated in the observation ofphase jumps in the fractional quantum Hall regime which canbe understood as arising nearly entirely from a distinct phys-ical mechanism — the anyonic braiding statistics. Notably,most of the above works have focused on the effects of bulk-edge coupling resulting from the detailed energetics at equi-librium of the quasiparticles occupying the bulk of the inter-arXiv:2509.09901v1  [cond-mat.mes-hall]  11 Sep 2025https://arxiv.org/abs/2509.09901v12CBB (T)VP (V)3.94 3.983.96 4.18 4.20 4.22 4.494.474.450.60.1IT / ISRC1.04.02.03.0ΔB = 8mTareadecr.(θ-θAB)/2πB (T)B (T) B (T)GXY (e2/h)1.01.20.9AB (T)4 6III4.494.474.450.0-0.2-0.4-0.60.0970.0940.1020.0840.0880.0950.0800.098τ (sec)041234.20 4.24B (T)2(a) (b) (c)(d) (e) (f) (g)VP (V)I T / ISRC1.40 1.500.20.50.30.4FIG. 2. Continuous evolution of the quasiparticle charging time at ν = −1. (a) Transmitted current IT across the interferometer as afunction of the plunger gate voltage, VP , and magnetic field B, measured at a fixed electron density, with VB = 0.3V and VC = −0.580Vwithin the ν = −1 plateau. The interference is Aharonov-Bohm dominated throughout. Continuous “soft” phase slips at B ≈ 3.96T evolveinto (b) “noisy” phase slips near B ≈ 4.2T and finally (c) “hard” phase slips for B ≈ 4.47T. (d) Measurement of Rxy on the W side ofthe device at VC = −0.580V and VB = 0.3V. The gray boxes mark the field ranges corresponding to panels a, b, and c. (e) Trace of thetransmitted current measured over 36 seconds across one of the noisy lines at the point marked in panel b. We observed two-state switchingnoise corresponding to two phase-shifted interference curves. (f) Evolution of the quasiparticle switching rate as a function of magnetic fieldin the “noisy” regime. Plotted values are the average of the state 1-to-state 2 and the state 2- to state 1 switching times τ ≡ 12(τ12 + τ21) at thepoint where these values are most nearly equal, corresponding to charge degeneracy. The error bar is the absolute difference |τ12 − τ21|. (g)The phase θ as a function of magnetic field, extracted from the Fourier transform of panel c. The median phase difference between successivetraces is subtracted to remove the smooth evolution of the Aharonov-Bohm phase. The eight discrete jumps shown in the figure have an averagevalue of ∆θ/2π = −0.092 and standard deviation of 0.008.ferometer. Here, we emphasize a heretofore underappreciatedaspect of the physics underlying the Fabry-Pérot interferom-eter, which is that the interference phase can also be used toprobe the dynamics of the quasiparticles through the time de-pendence of the quasiparticle occupation, Nqp(t). While afew theoretical works have considered the effect that slowor random-in-time changes in Nqp may have on the inter-ference signal [21–23], experimental observations have beenlimited to the Coulomb-blockaded regime[24, 25]. Here wefocus on the effects of a slowly time-dependent Nqp near theAharonov-Bohm dominated limit.Our experimental system consists of a dual graphite-gatedmonolayer graphene interferometer, with a patterned top gatedefining the edge trajectory as illustrated in Fig. 1a. The topgraphite gate is patterned before stacking via local anodic ox-idation lithography[26] (see Fig. 1b) into 6 separate top gates–the plunger (P) gate, which primarily controls the interfer-ometer area, the center (C) gate which is used to define thebulk filling factor in the cavity and on either side of the deviceand four additional gates (NW, SW, NE, and SE) which are de-pleted in order to define two quantum point contacts (QPCs) atwhich the edge states are partially back-scattered to completethe interference loop. The top- and bottom- hBN spacers are40 nm and 50 nm thick, respectively, and the nominal area ofthe interference cavity is 0.80± 0.10 µm2.Fig. 1c shows the two terminal conductance at bulk fill-ing ν = −1 as a function of VP with the exterior of the in-terferometer over-depleted such that νexterior > 0. The highdegree of coherence is apparent from the conductance oscil-lations which are narrowly peaked rather than sinusoidal, in-dicating the presence of higher harmonics in the interferencesignal. The velocity of the edge modes can be extracted[1, 27]from the period of the oscillation pattern as a function of thedc source-drain voltage, e∆VSD = 2hvL , where v is the edgevelocity and L is the loop perimeter, and ∆VSD is the periodof the oscillations as a function of the source drain difference.In this measurement VS and VD are the individual voltages onthe source and drain electrodes which we vary independently.Estimating L ≈ 3.36µm from the lithographically definedperimeter, the measured period ∆VSD = 535µV gives anedge velocity v = 2.17 × 105m/s, consistent with previousfindings in graphene systems [12, 13, 15].Fig. 2a shows an interference plot acquired for ν = −1.Since ∂AI/∂VP < 0 for ν = −1, we orient 2(a-c) to be right-3handed with respect to the B − AI plane. The lines of con-stant phase have an negative slope consistent with “Aharonov-Bohm” dominated behavior, in which AI is approximatelyfixed as B is