††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Anyonic statistics and slow quasiparticle dynamics in a graphene fractional
quantum Hall interferometer

Noah L. Samuelson  Department of Physics, University of California at Santa
Barbara, Santa Barbara CA 93106, USA  Liam A. Cohen  Department of Physics,
University of California at Santa Barbara, Santa Barbara CA 93106, USA  Will
Wang  Department of Physics, University of California at Santa Barbara, Santa
Barbara CA 93106, USA  Simon Blanch  Department of Physics, University of
California at Santa Barbara, Santa Barbara CA 93106, USA  Takashi Taniguchi
International Center for Materials Nanoarchitectonics, National Institute for
Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan  Kenji Watanabe
Research Center for Functional Materials, National Institute for Materials
Science, 1-1 Namiki, Tsukuba 305-0044, Japan  Michael P. Zaletel  Department
of Physics, University of California, Berkeley, California 94720, USA
Material Science Division, Lawrence Berkeley National Laboratory, Berkeley,
California 94720, USA  Andrea F. Young<EMAIL_ADDRESS>Department of
Physics, University of California at Santa Barbara, Santa Barbara CA 93106,
USA

###### Abstract

Anyons are two dimensional particles with fractional exchange statistics that
emerge as elementary excitations of fractional quantum Hall phasesLeinaas and
Myrheim (1977); Wilczek (1982); Arovas  _et al._ (1984); Halperin (1984);
Goldin  _et al._ (1980). Experimentally, anyonic statistics manifest directly
in the edge-state Fabry-Pérot interferometer geometryKivelson (1990); Chamon
_et al._ (1997); Kim (2006); Halperin  _et al._ (2011), where the presence of
$N_{qp}$ localized anyons in the interferometer bulk contributes a phase
$N_{qp}\theta_{a}$ to the observed interference pattern, where $\theta_{a}$ is
twice the statistical exchange phaseRead and Das Sarma (2024). Here, we report
a measurement of $\theta_{a}$ in a monolayer graphene Fabry-Pérot
interferometer at $\nu=1/3$. We find a preponderance of phase slips with
magnitudes $\Delta\theta\approx 2\pi/3$, confirming the result of past
experiments in GaAs quantum wellsNakamura  _et al._ (2020, 2023) and
consistent with expectations for the tunneling of Abelian anyons into the
interferometer bulk. In contrast to prior work, however, single anyon
tunneling events manifest as instantaneous and irreversible phase slips,
indicative of quasiparticle equilibration times exceeding 20 minutes in some
cases. We use the discrepancy between the quasiparticle equilibration rate and
our measurement speed to vary the interferometer area $A_{I}$ and $N_{qp}$
independently, allowing us to precisely determine the interferometer phase and
monitor the entry and exit of individual anyons to the interferometer loop in
the time domain. Besides providing a replication of previous interferometric
measurements sensitive to $\theta_{a}$ in GaAs Nakamura  _et al._ (2020), our
results bring anyon dynamics into the experimental regime and suggest that the
average ‘topological charge’ of a mesoscopic quantum Hall device can be held
constant over hour long timescales.

##  I Introduction

When an Abelian anyon is brought along a closed trajectory encircling $N_{qp}$
localized anyons, its wavefunction accumulates a phase

$\frac{\theta}{2\pi}=\frac{e^{\ast}}{e}\frac{A_{I}B}{\Phi_{0}}+N_{qp}\frac{\theta_{a}}{2\pi},$
(1)

where $A_{I}$ is the area of the loop, $B$ is the applied magnetic field,
$\theta_{a}$ is twice the exchange phase, and $e^{\ast}$ is the quasiparticle
charge. Quantum Hall edge state Fabry-Pérot interferometers exploit the
contrast between localized anyons in the bulk and propagating anyonic
quasiparticles along the chiral edge modes to directly observe this
phaseKivelson (1990); Chamon  _et al._ (1997); Kim (2006); Halperin  _et al._
(2011). In a Fabry-Pérot interferometer, delocalized quasiparticles enter the
cavity via a quantum point contact (QPC) and propagate along the edge to a
second QPC; they can then exit the cavity immediately or complete an integer
number of additional circuits before exiting. Trajectories differing by the
number of circuits give an interference contribution to the conductance,
$\delta G$, that is periodic in $\theta$ and can be measured as a function of
$B$, $A_{I}$, or $N_{qp}$. The clearest signature of anyonic statistics is
expected if $N_{qp}$ changes discretely while keeping $A_{I}$ and $B$ fixed;
the resulting jump in $\theta$ then gives $\theta_{a}$ directly.

Fabry-Pérot interferometers have been investigated in GaAs heterostructures
for nearly two decadesCamino  _et al._ (2005, 2007); Zhang  _et al._ (2009);
McClure  _et al._ (2009); Ofek  _et al._ (2010); Choi  _et al._ (2015); Sivan
_et al._ (2016); Nakamura  _et al._ (2019, 2020, 2023); Willett  _et al._
(2023). These experiments have revealed that Coulomb interactions may prevent
observation of the anyon phaseHalperin  _et al._ (2011). Specifically, as
charge enters the bulk of the interferometer, Coulomb repulsion may cause a
change in $A_{I}$, leading to an observable phase shift even for fermionic
quasiparticles. If the “bulk-edge coupling” is large, this change in the
Aharonov-Bohm phase can completely obscure the contribution of $\theta_{a}$.
Recently, a breakthrough in the design of GaAs heterostructures led to the
observation of phase shifts that agree quantitatively with the expected
$\theta_{a}=2\pi/3$ in the $\nu$ = 1/3 stateNakamura  _et al._ (2019, 2020,
2022, 2023). Graphene heterostructures are a natural venue in which to extend
these results owing to the large fractional quantum Hall energy gaps observed
at both odd-Dean  _et al._ (2011); Polshyn  _et al._ (2018); Zeng  _et al._
(2019); Dean  _et al._ (2020) and even-denominator filling factorsZibrov  _et
al._ (2017); Li  _et al._ (2017); Zibrov  _et al._ (2018); Hu  _et al._
(2023); Assouline  _et al._ (2024). Moreover, the nearby graphite gates in
typical dual-gated geometries ensure a high degree of screening, suppressing
bulk-edge coupling. Indeed, measurements of Fabry-Pérot interferometers in
graphene have uniformly observed Aharonov-Bohm dominated interferenceDéprez
_et al._ (2021); Ronen  _et al._ (2021); Zhao  _et al._ (2022); Fu  _et al._
(2023); Kim  _et al._ (2024); however, no measurements of the anyon phase at
fractional filling have been reported.