increased at fixed VP [3]. The measured mag-netic field period of 8mT yields an inferred area of 0.52 µm2,slightly smaller than the nominal lithographic area.Notably, the negatively-sloped lines are not perfectlystraight, but instead are interrupted by smooth phase slipswhere the phase evolves with B more rapidly. We inter-pret these as corresponding to the addition or subtraction ofa quasiparticle; in other words, at these points in the VP − Bplane, it becomes energetically favorable to add or expel asingle quasiparticle from the interferometer. The resultingchange in phase is due to the bulk-edge coupling.Figs. 2a-c show the evolution of the phase slip behav-ior with increasing B. The data are acquired with VP asthe fast axis, and all other gate voltages are held constantso that increasing B brings the density closer to the centerof the ν = −1 plateau (see Fig. 2d). The phase slips un-dergo a qualitative change in behavior with increasing B . AtB ≈ 4.2T , (Fig. 2b), phase slips are associated with notice-able increase in the noise of the measured current near thephase slip positions. Notably, in this regime phase slips ondifferent constant-flux lines occur across lines of finite slopein the VP -B plane, highlighted by the white dashed line inFig. 2b. At B ≈ 4.45T , (Fig. 2c), the noise associated withthe phase slips vanishes, as does the slope of the phase sliplines in the VP -B plane. Instead, sharp discontinuities occurbetween successive constant-B data traces.To quantitatively understand the evolution of the interfer-ence patterns between these regimes, we investigate the in-creased noise observed in Fig. 2b. Fig. 2e illustrates the con-ductance measured while sweeping the VP gate across oneof these noisy lines at constant B, taken over the course of36 seconds. The noise takes the form of two-state randomtelegraph noise with a few-second timescale, with the sig-nal switching between two curves with a small phase offset.These ‘noisy’ slips provide a natural connection between thesmooth slips of Fig. 2a and ‘hard’ slips of Fig. 2c. In panel a,the quasiparticle occupation near a phase slip fluctuates fasterthan can be resolved in our measurement. This results in anapparently gradual phase shift, as the measured current nearthe charge-degeneracy point is a time-average of the phase-shifted curves corresponding to the two distinct quasiparticleoccupation numbers. In panel b, the quasiparticle occupa-tion remains constant over the averaging time for each point,switching only occasionally. As a result, configurations bothwith and without an additional quasiparticle are accessible inthe same range of VP .The ability to resolve individual quasiparticle tunnelingevents in the time domain allows us to directly measure thecharacteristic time τ describing quasiparticle entry into theinterferometer bulk and quantify its evolution with B. Thisis done quantitatively for the 60mT field range presented inFig. 2b. As described in the supplementary material, points ofcharge degeneracy can be identified by comparing the averagetime spent in the Nqp and Nqp + 1 state; charge degeneracyoccurs when these times are equal, with the switching timeτ given by the average time between events. The quantity τmay be understood within a simple circuit model where thebulk and edge are connected by an effective resistance R, andτ = R · Cbulk where Cbulk ≈ 1 fF is the capacitance of theinterferometer bulk. Fig. 2f shows τ measured across a nar-row range of magnetic field where τ falls in an experimentallyconvenient range. τ changes over an order of magnitude veryrapidly as the plateau center is approached; remarkably, theimplied R > 1015Ω when τ > 1 s.The increase in τ shown in Fig. 2f provides a natural ex-planation for the seemingly instantaneous nature of the phaseslips in Fig. 2c. When τ becomes much larger than theT ≈ 7s period of the constant-B traces, the delay betweentuning VP and B across a charging threshold and the actualexit of a quasiparticle from the interferometer bulk will benoticeable experimentally. In this picture, the quasiparticlesin the interferometer bulk are far from equilibrium with theedge, and the quasiparticles exit with some random delay rel-ative to when the equilibration line is crossed, but then do notfluctuate in again before another threshold is crossed and thequasiparticle number again increases. This explains both theirregular spacing between phase slips in Fig. 2c, and the ap-parent disappearance of the slope observed connecting phaseslips at different values of B. Due to the sudden nature of thephase slips, their magnitude can be measured with high accu-racy. Fig. 2g shows the evolution of the interferometer phaseθ as a function of B extracted from Fig. 2c, after subtract-ing the smooth evolution due to the Aharonov-Bohm phase.The magnitudes of the individual phase slips are marked, withδθ ∈ 2π × (0.080− 0.102) for the 8 individual slips found inthe measured magnetic field range.There are two generic possibilities concerning which statesquasiparticles occupy in the interferometer bulk. In one sce-nario, the