Figure 1:  Fabry-Pérot interference in the $\nu=1/3$ state (A) Schematic of
the dual-graphite gated edge state Fabry-Pérot interferometer. The gates
defining the interferometer are labeled C, NW, SW, NE, SE and P. Edge states
are formed in the monolayer graphene around the center-gated region (C) and
enter the interferometer via two QPCs tuned by the NW/SW and NE/SE gates.
There is also a global graphite back gate. (B) Hall conductance $G_{XY}$
measured on the west side of the device as a function of the voltage $V_{TOP}$
applied to all top gates together, with a fixed back gate voltage
$V_{BG}=$-1.5\text{\,}\mathrm{V}$$. (C) Conductance through each QPC vs. the
DC source-drain bias $V_{SD}$ (applied as an added voltage on the source
electrode), with the C region fixed at the density corresponding to the marked
point in the 1/3 plateau in panel B. The blue trace was taken with the NW/SW
regions at filling 1/3, and the NE/SE regions depleted. The red trace was
taken with the NE/SE regions at filling 1/3 and the NW/SW regions depleted
(see Supplement for the individual QPC pinch-off curves). (D) $G_{D}$ measured
across the interferometer at $\nu=1/3$, with both QPCs partially transmitting.
The magnetic field is swept from low to high. (E) The phase extracted from the
Fourier transform of the data in panel D. A continuous Aharonov-Bohm phase is
subtracted, and we adopt the convention $\Delta\theta\in(-\pi,\pi)$.

##  II Fabry-Pérot Interference in $\nu=1/3$

In this work, we study a monolayer graphene gate-defined Fabry-Pérot
interferometer, shown schematically in Fig. 1a. The interferometer is
fabricated using anodic-oxidation lithography to define the gate structure in
a graphite layer, which is then combined using dry van der Waals assembly
techniques to form a device with six separately-gated regionsCohen  _et al._
(2023a). In our device (Fig. 1a), two pairs of gates (NE/SE and NW/SW) define
QPCs, while a plunger gate (P) provides additional control of the
interferometer area. The center gate (C) and a global graphite bottom gate are
used, together, to set the filling factor and, through fringe electric fields,
adjust the transmission through the QPCs. Transport data at
$B=$9\text{\,}\mathrm{T}$$ as a function of a common voltage applied to all
six top gates in Fig. 1b show well-developed plateaus at filling factor 1/3
and 2/3. We operate our interferometer within the 1/3 plateau at the indicated
point at $B=$9\text{\,}\mathrm{T}$$. We measure the transmission through the
interferometer via the diagonal conductance, $G_{D}\equiv
I_{\text{out}}/(V_{+}-V_{-})$ (see Fig. 1a). To confirm that our experiment is
probing chiral edge modes of the 1/3 state, we measure the source-drain bias
dependence of the two QPCs in the partial transmission regime individually. As
shown in Fig. 1c, both show strong suppression of $G_{D}$ at low bias, as
expected for tunneling between chiral Luttinger liquids at the QPCsWen (1991);
Kane and Fisher (1992); Fendley  _et al._ (1994); Kane and Fisher (1995);
Fendley  _et al._ (1995); Milliken  _et al._ (1996); Radu  _et al._ (2008);
Cohen  _et al._ (2023b). This behavior is neither expected nor observed in the
integer quantum Hall regime (see Supplement for a comparable measurement in
the $\nu=2$ state, showing no zero-bias suppression).

Fig. 1d shows $G_{D}$ at $\nu=1/3$ with both QPCs set to partial pinch-off as
a function of $V_{P}$ and $B$. The interference shows high-visibility
oscillations with lines of constant phase having a negative slope in the
$V_{P}-B$ plane, consistent with an Aharonov-Bohm dominated interference
phaseHalperin  _et al._ (2011). Following Eq. (1), we estimate the effective
interferometer area to be $A_{I}=3\Phi_{0}/\Delta B=0.69-0.83\mathrm{\mu
m^{2}}$ based on the $\Delta B\approx 15-$18\text{\,}\mathrm{m}\mathrm{T}$$
field period of the oscillations. This agrees with the nominal device area of
$0.74-0.83\mathrm{\mu m^{2}}$ of the patterned graphite gates, without
accounting for bulk-edge coupling, suggesting such coupling is small (see
supplement) Halperin  _et al._ (2011).

The edge velocity may be estimated from the period of the interference as a
function of a common-mode DC voltage applied to the transport leads, shown in
the supplement. Yamauchi  _et al._ (2009); McClure  _et al._ (2009); Chamon
_et al._ (1997). This gives an edge state capacitance $C_{I}=e/\Delta
V_{\mathrm{CM}}=e^{2}/$234\text{\,}\mu\mathrm{e}\mathrm{V}$$ from which the
edge velocity follows via the relation $\frac{hv}{\nu
L}=$234\text{\,}\mu\mathrm{e}\mathrm{V}$$. Taking
$L=$3.30(11)\text{\,}\mu\mathrm{m}$$ we obtain $v=6.2\pm 0.2\times
10^{4}\mathrm{m/s}$, comparable to prior estimates for the velocity of integer
quantum Hall edge states in graphene systemsDéprez  _et al._ (2021); Ronen
_et al._ (2021); Fu  _et al._ (2023); Werkmeister  _et al._ (2023) but several
times larger than measured in GaAs quantum wells under similar external
conditionsNakamura  _et al._ (2020).

In addition to the continuously-tuned Aharonov-Bohm phase giving rise to the
negative slope, the oscillations in Fig. 1d are punctuated by sharp ‘slips’
where the interference phase changes suddenly between $V_{P}$ traces taken at
consecutive values of $B$. The majority of the events are effectively
instantaneous with respect to the 30-second measurement time of each
individual $V_{P}$ trace, manifesting as a discontinuity between traces taken
at subsequent magnetic fields. To quantify the magnitude of the phase slips,
we compute the Fourier transform of $G_{D}$ with respect to $V_{P}$ for each
value of $B$ and extract the phase of the largest-magnitude peak, which
determines the oscillation phase $\theta$. Per Eq. (1), $\theta$ is expected
to contain both a smoothly-varying Aharonov-Bohm contribution as well as an
anyonic contribution proportional to $N_{qp}$. To isolate the latter, we take
a running trimmed-mean of the line-by-line phase difference and subtract it
from the measured phase.

The residual phase, $\theta-\theta_{AB}$, is plotted in Fig. 1e. We can only
determine phase differences $\mod$ $2\pi$, so we take the convention that the
phase slips lie in the interval $(-\pi,\pi)$. We mark the magnitude of each
phase slip calculated from the difference in average value between intervals
of stable phase. The statistical error in the phase slip measurements is
small; repeated measurements within a single interval of stable phase show a
standard deviation of $\sigma_{\theta}\approx.012$. A larger source of error
arises from the fact that the oscillations are not perfectly periodic in
$V_{P}$, generating different values of $\theta$ for different components of
the Fourier transform. We estimate that this error may be as high as $\pm
2\pi\times.04$ (see supplement). Most of the marked phase slips are
consistent, within this uncertainty, with $\theta_{a}=2\pi/3$. We interpret
each event as corresponding to the entry of a single charge-$e/3$ anyon into
the interferometer.

The small deviation in the phase slip magnitudes from the predicted value
suggests a low degree of bulk-edge coupling for many of the eventsRosenow and
Halperin (2007); Halperin  _et al._ (2011). To assess the plausibility of this
in our geometry, we estimate the bulk-edge coupling for an anyon entering a
localized impurity state a distance $R>d$ from the edge, where
$d=$45\text{\,}\mathrm{n}\mathrm{m}$$ is the gate distance. When a charge
$e/3$ fills such a state, the compressible edge will partially screen the
resulting potential, effectively increasing $A_{I}$. Accounting for the
electrostatics of a double-gated device (see supplement), this edge screening
will induce a charge

$\delta
Q_{I}\approx-\frac{e}{3}\frac{C_{I}}{\pi\sqrt{\epsilon_{xy}\epsilon_{z}}L}\exp\left(-\frac{\pi
R}{2d}\sqrt{\frac{\epsilon_{z}}{\epsilon_{xy}}}\right),$ (2)

where $\epsilon_{xy}=6.6$ and $\epsilon_{z}=3$ are the dielectric constants of
the hexagonal boron nitride dielectric, and $L$ is the perimeter of the
interferometer. Taking the measured values of $C_{I}$ and $L$, the bulk-edge
coupling contribution to the phase slip is $\delta\theta=2\pi\times\delta
Q_{I}/e\approx 2\pi\times 0.57\,e^{-R/42.5\mathrm{nm}}$. For
$R>$110\text{\,}\mathrm{n}\mathrm{m}$$, corresponding to approximately half of
our interferometer area, this correction is less than our experimental
uncertainty. Assuming impurity sites to be randomly distributed, then, it is
reasonable that the plurality of events will have undetectable bulk-edge
coupling.

However, not all of the $\nu=1/3$ phase slips are consistent with the
quantized value, suggesting that some impurity sites do trap quasiparticles
near the edge. For example, we observe several slips with
$\Delta\theta\approx\pm 2\pi\times 0.25$. From Eq. (2), we estimate the
position of the impurity trapping an anyon to be a distance
$R\approx$80\text{\,}\mathrm{n}\mathrm{m}$$ from the perimeter. Similarly, a
pair of slips near $B=$8.963\text{\,}\mathrm{T}$$ show magnitudes of
$-2\pi\times.194$ and $-2\pi\times 0.134$ whose sum, $-2\pi\times 0.328$,
agrees with the quantized value of $\theta_{a}$. This is consistent with an
anyon entering an impurity state $R\approx$60\text{\,}\mathrm{n}\mathrm{m}$$
from the edge before moving to an impurity state with negligible bulk edge
coupling shortly afterwards.

We may compare these results with previous experiments in GaAs quantum
wellsNakamura  _et al._ (2019, 2020). Near the center of the fractional
quantum Hall plateau, those experiments observed rare, nearly-quantized phase
slips. Outside of this hard-gap region, the interference signal shows a large
increase in the field period and a substantial reduction in the interference
amplitude. This latter behavior was interpreted as the entry of one $e/3$
anyon with every flux quantum, so as to keep the density constant. In this
thermodynamically compressible regime, the Aharonov-Bohm and statistical
phases cancel (on average), keeping $\theta(B)$ nearly constant. However, in
this regime no sharp phase slips associated with anyon entry or exit were
reported, a phenomenon understood to result from rapid thermal fluctuations in
the anyon number when states with $N_{qp}$ and $N_{qp}+1$ are close to
degeneracy. Notably, in this picture fast equilibration between the bulk anyon
occupation and the gapless edge plays a key role.

As in GaAs, we observe a region of continuous, Aharonov-Bohm driven phase
evolution between $B=$8.85\text{\,}\mathrm{T}$$ and
$B=$8.93\text{\,}\mathrm{T}$$ interrupted by a small number of nearly-
quantized phase slips. This is expected for a ‘hard’ gap with only rare
impurity states available to host localized anyonic quasiparticlesNakamura
_et al._ (2020). For $B>$8.93\text{\,}\mathrm{T}$$, in contrast, we observe a
cascade of events, where a phase slip occurs on average once per flux quantum,
as expected outside the hard gap where the system is thermodynamically
compressible. However, we observe no diminution of the interference amplitude
in this regime: each slip remains sharp and close to the quantized value
despite the average value of $\theta(B)$ remaining constant. Moreover, none of
the phase slips shown in Fig. 1d show an experimentally resolvable slope in
the $V_{P}-B$ plane. These findings contrast with experiments on GaAs, where
both $V_{P}$ and $B$ tune the anyon occupation in the vicinity of the charge
degenerate lines on which phase slips occurNakamura  _et al._ (2020).

The vertical nature of the phase slip lines is a consequence of the
instantaneous and irreversible nature of the quasiparticle addition events.
Indeed, repeated measurement of the interference in the regime of Fig 1d shows
that the location of the phase slips changes between measurements (see
supplement). An illustration of this is offered by Fig. 2a, which shows an
interference plot in a similar regime to Fig. 1d. In these plots, the $V_{P}$
is fixed while $V_{C}$ is swept rapidly to vary the interferometer area and
produce an interference pattern, while the magnetic field is ramped from
minimum to maximum over 33 minutes. As in Fig. 1d, a series of discrete slips
are observed. However, in this comparatively rapid measurement, phase slips do
not occur until the magnetic field has already been increased by
$70\text{\,}\mathrm{m}\mathrm{T}$, nearly 20 minutes after the beginning of
the measurement. This is evidenced by the continuous Aharonov-Bohm phase
evolution at low $B$, which reflects the changing flux through a loop of fixed
anyon charge and approximately fixed size. Reversing the direction of the $B$
sweep (Fig. 2b) reveals a symmetric, and thus hysteretic, behavior where phase
slips start only after the field is lowered sufficiently. Evidently, the
location of a given phase slip in our experiment is not an equilibrium
property, but depends on the history of the system. This suggests that the
anyon occupation $N_{qp}$ is out of equilibrium with the sample edge.

Hysteretic behavior of this type can be understood as a consequence of slow
charge dynamics. As illustrated in Fig. 2c, slow charging leads to a
discrepancy between the the equilibrium value of the chemical potential,
$\mu_{eq}$, which is set by the gate voltages and magnetic field, and the
actual chemical potential of the graphene layer $\mu(t)$. This effect is not
completely unexpected, as long charging times are a characteristic of quantum
Hall systemsGoodall  _et al._ (1985); Eisenstein  _et al._ (1992), including
within quantum Hall gaps in similarly fabricated graphene devicesYang  _et
al._ (2021). In the supplement, we use integer quantum Hall interference at
$\nu=-1$ at $B\approx$4\text{\,}\mathrm{T}$$ to measure the bulk quasiparticle
charging time, $\tau$, directly. We find that the charging time grows rapidly
as the chemical potential moves deeper into the transport plateau, with phase
slips (in the integer regime, arising completely from bulk-edge coupling)
transitioning from reversible to hysteretic as $\tau$ becomes longer than the
measurement time.

##  III Quasiparticle Dynamics in $\nu=1/3$

Figure 2:  Irreversible charging dynamics. (A) $G_{D}$ vs. $V_{C}$ as the
magnetic field is swept from 8.88T to 9T. The latter half of the measurement
shows many sudden phase jumps, beginning at $B=8.95T$ and continuing until the
end of the sweep. (B) $G_{D}$ vs. $V_{C}$ as the magnetic field is swept from
9T to 8.88T. Now, the sudden phase jumps differ in exact location, number, and
magnitude, not beginning until $B<8.97T$ and again continuing until the end of
the sweep. (C) Localized states within the fractional quantum Hall energy gap
have slow tunneling rates to the gapless edge; as a result, the chemical
potential of the electrically isolated bulk ($\mu_{bulk}$) lags the chemical
potential of the edge ($\mu_{edge}$) during a ramp of the magnetic field or
gate voltage. In this schematic, purple indicates filled states.  Figure 3:
Addition of individual e/3 quasiparticles at constant field (A) Repeated line
traces of the conductance $G_{D}$ plotted as a function of $V_{C}$. Traces are
vertically offset by an amount proportional to the time between the start of
each trace. The window over which $V_{C}$ is swept gradually increases over
time, favoring an increase in the number of quasiparticles. Interpreting each
phase slip as the addition of a single anyon, the traces are grouped into
distinct classes $\alpha$, $\beta$, and $\gamma$ corresponding to the
respective values of $N_{qp}\mod 3$. (B) The phases extracted from the Fourier
transform of each trace in panel A are plotted in red. Each phase jump is
assumed to lie in the interval $(-\pi,\pi)$. Blue points correspond to the
magnitude of each sudden jump in $\theta$. (C) Collections of line traces from
each set of three adjacent classes $\alpha_{i}$, $\beta_{i}$, $\gamma_{i}$
plotted over the region where the data overlap in $V_{C}$. The stability of
the phase is apparent given the overlap between traces within a given class,
while the classes are offset from each other by a $\sim 2\pi/3$ phase shift.
(D) Collections of line traces from classes which are separated by three phase
slips. These pairs of classes have an indistinguishable phase, but small
differences in the exact shape of the curves are apparent.

The slow charging allows quasiparticle dynamics to be studied at fixed
magnetic field, using only a single gate voltage to measure both the
interference phase and quasiparticle dynamics. Fig. 3a shows data taken at
$B=9T$ as a function of $V_{C}$. In this measurement, $V_{C}$ is swept over a
range spanning $\sim 10$ oscillations in $\approx 30$ seconds. The range of
this raster is adjusted from trace to trace, so that the average value of
$V_{C}$ increments slowly over $\sim$ 30 minutes. Most successive traces show
identical oscillatory patterns, but a pattern of abrupt phase slips is again
evident. Fig. 3b shows the phase $\theta$ of each trace extracted from the
discrete Fourier transform. Notably, no background subtraction is necessary at
constant $B$. We observe 18 phase slips over this range with a mean value of
$2\pi\times 0.334$ and a standard deviation of $2\pi\times 0.038$.

We interpret each of these slips as the entry of a single anyon into the
interferometer. We correspondingly collect traces into groups where the phase
is the same, and label these by a Greek letter $\\{\alpha,\beta,\gamma\\}$
corresponding to $N_{qp}\mod 3$. A numerical subscript distinguishes traces
with the same $N_{qp}\mod 3$ that are separated by 3 phase slips; assuming
each phase slip corresponds to entry of a single fractionally-charged
quasiparticle, these traces correspond to charge configurations differing by
an integer number of whole electron charges in the interferometer. Fig. 3c
compares traces $\\{\alpha_{i},\beta_{i},\gamma_{i}\\}$ for $i=\\{1,...6\\}$.
Each comparison shows a clear ‘triple-helix’ pattern arising from the $2\pi/3$
relative phase shift between each set of curves.

Within an individual panel of Fig. 3c, all curves are measured under identical
conditions of magnetic field and applied gate voltages. As might be expected,
traces taken at different times but for the same total charge show excellent
reproducibility. A less obvious question is whether addition of a whole
electron modifies the $G_{D}-V_{C}$ relation; in particular, if that electron
is added deep in the bulk, it is not expected to detectably affect the
interferometer phase. Indeed, comparison of data separated by three quantized
phase slips show no discernible change in the oscillation phase, as
illustrated in Fig. 3d. Notably, however, traces differing by a whole
electron—e.g., $\alpha_{3}$ and $\alpha_{4}$—are distinguishable through
deviations in the precise amplitude of individual interference fringes. This
effect is robust across nearly all groups of traces separated by a $2\pi$
phase shift in Fig. 3 (see supplement). We speculate that this ‘charge
fingerprint’ encoded in the fringe intensities arises from the exponential
sensitivity of transmission through our quantum point contacts to their
electrostatic environment—including the electrical potential generated by the
addition of even a gate-screened electrical charge in the bulk.