quasiparticles may enter a compressible puddle witha high density of states in the center of the interferometer inwhich case each added quasiparticle will contribute nearly thesame degree of bulk-edge coupling as the last. As a conse-quence, each individual quasiparticle is expected to producea nearly identical phase slip magnitude. τ in this case de-pends on the separation of the puddle from the edge: as the in-compressible strip of quantum Hall fluid separating the puddlefrom the compressible edge grows (increasing the resistanceacross it), τ = RCbulk becomes larger. Alternatively, eachquasiparticle may be pinned by disorder to a distinct localtrapping potential. Individual quasiparticle states may thenexhibit widely varying degrees of bulk-edge coupling, de-pending on the proximity of the pinning site to the edge. Thedata in Fig. 2a-c are consistent with the first scenario wherequasiparticles occupy a compressible puddle in the bulk, bothbecause the phase slip magnitudes are relatively uniform andbecause the timescale τ increases towards the plateau centerwhere the bulk puddle is expected to shrink.The two scenarios can be distinguished more directly bymeasuring the capacitance of a given quasiparticle state to44.194.18B (T)4.253.75VNE/SE (V)2.04.01.0VP (V)4.194.18 B (T)(c)4.194.18 B (T)2.02.4VNW/SW (V)G2T (e2/h)0.10.6ν = -13.0(d)(e)P NENWSW SE(a)Nqp(b)ℓNEℓSEℓNENN+1N - 1ΔViViBFIG. 3. Dependence of the noisy phase slip lines on the varioussurrounding top gates (a) Schematic of the edge configuration inν = −1 with labels of the surrounding top graphite gates. If quasi-particles occupy a compressible puddle in the center, shown in blue,the capacitive coupling between the puddle and each surroundinggate per unit length should be comparable. (b) Schematic of thephase diagram as quasiparticle occupation of the central puddle ischanged. Again note that the direction of the y-axis is oriented suchthat the Aharonov-Bohm lines have negative slope while the bulkis hole-doped. The noisy phase slip lines have the opposite sign ofslope, and separate regions with fixed Nqp. For a set of uniformlyspaced lines corresponding to charging of the central puddle, thespacing between lines gives the capacitive lever arm of the puddle torespective gate. (c-e) Aharonov-Bohm plots acquired in the regimeof Fig. 2 (b) showing noisy phase slip lines. Measurements are per-formed at T = 40mK, with the bulk density in ν = −1. The twoterminal conductance G2T is plotted, without subtracting the seriesresistances from the in-line RC filters. The y-axis (fast axis) of eachplot corresponds to a different top gate(s), and the magnetic field isswept from low to high. The noisy phase slips lines, marked withdashed white lines, all show slopes with respect to all surroundingtop gates in proportion with the length along which they border theperimeter of the interferometer.the distinct top gate electrodes. For a compressible puddleof quasiparticles, the occupation will be coupled to each topgate by a capacitance proportional to the gate length along theinterferometer edge (see Fig. 3a); in contrast, for a stronglylocalized quasiparticle state the capacitances will differ de-pending on the location of the trapping potential. We testthis hypothesis experimentally by measuring the interferencepattern as a function of different combinations of top gatevoltages in Fig. 3c-e. We focus on the ‘noisy’ phase slipregime where phase slip positions are evident but chargingevents are reversible on the time scale of the gate-magneticfield scans. The capacitance between the puddle and a givengate can be determined by the spacing of phase slips at con-stant B, ∆Vi = e/Ci. Figs. 3c-e show data for, respec-tively, the P gate, both NW and SW gates, and both NEand SE gates. We expect the gate-edge-to-puddle capacitanceCi = e/∆Vi to scale linearly with the length of each gateedge ℓi, which we can estimate from the AFM topographyscan of the top gate as {ℓP , ℓNW/SW , ℓNE/SE} = {0.47 ±0.07, 1.42± 0.10, 1.47± 0.14} µm. From the correspondingdata in Fig. 3c-e, we find {∆VP ,∆VNW/SW ,∆VNE/SE} ={1.63 ± 0.31, 0.48 ± 0.12, 0.71 ± 0.16} V, which yields aproduct which is constant within uncertainty ℓi × ∆Vi ={0.77±0.26, 0.68±0.22, 1.04±0.33}µm ·V, consistent witha single puddle of charge capacitively coupled to the entireedge.The magnitude of the phase slips can be compared withthe theory of bulk-edge coupling[8], which was developed forthe scenario where quasiparticles enter a compressible bulkpuddle. In this theory, the phase evolution is parameterizedby three quantities, KIL,KI , and KL whereE =KI2(δnI)2 +KL2(δnL)2 +KILδnIδnL. (2)Here E is the total energy, δnI is the charge on the interferingedge mode, and δnL is the charge in the bulk of the interfer-ometer. The edge charge δnI is continuous and adjusts so asto minimize E given δnL such that δnI = −δnLKIL/KI .The interference phase is proportional to the total charge con-tained in the interferometer[8]; as a result, for each integerquasiparticle removed from or added to the bulk, the predictedmagnitude of the phase shift is then δθ = ±2πKIL/KI . Ingeneral, KIL will depend on the microscopic nature of thestate to which bulk quasiparticles are added. For the casewhere quasiparticles enter a compressible puddle evenly dis-tributed across the bulk, an estimate of the puddle-edge elec-trostatic coupling when accounting for screening by the gatesgives KIL ≤ 12e2/Cgbulk, where Cgbulk = ϵzAId/2 ≈ 1.0 fF isthe estimated geometric capacitance of the bulk puddle (seesupplementary information). The upper bound is achieved ifthe puddle extends to the edge; if the puddle is displaced in-wards, the coupling is reduced due to screening of the puddle-edge coupling by the gates. Experimentally, KI = hv/L =0.27meV follows from the edge velocity. The bulk-edge cou-pling KIL = KI × δθ/(2π) ≈ 27µeV can then be obtainedfrom the phase slip magnitudes of Fig. 2g. Together, thesegive KIL ≈ 0.17e2/Cgbulk. This value suggests a puddle ofcharge separated from the edge by a width of approximately35 nm (see SI). This separation is large compared to the mag-netic length (ℓB ≈ 13 nm). Tunneling across such a barrier isexpected to be strongly suppressed, consistent with the longtimescale τ we measure for the quasiparticle tunneling be-tween edge and bulk.52.031.99 B (T)3.24.0VNW/SW (V)2.031.99 B (T) 2.031.99 B (T)0.50.0VNE (V)0.50.0VSE (V)1.00.0VP (V)2.012.00 B (T)ν = -2, inner edge G2T (e2/h)0.65 0.95(a) (b)(c) (d)FIG. 4. Relative slopes of a soft phase slip line to each surround-ing top gate (a-d) Aharonov-Bohm plots acquired at T = 40mK,with the bulk density in ν = −2 and the inner edge mode partiallytransmitted. The two terminal conductance G2T is plotted, withoutsubtracting the series resistances from the in-line RC filters. The y-axis (fast axis) of each plot corresponds to a different top gate(s), withotherwise the gate voltages fixed to VBG = 0V, VCG = −0.234V,VNW/SW = 3.025V, VNE/SE = 0V, VP = 0.1V, and themagnetic field is swept from low to high. The otherwise smoothAharonov-Bohm phase evolution (constant-phase lines having a pos-itive slope since ν < 0) is punctuated by a single reproducible phaseslip at around 2.005T, marked with a dashed white line. The slopeof this phase slip line in the V -B plane depends dramatically onwhich gate is swept, implying the relevant quasiparticle state is lo-cated somewhere near the x marked in the insets.Interestingly, while the data acquired in Figs. 2-3 are inagreement with a picture where quasiparticles fill a compress-ible puddle in the bulk, we find signatures of the alterna-tive regime proposed above (wherein added quasiparticles arepinned to local disorder sites) at a different experimental setpoint. Fig. 4 shows measurements at B ≈ 2T with the bulk atfilling factor ν = −2. Note that from the gradual occurrenceof the phase slip in Fig. 4a, we infer that τ is sufficientlyshort that phase slip is reversible on measurement timescales.As shown in Figs. 4a-b, the value of B where the phaseslip occurs is independent of VP and VNW/SW . In contrast,Figs. 4c-d show a weak- and strong dependence of the phaseslip magnetic field on VNE and VSE , respectively. This im-plies that the quasiparticle is added to a localized state locatedclosest to the SE-gate edge of the interferometer cavity, andsecond-closest to the NE-gate edge, but far enough from theP, NW, and SW gates that the capacitive coupling is negligi-ble. This physical picture is also consistent with the large ob-served magnitude δθ = 2π × 0.24 of this phase slip, as mightbe expected for the charging of a single defect—for exam-ple, caused by a charge impurity in the hBN dielectric[28]—localized very near one of the sample edges. Although wehave not studied the nature of bulk quasiparticle states system-atically as a function of ν and B (it is possible, for example,that a similar regime of charging localized defect states alsoexists at ν = −1), our results suggest that which quasiparti-cle states are occupied involves a complex interplay betweenquasiparticle interactions in the bulk and the nature of disorderin van der Waals heterostructures.In conclusion, we have shown that Fabry-Pérot interferom-eters in the quantum Hall regime can be used as a sensitiveprobe of quasiparticle dynamics. In particular, we have shownthat phase slips are characterized by a timescale, τ , related tothe ‘RC time’ to charge a given localized bulk quasiparticlestate. Experimentally, τ can become long on typical labora-tory timescales in our graphene devices, so that the stochasticdynamics of individual charges manifest as sudden and seem-ingly random phase slips in our quasi-DC electronic transportmeasurements of the interferometer phase. We note that de-spite the seeming lack of reproducibility of individual charg-ing events in this regime, stochastic behavior is precisely theexpectation for a clean quantum Hall system in the low tem-perature limit, given the localaization of the bulk states. Inaddition, the protocol we introduce to distinguish betweenquasiparticles which are trapped at local defect sites and thosewhich enter a bulk compressible puddle by measuring the rela-tive lever arms of