Because the pattern of oscillation amplitudes provides information about the
charge configuration not contained in the oscillation phase alone, it allows
us, under certain conditions, to determine the absolute phase (interpreted as
the total electron charge in the interferometer Feldman and Halperin (2022))
rather than just the phase modulo $2\pi$. This allows us to distinguish
quasiparticle entry and exit under static experimental conditions in the time
domain. Fig. 4a shows the result of ramping the magnetic field to a set point
and then, starting from $t=0$, leaving it constant for 35 minutes while
continuing to repeatedly ramp $V_{C}$ over a small
($\sim$1.5\text{\,}\mathrm{m}\mathrm{V}$$) range. Several phase slips are
visible over the course of the experiment, separating regions of stable phase
which we label I-VII in Fig. 4b. Were the phase jumps assumed to fall into the
interval $(-\pi,\pi)$, their nearly consistent magnitudes could lead to the
assumption that every jump corresponds to the addition of one quasiparticle,
as is energetically favored by the reduction in total flux at the beginning of
the measurement.

However, a close examination of the measured conductance (Fig. 4c) shows the
fringe intensity patterns in regions II and V to be indistinguishable; regions
I and IV are also identical after noting that the phase slip occurs midway
through the single trace that comprises region IV. This leads us to identify
these regions with a return to the same charge configuration as well as the
same absolute phase. In fact, equating the regions which have an
indistinguishable ‘charge fingerprint’ necessitates that at least one of the
first 3 phase slips must correspond to a removal of two quasiparticles, rather
than the naïve interpretation of every slip as the addition of a single
quasiparticle. Fig. 4C shows two possible corrected traces of $\theta(t)$,
with the light and dark blue traces differing from the uncorrected $\theta$ by
a shift of $-2\pi$ at the second or third phase slip, respectively.

The time dependence of the quasiparticle number implied by either of these
possibilities reconciles our earlier observation of the identical ‘charge
fingerprints’ in both regions I and IV, and regions II and V. The information
required to discriminate between the two possibilities is provided by the
magnitude of the phase slip from region II to region III: here, $\Delta\theta$
is significantly larger than $\theta_{a}$, whereas a finite amount of bulk-
edge coupling should only serve to decrease the magnitude of the phase jump.
(All of the other phase slips show a value slightly less than $\theta_{a}$).
This suggests that the slip separating regions II and III should be associated
with a decrease in the number of quasiparticles by two, as the change in phase
is then $\Delta\theta=-4\pi/3+\delta\theta_{BE}$.

The observed relaxation dynamics depend strongly on the preparation and
operating regimes of the system. As shown in the supplementary material,
measurements in similar ranges to those shown in Figs. 3a and 4a can be made
to show zero phase slips over several hours for suitable choice of state
preparation, despite the flux in the interferometer loop being modulated by
several flux quanta. These results imply that for certain impurities, at
least, equilibration times are considerably in excess of twenty minutes. As
described in the supplementary material, other preparation-dependent anomalies
may arise: running the measurement sequence of Fig. 3a in reverse immediately
following that measurement reveals no phase slips as $V_{C}$ is decreased
across the same range, but for a different preparation history, a sequence of
phase slips is observed as a function of decreasing $V_{C}$ with magnitude
most easily attributed to the correlated entry of two quasiholes near the
sample edge per phase slip.

Figure 4:  Time domain reconstruction of the quasiparticle occupation (A)
Repeated measurement of $G_{D}$ over a fixed window of $V_{C}$. At the
beginning of the measurement, the magnetic field is swept from 9.0T to 8.95T,
where it is subsequently held constant for 35 min. (B) The phase, $\theta$,
extracted from the Fourier transform of the data in panel A. Each jump is
assumed to be in the interval $\Delta\theta\in(-\pi,\pi)$. For traces in which
a phase jump is observed in the middle of the line trace, two points are
plotted as open circles, corresponding to the two values of phase extracted
before and after the jump. Inset: Line traces of $G_{D}$ taken from regions II
and V, showing the indistinguishability not only in phase but also in
amplitude. Line traces taken from regions I and IV show a similar overlap,
after the phase jump occurs mid-line in trace IV. (C) Two versions of the
corrected phase, $\theta$, identical to the data from panel B except with a
shift of $-2\pi$ applied to either the jump from II to III (dark blue), or the
jump from III to IV (light blue). Either modification brings the time
dependence of $\theta$ into accordance with the interpreted changes in
$N_{qp}$ (marked for each curve) by matching the ’charge fingerprints’ of
regions I and IV, and regions II and V.

## Discussion

The exceptionally slow quasiparticle dynamics of our interferometer open new
routes to quantitatively probe the physics of fractionalized phases at the
single anyon level. Measuring few-anyon dynamical processes via the response
of the interferometric phase to both $\theta_{a}$ and Coulomb effects may give
new insight into states where inter-quasiparticle correlations are important,
such as in the formation dynamics of anyonic Wigner crystal states, as well as
the hierarchical fractional quantum Hall states.

The observed time-dependence of $N_{qp}$ also raises questions about the
dynamics in the 1/3 state that we have not addressed here. One question
concerns the mechanism of the time-dependent fluctuations in $N_{qp}$, as
observed in Fig. 4C. For instance, these events may be driven by some internal
mechanism, such as random quantum tunneling of quasiparticles between sharply-
confined disorder sites, or an external effect such as the random absorption
of high-energy photons.

Recent advances in spatial imaging techniques, allowing high resolution charge
sensing in dual-gated devicesLi  _et al._ (2021), may be used to help resolve
these questions. Applied to interferometer devices of a similar construction
to ours, this may allow direct correlation between the real-space distribution
of localized anyons and the interferometric phase. Such experiments could also
provide information on the consequences of the real-space disorder present in
van der Waals devices on both equilibrium and non-equilibrium models of bulk-
edge coupling.

Graphene heterostructures also exhibit even denominator fractional quantum
Hall states expected to host non-Abelian anyonsKi  _et al._ (2014); Zibrov
_et al._ (2017); Li  _et al._ (2017), raising the question of whether
interferometry experiments of this type will be similarly enlightening in such
states. In bilayer graphene, energy gaps for the even denominator states have
been found to be comparable in size to those of the 1/3 state studied in the
present workAssouline  _et al._ (2024); Hu  _et al._ (2023), likely leading to
similarly slow dynamics for the charged e/4 quasiparticles in those states. A
key outstanding question concerns the timescale for motion of the charge-
neutral excitations that encode the fermion parity. While this timescale must
ultimately be determined by experiment, the exceptionally long timescales for
charge motion reported here lend hope that unambiguous detection of non-
Abelian statistics may be within reach.

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## Acknowledgements

The authors acknowledge helpful discussions with T. Wang, S. Kivelson and S.
Das Sarma. Work at UCSB was primarily supported by the Office of Naval
Research under award N00014-23-1-2066. Development of fabrication based on
anodic oxidation lithography were supported by the Air Force Office of
Scientific Research under award FA9560-20-1-0208. AFY acknowledges additional
support by the Gordon and Betty Moore Foundation EPIQS program under award
GBMF9471. LC and NS received additional support from the Army Research Office
under award W911NF20-1-0082. MZ was supported by the U.S. Department of
Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences
and Engineering Division under Contract No. DE-AC02-05-CH11231 (Theory of
Materials program KC2301). K.W. and T.T. acknowledge support from JSPS KAKENHI
(Grant Numbers 19H05790, 20H00354 and 21H05233).

## References

  * Leinaas and Myrheim (1977) J. M. Leinaas and J. Myrheim, Il Nuovo Cimento B 37, 1 (1977).
  * Wilczek (1982) F. Wilczek, Physical Review Letters 49, 957 (1982).
  * Arovas  _et al._ (1984) D. Arovas, J. R. Schrieffer, and F. Wilczek, Physical Review Letters 53, 722 (1984).
  * Halperin (1984) B. I. Halperin, Physical Review Letters 52, 1583 (1984), publisher: American Physical Society.
  * Goldin  _et al._ (1980) G. A. Goldin, R. Menikoff, and D. H. Sharp, Journal of Mathematical Physics 21, 650 (1980), publisher: American Institute of Physics.
  * Kivelson (1990) S. Kivelson, Physical Review Letters 65, 3369 (1990).
  * Chamon  _et al._ (1997) C. d. C. Chamon, D. E. Freed, S. A. Kivelson, S. L. Sondhi, and X. G. Wen, Physical Review B 55, 2331 (1997).
  * Kim (2006) E.-A. Kim, Physical Review Letters 97, 216404 (2006), publisher: American Physical Society.
  * Halperin  _et al._ (2011) B. I. Halperin, A. Stern, I. Neder, and B. Rosenow, Physical Review B 83, 155440 (2011).
  * Read and Das Sarma (2024) N. Read and S. Das Sarma, Nature Physics 20, 381 (2024).
  * Nakamura  _et al._ (2020) J. Nakamura, S. Liang, G. C. Gardner, and M. J. Manfra, Nature Physics 16, 931 (2020).
  * Nakamura  _et al._ (2023) J. Nakamura, S. Liang, G. Gardner, and M. Manfra, Physical Review X 13, 041012 (2023).
  * Camino  _et al._ (2005) F. E. Camino, W. Zhou, and V. J. Goldman, Phys. Rev. B 72 (2005).
  * Camino  _et al._ (2007) F. E. Camino, W. Zhou, and V. J. Goldman, Physical Review Letters 98, 076805 (2007).
  * Zhang  _et al._ (2009) 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).
  * McClure  _et al._ (2009) D. T. McClure, Y. Zhang, B. Rosenow, E. M. Levenson-Falk, C. M. Marcus, L. N. Pfeiffer, and K. W. West, Physical Review Letters 103, 206806 (2009).
  * Ofek  _et al._ (2010) N. Ofek, A. Bid, M. Heiblum, A. Stern, V. Umansky, and D. Mahalu, Proceedings of the National Academy of Sciences 107, 5276 (2010).
  * Choi  _et al._ (2015) H. K. Choi, I. Sivan, A. Rosenblatt, M. Heiblum, V. Umansky, and D. Mahalu, Nature Communications 6 (2015), 10.1038/ncomms8435.
  * Sivan  _et al._ (2016) I. Sivan, H. K. Choi, J. Park, A. Rosenblatt, Y. Gefen, D. Mahalu, and V. Umansky, Nature Communications 7, 12184 (2016).
  * Nakamura  _et al._ (2019) J. Nakamura, S. Fallahi, H. Sahasrabudhe, R. Rahman, S. Liang, G. C. Gardner, and M. J. Manfra, Nature Physics 15, 563 (2019).
  * Willett  _et al._ (2023) R. Willett, K. Shtengel, C. Nayak, L. Pfeiffer, Y. Chung, M. Peabody, K. Baldwin, and K. West, Physical Review X 13, 011028 (2023).
  * Nakamura  _et al._ (2022) J. Nakamura, S. Liang, G. C. Gardner, and M. J. Manfra, Nature Communications 13, 344 (2022).
  * Dean  _et al._ (2011) C. R. Dean, A. F. Young, P. Cadden-Zimansky, L. Wang, H. Ren, K. Watanabe, T. Taniguchi, P. Kim, J. Hone, and K. L. Shepard, Nature Physics 7, 693 (2011).
  * Polshyn  _et al._ (2018) H. Polshyn, H. Zhou, E. M. Spanton, T. Taniguchi, K. Watanabe, and A. F. Young, Physical Review Letters 121, 226801 (2018).
  * Zeng  _et al._ (2019) Y. Zeng, J. I. A. Li, S. A. Dietrich, O. M. Ghosh, K. Watanabe, T. Taniguchi, J. Hone, and C. R. Dean, Physical Review Letters 122, 137701 (2019).
  * Dean  _et al._ (2020) C. Dean, P. Kim, J. I. A. Li, and A. Young, in  _Fractional Quantum Hall Effects: New Developments_ (World Scientific, Singapore, 2020) pp. 317–375.
  * Zibrov  _et al._ (2017) A. A. Zibrov, C. Kometter, H. Zhou, E. M. Spanton, T. Taniguchi, K. Watanabe, M. P. Zaletel, and A. F. Young, Nature 549, 360 (2017).
  * Li  _et al._ (2017) J. I. A. Li, C. Tan, S. Chen, Y. Zeng, T. Taniguchi, K. Watanabe, J. Hone, and C. R. Dean, Science , eaao2521 (2017).
  * Zibrov  _et al._ (2018) A. A. Zibrov, E. M. Spanton, H. Zhou, C. Kometter, T. Taniguchi, K. Watanabe, and A. F. Young, Nature Physics 14, 930 (2018).
  * Hu  _et al._ (2023) Y. Hu, Y.-C. Tsui, M. He, U. Kamber, T. Wang, A. S. Mohammadi, K. Watanabe, T. Taniguchi, Z. Papic, M. P. Zaletel, and A. Yazdani, “High-Resolution Tunneling Spectroscopy of Fractional Quantum Hall States,”  (2023), arXiv:2308.05789 [cond-mat].
  * Assouline  _et al._ (2024) A. Assouline, T. Wang, H. Zhou, L. A. Cohen, F. Yang, R. Zhang, T. Taniguchi, K. Watanabe, R. S. Mong, M. P. Zaletel, and A. F. Young, Physical Review Letters 132, 046603 (2024).
  * Déprez  _et al._ (2021) C. Déprez, L. Veyrat, H. Vignaud, G. Nayak, K. Watanabe, T. Taniguchi, F. Gay, H. Sellier, and B. Sacépé, Nature Nanotechnology 16, 555 (2021).
  * Ronen  _et al._ (2021) 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).
  * Zhao  _et al._ (2022) L. Zhao, E. G. Arnault, T. F. Q. Larson, Z. Iftikhar, A. Seredinski, T. Fleming, K. Watanabe, T. Taniguchi, F. Amet, and G. Finkelstein, Nano Letters 22, 9645 (2022).
  * Fu  _et al._ (2023) H. Fu, K. Huang, K. Watanabe, T. Taniguchi, M. Kayyalha, and J. Zhu, Nano Letters 23, 718 (2023).
  * Kim  _et al._ (2024) 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].
  * Cohen  _et al._ (2023a) 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 (2023a).
  * Wen (1991) X.-G. Wen, Physical Review B 44, 5708 (1991).
  * Kane and Fisher (1992) C. L. Kane and M. P. A. Fisher, Physical Review Letters 68, 1220 (1992).
  * Fendley  _et al._ (1994) P. Fendley, H. Saleur, and N. P. Warner, Nuclear Physics B 430, 577 (1994).
  * Kane and Fisher (1995) C. L. Kane and M. P. A. Fisher, Physical Review B 51, 13449 (1995).
  * Fendley  _et al._ (1995) P. Fendley, A. W. W. Ludwig, and H. Saleur, Physical Review Letters 74, 3005 (1995).
  * Milliken  _et al._ (1996) F. P. Milliken, C. P. Umbach, and R. A. Webb, Solid State Communications 97, 309 (1996).
  * Radu  _et al._ (2008) I. P. Radu, J. B. Miller, C. M. Marcus, M. A. Kastner, L. N. Pfeiffer, and K. W. West, Science 320, 899 (2008).
  * Cohen  _et al._ (2023b) L. A. Cohen, N. L. Samuelson, T. Wang, T. Taniguchi, K. Watanabe, M. P. Zaletel, and A. F. Young, Science 382, 542 (2023b).
  * Yamauchi  _et al._ (2009) Y. Yamauchi, M. Hashisaka, S. Nakamura, K. Chida, S. Kasai, T. Ono, R. Leturcq, K. Ensslin, D. C. Driscoll, A. C. Gossard, and K. Kobayashi, Physical Review B 79, 161306 (2009).
  * Werkmeister  _et al._ (2023) 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].
  * Rosenow and Halperin (2007) B. Rosenow and B. I. Halperin, Physical Review Letters 98, 106801 (2007).
  * Goodall  _et al._ (1985) R. K. Goodall, R. J. Higgins, and J. P. Harrang, Phys. Rev. B 31, 6597 (1985).
  * Eisenstein  _et al._ (1992) J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 68, 674 (1992).
  * Yang  _et al._ (2021) F. Yang, A. A. Zibrov, R. Bai, T. Taniguchi, K. Watanabe, M. P. Zaletel, and A. F. Young, Physical Review Letters 126, 156802 (2021).
  * Feldman and Halperin (2022) D. E. Feldman and B. I. Halperin, Physical Review B 105, 165310 (2022), publisher: American Physical Society.
  * Li  _et al._ (2021) H. Li, S. Li, E. C. Regan, D. Wang, W. Zhao, S. Kahn, K. Yumigeta, M. Blei, T. Taniguchi, K. Watanabe, S. Tongay, A. Zettl, M. F. Crommie, and F. Wang, Nature 597, 650 (2021).
  * Ki  _et al._ (2014) D.-K. Ki, V. I. Fal’ko, D. A. Abanin, and A. F. Morpurgo, Nano Letters 14, 2135 (2014).
  * von Keyserlingk  _et al._ (2015) C. von Keyserlingk, S. Simon, and B. Rosenow, Physical Review Letters 115, 126807 (2015).