various gates to the quasiparticle occupationmay be useful for future experiments in the fractional quan-tum Hall states which need to reliably distinguish the effectsof bulk-edge coupling and anyonic statistics. In particular,the presence of a measurable tunneling time τ offers an in-triguing possibility in the fractional quantum Hall regime; onecan measure the magnitude of the observed phase-slips, in thebulk quasiparticle puddle, as a function of the tunneling time.As the droplet shrinks, and τ increases, we expect the bulk-edge coupling to be suppressed, only leaving the contributionfrom the anyon phase in the interference signal for each addedquasiparticle.A possible future improvement on this experiment wouldbe to utilize wide bandwidth radio-frequency impedance re-flectometry to greatly decrease the readout time. In particular,for investigating the interference signatures of potential non-abelian quantum Hall states, such as the even-denominatorstates in bilayer graphene, this capability may be critical. Inthese states the quasiparticle dynamics are likely to be consid-erably faster and may simply lead to complete dephasing ina quasi-DC measurement as presented in this work. 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Ro-nen, “Aharonov-Bohm interference and the evolution of phasejumps in fractional quantum Hall Fabry-Perot interferometersbased on bi-layer graphene,” (2024), arXiv:2402.12432 [cond-mat].[20] J. Kim, H. Dev, A. Shaer, R. Kumar, A. Ilin, A. Haug,S. Iskoz, K. Watanabe, T. Taniguchi, D. F. Mross, A. Stern,and Y. Ronen, “Aharonov-bohm interference in even-denominator fractional quantum hall states,” (2024),http://arxiv.org/abs/2412.19886 arXiv:2412.19886 [cond-mat.mes-hall].[21] C. L. Kane, Phys. Rev. Lett. 90, 226802 (2003).[22] E. Grosfeld, S. H. Simon, and A. Stern, Phys. Rev. Lett. 96,226803 (2006).[23] B. Rosenow and S. H. Simon, Phys. Rev. B 85, 201302 (2012).[24] N. C. van der Vaart, M. P. de Ruyter van Steveninck, L. P.Kouwenhoven, A. T. Johnson, Y. V. Nazarov, C. J. P. M. Har-mans, and C. T. Foxon, Phys. Rev. Lett. 73, 320 (1994).[25] N. C. van der Vaart, L. P. Kouwenhoven, M. P. de Ruyter vanSteveninck, Y. V. Nazarov, C. J. P. M. Harmans, and C. T.Foxon, Phys. Rev. B 55, 9746 (1997).[26] L. A. Cohen, N. L. Samuelson, T. Wang, K. Klocke, C. C.Reeves, T. Taniguchi, K. Watanabe, S. Vijay, M. P. Zaletel, andA. F. Young, Nature Physics 19, 1502 (2023).[27] D. T. McClure, Y. Zhang, B. Rosenow, E. M. Levenson-Falk,C. M. Marcus, L. N. Pfeiffer, and K. W. West, Physical ReviewLetters 103, 206806 (2009).[28] C.-L. Chiu, T. Wang, R. Fan, K. Watanabe, T. Taniguchi,X. Liu, M. P. Zaletel, and A. Yazdani, “High spatial res-olution charge sensing of quantum hall states,” (2024),http://arxiv.org/abs/2410.10961 arXiv:2410.10961 [cond-mat.mes-hall].Supplemental InformationN. L. Samuelson,1, ∗ L. A. Cohen,1, ∗ W. Wang,1 S. Blanch,1 T. Taniguchi,2 K. Watanabe,3 M. P. Zaletel,4, 5 and A. F. Young1, †1Department of Physics, University of California at Santa Barbara, Santa Barbara CA 93106, USA2International Center for Materials Nanoarchitectonics,National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan3Research Center for Functional Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan4Department of Physics, University of California, Berkeley, California 94720, USA5Material Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA(Dated: September 15, 2025)SAMPLE FABRICATIONThe van der Waals stack was fabricated using the van der Waals dry-transfer process with a polycarbonate film on a PDMSdome. Before stacking, electrode-free AFM anodic oxidation lithography is performed on the top graphite to create the inter-ferometer gate structure [27]. After stacking, the device is detached onto a conductively-doped Si substrate with 285 nm ofthermal SiO2 and the polycarbonate is dissolved in chloroform. An aluminum etch mask is defined via E-Beam Lithography(EBL) of a PMMA A4 495K/A2 950K bilayer resist before e-beam evaporation of 40 nm of aluminum. The stack is etchedusing a CHF3/O2 plasma reactive ion etch for several minutes. The aluminum mask is removed by etching in a <3% TMAHsolution (AZ300MIF) for 20 minutes. Then, a contact electrode pattern is defined using a PMMA 950K A8 mask and anotherEBL exposure, a 30 second CHF3/O2 RIE plasma etch is performed to clean the exposed device edges, and edge contacts aredeposited by e-Beam evaporation of 5/15/150nm of Cr/Pd/Au.(a)NW NESW SEPCA = 0.80um2L = 3.36umNENWCGPBGSW SEC1C2C3C4C5C6C7C8AFM Cuts(b)FIG. S1. Device Image and AFM Cut Gate (a) Optical micrograph of the device. Edge contacts to the monolayer are labeled C1-8. Contactsto the gates are marked NW, SW, NE, SE, P, CG, and BG. Scale bar (upper right) is 10µm. (b) AFM topograph of the top graphite gate afterpick-up with an hBN flake (mid-stack). The lithographic area, defined by the perimeter drawn in the red dashed line, is 0.80± 0.10µm2, andthe perimeter is 3.36±0.31µm. Cbulk ≡2ϵ0ϵ⊥Ad≈ 1.0 fF, the geometric capacitance of the interferometer bulk, can be estimated from theseparameters, with d = 45nm being average of the top- and bottom- gate dielectric thicknesses of 40 and 50 nm respectively. The perpendiculardielectric constant of hBN is taken as ϵ⊥ ≈ 3.25.2MEASUREMENT PARAMETERSExperiments were performed in a dry dilution refrigerator at a base temperature of 55mK. Electronic RC filters on alltransport lines are used to lower the electron temperature. Transport was measured at 17.7777Hz with SR860 lock-in amplifiers.A Basel Precision Instruments SP983c high stability I to V converter (IF3602) is used to amplify the current signals, while thevoltages are measured directly with the SR860 using no additional pre-amplification.Fixed Gate Voltages and Magnetic Fields for Main Text FiguresMain Text Figure VC VBG VNW/SW VNE/SE VP BFig. 1c -0.580V 0.300V 2.20V 4.00V - 4.0TFig. 1d -0.580V 0.300V 2.20V 4.00V 2.20V 4.0TFig. 2a-b -0.580V 0.300V 2.20V 4.00V - -Fig. 2c (g) -0.580V 0.300V 2.50V 4.00V - -Fig. 2d -0.580V 0.300V 0.0V 0.0V 0.0V -Fig. 3c-e -0.580V 0.300V 2.20V 4.00V 2.00V -Fig. 4a-d -0.234V 0.0V 3.025V 0.0 0.1 -FIG. S2. Fixed voltage set points and magnetic fields for main text figures For figures 3 and 4, the fixed voltages specified apply for eachpanel in which the gate is not varied as the fast axis for the measurement.The two terminal conductance across the device, G2T , is measured in figures 1c, 3 and 4, with an ac voltage bias of 25µV,25µV, and 10µV, respectively, without subtracting the in-line series resistance of 23.4 kΩ from the RC filters.A two-ground configuration is employed for the measurements of the transmission coefficient, IT /ISRC , in Fig. 2: A 0.9 nAac current bias is applied at contact C6, and the contact C8 is grounded to sink the current reflected by the interferometer. Thetransmitted current is measured on the opposite side of the device, on contact C4. A constant −12V is applied to the Si substratethroughout these measurements to maintain hole-doping of the contact regions.ANALYSIS OF THE SOURCE-DRAIN AND COMMON-MODE VOLTAGE DEPENDENCEδI/δVSD (e2/h)1.00.0V S - VD (mV)~~-2 0 2 4 6-110VD (mV)~RFRFRDUTδVSDδIδVSVD~VS~~IDC VSVD(a)(b)(c)(d) (f)(e)dI/dVSD (e2/h)1.00.0VCM (mV)V SD (mV)0 2 40-0.40.4 535 μVdI/dVCM (e2/h)0.2-0.2VCM (mV)V SD (mV)0 2 40-0.40.4dI/dVS (e2/h)1.00.0VCM (mV)V SD (mV)0 2 40-0.40.4dI/dVD (e2/h)0.0-1.0VCM (mV)V SD (mV)0 2 40-0.40.4FIG. S3. Source-drain and common-mode DC voltage dependence of the interference in ν = −1 (a) Equivalent circuit model for themeasurement of the DC voltage dependence of the interference (b) The four-terminal conductance ignoring the contribution of the filters, asa function of the applied DC voltages ṼD and ṼS − ṼD . (c-f) dI/dVSD , dI/dVCM , dI/dVS , and dI/dVD numerically calculated from theDC current, as a function of the corrected dc voltages VSD = VS − VD and VCM = 12(VS + VD), accounting for the finite filter resistance.The period in VSD (marked) is 535µV.3A schematic for the measurement circuit is shown in Fig. S3a. Each of the RC filters on the source and drain contacts providesa resistance of RF = 11.7kΩ (the capacitors are not illustrated). A fixed ac voltage bias of δṼS = 138µV is applied, in additionto varying DC source and drain voltages ṼS and ṼD. The AC current into the drain, δI is measured, along with the diagonalvoltage drop δVSD. The sample resistance RDUT is unknown and varies with the (unmeasured) voltages VSD = VS − VD andVCM = 12(VS + VD).The finite resistance of the in-line RC filters on each transport contact leads to an actual voltage drop, VSD = VS − VDacross the sample which is significantly less than the applied DC voltage, ṼS − ṼD. The DC current and voltage are not directlymeasured — instead, we calculate them by integrating the measured ac voltage as a function of the applied ṼS :VSD =∫ ṼS−ṼDṼDδVSDδṼSd(Ṽ ′S − Ṽ ′D)The DC current and DC common mode voltage can be obtained straightforwardly from VSD:I =ṼS − ṼD − VSD2RFVCM =12VSD + ṼD + IRFWe can then take numerical derivatives of the calculated DC current in any chosen direction. These derivatives are shown inFig. S3c-f.The period of the conductance dI/dVSD is used to determine the edge velocity by its relation to the dynamical phase accruedby particles injected at a finite energy: e∆VSD = 2hvL , where L is the perimeter of the interferometer. The common-mode perioddiffers from this — we note that in the measured regime (the same regime as in Fig. ??a, near a large number of soft phase slips),we expect the bulk to have a non-negligible compressibility, which means that the possible addition of quasiparticles cannot beignored as the voltage VCM is swept. This potentially makes the interpretation of the period ∆VCM more complicated.ESTIMATION OF KIL FOR A COMPRESSIBLE PUDDLEHere we justify the estimate KIL ≲ 12e2Cgbulk, where Cgbulk = ϵzAId/2 is the geometric capacitance of the bulk. The upper boundis expected when the density of added quasiparticles is uniform and approaches right up to the edge of the interferometer; if thepuddle is confined inwards, KIL will be decreased exponentially due to screening of the bulk-edge interaction by the gates, asanalyzed below.Recall