Supplementary Information

##  IV Materials and Methods

###  IV.1 S1. Sample Fabrication

Samples 1 and 2 (optical images shown in Fig. S1A,C) were both fabricated
using the standard van der Waals dry-transfer process combined with electrode-
free AFM anodic oxidation lithography of the graphite top gates before
stacking, as described in reference Cohen  _et al._ (2023a). The structures
patterned into each top graphite gate are shown in the AFM topographs of Fig.
S1B,D. These scans were acquired in the middle of the stacking process,
imaging the exposed graphite top gate after picking it up. After stacking, the
device is transferred onto a conductively-doped Si substrate with
$285\text{\,}\mathrm{n}\mathrm{m}$ of thermally-grown SiO2 on the surface.

A mask is defined via E-Beam Lithography (EBL) of a polymethyl methacrylate A4
495K/A2 950K bilayer resist, and $40\text{\,}\mathrm{n}\mathrm{m}$ of aluminum
is deposited onto the device to be used as a hard etch mask. The whole stack
is then etched through using a CHF3/O2 plasma reactive ion etch for several
minutes. The aluminum mask is removed by submersion in a $<$3% TMAH solution
(AZ300MIF) for 20 minutes. Then, a contact electrode pattern is defined using
a PMMA 950K A8 mask and another EBL exposure, a 30 second CHF3/O2 RIE etch is
performed to clean the expose device edges, and edge contacts are deposited by
E-Beam evaporation of 5/15/150nm Cr/Pd/Au layers.

Figure S1: Device Images and AFM Cut Gates (a) Optical micrograph of device 1,
the device studied in the main text. Electrical edge contacts to the monolayer
are labeled as C1-10. Electrical edge contacts to the gate layers, NW, SW, NE,
SE, P, CG, and BG are marked respectively. Scale bar inset is
$10\text{\,}\mu\mathrm{m}$. (b) AFM topograph on a transfer slide of the top
gate layer after pick-up with an hBN flake. The gate regions shown in panel a,
as well as in Fig. 1a, are marked. The measured lithographic area, defined by
the perimeter drawn in the red dashed line, is measured to be
$0.79(5)\text{\,}\mu\mathrm{m}^{2}$, and the perimeter is measured to be
$3.3(1)\text{\,}\mu\mathrm{m}$. The uncertainty comes from the width of the
etched region in the graphite top gate (it is unknown as to where exactly the
edge state is located). (c) Optical micrograph of device 2, a nearly identical
device as studied in the main text which we used to perform experiments in the
integer quantum Hall regime. Electrical edge contacts to the monolayer are
labeled as C1-8. Electrical edge contacts to the gate layers, NW, SW, NE, SE,
P, CG, and BG are marked respectively. Scale bar inset is
$10\text{\,}\mu\mathrm{m}$. (d) AFM topograph on a transfer slide of the top
gate layer after pick-up with an hBN flake for the device in panel c. The gate
regions shown in panel c are marked. The measured lithographic area, defined
by the perimeter drawn in the red dashed line, is measured to be
$0.76(5)\text{\,}\mu\mathrm{m}^{2}$, and the perimeter is measured to be
$3.25(10)\text{\,}\mu\mathrm{m}$. The uncertainty comes the same source as in
panel b.