the couplings K are defined through the phenomenological charging energy E = δn2LKL/2+δnLδnIKIL+δn2IKI/2,where δnI/L is the charge added to the edge / bulk. KIL is then determined by the following: when charge density δnL/AI isadded to the bulk, what is the resulting potential produced at the edge? When the radius of curvature is large compared to d, wecan treat the added charge as an infinite half-plane, with charge distribution δn(x, y) = θ(x)eδnL/AI . Deep in the bulk, x ≫ 0,this distribution produces potential ϕ = eδnL/(AIcg), where cg = ϵz/(d/2) is the capacitance per unit area to the double gates.On the other hand, for x ≪ 0 ϕ = 0. By reflection symmetry, ϕ = eδnL/2AIcg at x = 0. For an edge at x = 0, we thusconclude KIL = eϕ(x = 0)/δnL ≈ e2/2Cgbulk.To determine the correction when the puddle is displaced inwards, we can solve for the full spatial dependence ϕ(x). We firstsolve Poisson’s equation in the presence of the double gate to conclude a line charge ρ at x = 0 produces a potentialϕ(x) = − ρ2π√ϵxyϵzlog(tanh(πx√ϵz/ϵxy/4d)) (1)where d is the gate distance and ϵxy,z is the anisotropic hBN dielectric constant. The total potential produced at the x = 0 edgeby the x > w puddle of density δnL/AI is thusϕIL = −eδnL/AI2π√ϵxyϵz∫ ∞wdy log(tanh(πy√ϵz/ϵxy/4d)) (2)4The geometric estimate is thus KIL = eϕIL/δnL, from which it follows thatKIL/(e2/AIcg) = −12π√ϵxyϵz∫∞wdy log(tanh(πy√ϵz/ϵxy/4d))d2ϵz(3)= − 4π2∫ ∞π√ϵz/ϵxyw/4dds log(tanh(s)) (4)When w → 0, we obtain the limit KIL/(e2/AIcg) → 1/2. For w ≫ d, the bulk edge coupling falls of exponentially due to thescreening from the gates, and asymptotically we findlimw≫dKIL/(e2/Cgbulk) →4π2e−π√ϵz/ϵxyw/2d (5)Taking ϵz = 3.25, ϵxy = 6.6, and (e2/AIcg)/KIL = 5.8, we estimate w/d = 0.78, so w = 35nm.EXTRACTION OF THE QUASIPARTICLE SWITCHING RATEWe find that the timescale of the switching noise, which we interpret as the quasiparticle relaxation time τ , can vary over awide range of scales, leading to the qualitative distinction between soft and hard phase slips discussed in the main text. In Fig. 2a-c, this timescale goes from less than 10ms (the time to acquire a single pixel in the measurement) to greater than 7 s (the timeto acquire a complete line trace). This dramatic increase happens over a range of only 500mT. In the range where τ ≈ 1 s, itis experimentally convenient to extract the timescale precisely and determine quantitatively the dependence on magnetic field inthis regime. Here we describe the process used to extract τ from the interference data in this “switchy” crossover regime fromsoft to hard phase slips, at the field points plotted in Fig. 2f.We analyze the telegraph noise using a two-state switching model, where our readout is the transmitted current, IT , acrossthe interferometer at distinct quasiparticle charge-degenerate lines for several values of magnetic field. Fig. S4a shows ameasurement of IT illustrating one such quasiparticle charge-degenerate lines. Precisely at a charge-degeneracy point, weexpect the tunneling rate to have a single characteristic time τ . In general, however, the ratio P1→2/P2→1 = e−∆/kbT , whereP1→2 is the probability for an electron to hop from state 1 to state 2, P2→1 is the probability for the electron to hop from state 2to 1, and ∆ is the detuning away from the charge degeneracy point in energy. As a result, the location of the charge degeneracypoint in VP − B plane must be determined systematically by measuring the switching noise as a function of this detuning bysweeping the plunger gate, and only at the charge degenerate point exactly will we obtain a single timescale, which we willdefine as τ ≡ τ12 ≈ τ21.Fig. S4B shows the characteristic values of the transmitted current IT between which the signal switches, extracted from 20-minute measurements of the transmitted current while fixed at each point, along the illustrated line from A to B in Fig. S4A.The charge degenerate point is determined by plotting the fraction of time spent in state 1, P (1), as function of the plunger gatevoltage, plotted in Fig. S4C.Fig. S4G-L show histograms of the two dwell times, τ1 and τ2, for different plunger gate voltages near a phase slip line,showing a strong dependence of the relative rates on detuning from charge degeneracy. For a given B, we identify the the valueof VP where τ12 ≈ τ21 as the charge degeneracy point, and plot τ ≡ (τ12+τ21)/2 in Fig. 2f, with the difference ∆τ ≡ |τ12−τ21|as the error bar. We find that τ increases by nearly one order of magnitude over a 45mT range of magnetic fields.∗ These authors contributed equally to this work† andrea@physics.ucsb.edu[1] C. d. C. Chamon, D. E. Freed, S. A. Kivelson, S. L. Sondhi, and X. G. Wen, Physical Review B 55, 2331 (1997).[2] B. Rosenow and A. Stern, Physical Review Letters 124, 106805 (2020).[3] B. I. Halperin, A. Stern, I. Neder, and B. Rosenow, Physical Review B 83, 155440 (2011).[4] D. T. McClure, W. Chang, C. M. Marcus, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 108, 256804 (2012).[5] Y. Zhang, D. T. McClure, E. M. Levenson-Falk, C. M. Marcus, L. N. Pfeiffer, and K. W. West, Physical Review B 79, 241304 (2009).[6] N. Ofek, A. Bid, M. Heiblum, A. Stern, V. Umansky, and D. Mahalu, Proceedings of the National Academy of Sciences 107, 5276(2010).[7] I. Sivan, H. K. Choi, J. Park, A. Rosenblatt, Y. Gefen, D. Mahalu, and V. Umansky, Nature Communications 7, 12184 (2016).[8] J. Nakamura, S. Liang, G. C. Gardner, and M. J. Manfra, Nature Communications 13, 344 (2022).5FIG. S4. Analysis of the quasiparticle switching rate in the switching-noise regime in ν = −1 (A) Measurement of the interference patternin the transmitted current across a single charge degeneracy line in the switching noise regime of ν = −1. (B) The median current in each ofthe two stable configurations in the switching noise regime. The median value of IT in each state, plotted in red and blue, are extracted from20 minute measurements of the current at each fixed VP , by splitting the histogram of the acquired data into two halves at the midpoint andtaking the median value of current for each half. (C) The fractional probability to occupy state 1 at each plunger gate voltage in B, acquiredby simply taking the number of data points lying on either side of the halfway-point threshold between the two stable values of current. Thecharge degeneracy point is defined as the point at which P(1) crosses 0.5. (D,E,F) Three examples of the time-dependent current data usedto extract the dwell times τ12 and τ21. G,H,I Histograms of the dwell times in state 1, t12, each showing an exponential distribution with acharacteristic time τ12 that decreases as the charge degeneracy line is crossed from left to right. (J,K,L) Histograms of the dwell times in state2, t21, showing that the characteristic time τ21 increases as the charge degeneracy line is crossed from left to right.[9] B. Rosenow and B. Halperin, Physical review letters 98, 106801 (2007).[10] S. Ngo Dinh and D. A. Bagrets, Phys. Rev. B 85, 073403 (2012).6[11] J. Nakamura, S. Fallahi, H. Sahasrabudhe, R. Rahman, S. Liang, G. C. Gardner, and M. J. Manfra, Nature Physics 15, 563 (2019).[12] J. Nakamura, S. Liang, G. C. Gardner, and M. J. Manfra, Nature Physics 16, 931 (2020).[13] C. Déprez, L. Veyrat, H. Vignaud, G. Nayak, K. Watanabe, T. Taniguchi, F. Gay, H. Sellier, and B. Sacépé, Nature Nanotechnology 16,555 (2021).[14] Y. Ronen, T. Werkmeister, D. Haie Najafabadi, A. T. Pierce, L. E. Anderson, Y. J. Shin, S. Y. Lee, Y. H. Lee, B. Johnson, K. Watanabe,T. Taniguchi, A. Yacoby, and P. Kim, Nature Nanotechnology 16, 563 (2021).[15] J. Nakamura, S. Liang, G. Gardner, and M. Manfra, Physical Review X 13, 041012 (2023).[16] H. Fu, K. Huang, K. Watanabe, T. Taniguchi, M. Kayyalha, and J. Zhu, Nano Letters 23, 718 (2023).[17] T. Werkmeister, J. R. Ehrets, Y. Ronen, M. E. Wesson, D. Najafabadi, Z. Wei, K. Watanabe, T. Taniguchi, D. E. Feldman, B. I. Halperin,A. Yacoby, and P. Kim, “Strongly coupled edge states in a graphene quantum Hall interferometer,” (2023), arXiv:2312.03150 [cond-mat].[18] N. L. Samuelson, L. A. Cohen, W. Wang, S. Blanch, T. Taniguchi, K. Watanabe, M. P. Zaletel, and A. F. Young, “Anyonic statistics andslow quasiparticle dynamics in a graphene fractional quantum hall interferometer,” (2024), arXiv:2403.19628 [cond-mat.mes-hall].[19] T. Werkmeister, J. R. Ehrets, M. E. Wesson, D. H. Najafabadi, K. Watanabe, T. Taniguchi, B. I. Halperin, A. Yacoby, and P. Kim, “Anyonbraiding and telegraph noise in a graphene interferometer,” (2024), arXiv:2403.18983 [cond-mat.mes-hall].[20] J. Kim, H. Dev, R. Kumar, A. Ilin, A. Haug, V. Bhardwaj, C. Hong, K. Watanabe, T. Taniguchi, A. Stern, and Y. Ronen, “Aharonov-Bohm interference and the evolution of phase jumps in fractional quantum Hall Fabry-Perot interferometers based on bi-layer graphene,”(2024), arXiv:2402.12432 [cond-mat].[21] J. Kim, H. Dev, A. Shaer, R. Kumar, A. Ilin, A. Haug, S. Iskoz, K. Watanabe, T. Taniguchi, D. F. Mross, A. Stern, and Y. Ronen,“Aharonov-bohm interference in even-denominator fractional quantum hall states,” (2024), arXiv:2412.19886 [cond-mat.mes-hall].[22] C. L. Kane, Phys. Rev. Lett. 90, 226802 (2003).[23] E. Grosfeld, S. H. Simon, and A. Stern, Phys. Rev. Lett. 96, 226803 (2006).[24] B. Rosenow and S. H. Simon, Phys. Rev. B 85, 201302 (2012).[25] N. C. van der Vaart, M. P. de Ruyter van Steveninck, L. P. Kouwenhoven, A. T. Johnson, Y. V. Nazarov, C. J. P. M. Harmans, and C. T.Foxon, Phys. Rev. Lett. 73, 320 (1994).[26] N. C. van der Vaart, L. P. Kouwenhoven, M. P. de Ruyter van Steveninck, Y. V. Nazarov, C. J. P. M. Harmans, and C. T. Foxon, Phys.Rev. B 55, 9746 (1997).[27] L. A. Cohen, N. L. Samuelson, T. Wang, K. Klocke, C. C. Reeves, T. Taniguchi, K. Watanabe, S. Vijay, M. P. Zaletel, and A. F. Young,Nature Physics 19, 1502 (2023).[28] D. T. McClure, Y. Zhang, B. Rosenow, E. M. Levenson-Falk, C. M. Marcus, L. N. Pfeiffer, and K. W. West, Physical Review Letters103, 206806 (2009).[29] C.-L. Chiu, T. Wang, R. Fan, K. Watanabe, T. Taniguchi, X. Liu, M. P. Zaletel, and A. Yazdani, “High spatial resolution charge sensingof quantum hall states,” (2024), arXiv:2410.10961 [cond-mat.mes-hall].