###  IV.2 Voltages for Main Text Figures

Main Text Figure | $V_{CG}$ | $V_{BG}$ | $V_{NW/SW}$ | $V_{NE/SE}$ | $V_{P}$
---|---|---|---|---|---
Fig. 1c (blue trace) | 2.23V | -2V | 2.23V | -4.1V | 2.23V
Fig. 1c (red trace) | 2.23V | -2V | 0.5V | 2.23V | 2.23V
Fig. 1d | 2.23V | -2V | 0.5V | -4.1V | N/A
Fig. 2a | N/A | -2V | 1.0V | -3.2V | 2V
Fig. 3 | N/A | -2V | 1.0V | -3.2V | 2V
Fig. 4 | N/A | -2V | 0.3V | -3.94V | 2V
Figure S2: Fixed voltage set points for main text figures

###  IV.3 S2. Measurement parameters and interferometer characterization

Experiments were performed in a dry dilution refrigerator at a base
temperature of $18\text{\,}\mathrm{m}\mathrm{K}$ (on device 1) and
$55\text{\,}\mathrm{m}\mathrm{K}$ (on device 2). Electronic filters are
installed on all transport lines to lower the electron temperature. Transport
was measured at $10.3875\text{\,}\mathrm{H}\mathrm{z}$ (device 1) /
$17.7777\text{\,}\mathrm{H}\mathrm{z}$ (device 2) using SR860 lock-in
amplifiers. A Basel Precision Instruments SP983c high stability I to V
converter (IF3602) is used to measure the current signals, while the voltages
are measured directly with the SR860 using no additional pre-amplification.

For the diagonal conductance measurements performed on device 1, in the
fractional quantum Hall regime, an ac excitation is applied to contacts C6,
C7, C8, and C9 and the resulting current $I_{out}$ is measured on contact C1.
The diagonal voltage drop $V_{+}-V_{-}$ is measured between contacts C10 and
C2. A constant $10\text{\,}\mathrm{V}$ is applied to the conductively-doped Si
substrate throughout all measurements to maintain electron-doping in the
single- and un-gated regions of each contact. The excitation voltage is
$10\text{\,}\mu\mathrm{V}$ for the data in Fig. 1D and Fig. 4, and
$20\text{\,}\mu\mathrm{V}$ for Fig. 2A-B, and Fig. 3.

For the measurements of device 2 in the integer quantum Hall regime, a two-
ground configuration is employed. A $0.9\text{\,}\mathrm{n}\mathrm{A}$ ac
current bias is applied at contact C6, and the contact C8 is grounded to sink
the current reflected by the interferometer. The transmitted current is
measured on the opposite side of the device, on contact C4. A constant
$-12\text{\,}\mathrm{V}$ is applied to the Si substrate throughout these
measurements to maintain hole-doping of the contact regions.

As described in the main text, we perform several standard characterizations
of our Fabry-Perot interferometer to confirm normal operation, shown in the
figures that follow.

Figure S3: Individual QPC pinch-off curves in $\nu=1/3$. Blue curve is $G_{D}$
vs. $V_{NE/SE}$ while $V_{NW/SW}$ is set such that $\nu_{NW/SW}=1/3$. Red
curve is $G_{D}$ vs. $V_{NW/SW}$ with $V_{NE/SE}$ set such that
$\nu_{NE/SE}=1/3$. Operating points for both QPCs relevant to the interference
data shown in Fig. 1d are identified by the inset arrows. Figure S4: Bias and
Common Mode Dependence of Interference in $\nu$ = 1/3 (a) $G_{D}$ versus
$V_{P}$ and $V_{SD}$ near $\nu=1/3$, B = 9T, $V_{CG}=2.27V$,
$V_{NW/SW}=-0.23V$, $V_{NESE}=-4.25V$. The interference shows a strong
suppression of $G_{D}$ near zero DC bias, characteristic of chiral Luttinger
liquid behavior near the QPCs. Additionally, the interference exhibits a
phase-shift resulting from an applied $V_{SD}$. Notably, the periodicity in
$V_{SD}$ is reduced for values of $V_{P}<0.5V$. (b) Extracted Edge velocity
from the periodicity in $V_{SD}$ in panel a versus $V_{P}$, here we use the
formula $hv_{edge}/\nu L=e\Delta V_{SD}$. As $V_{P}$ becomes enhanced the edge
velocity exhibits a 19% reduction coming from the decreasing sharpness of the
confinement potential near the plunger gate. (c) $\delta G_{D}$ as a function
of both the common mode voltage on the monolayer, $V_{CM}$, and the difference
between the voltages on the source and drain terminals relative to fridge
ground, $V_{S}-V_{D}$; this difference is taken by varying the voltage on
$V_{S}$ while keeping $V_{D}$ fixed. This data is taken in the same operating
regime as Fig. 1d. The periodicity in $V_{CM}$ directly tells us e/$C_{I}$,
and consequently the edge velocity as quoted in the main text. (d) Line cut
along the dashed black line in panel c. We can extract the peak-to-peak
distance between fringes versus $V_{CM}$, which yields a periodicity $\Delta
V_{CM}=$234\text{\,}\mu\mathrm{V}$$. Using the relation $hv_{edge}/\nu
L=e\Delta V_{CM}=e^{2}/C_{I}$, we find that
$v_{edge}=$6.2\text{\times}{10}^{4}\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$$.
Despite the fact the panels a-b and panels c-d are taken at slightly different
gate voltages we observe good agreement with the predicted edge velocity for
the $\nu=1/3$ edge mode using two different methods. Figure S5: Temperature
dependence in FQH (device 1) (a) $G_{D}$ versus $V_{C}$ in $\nu=1/3$, with
both QPCs at partial pinch off, showing clear oscillations for a range of
temperatures measured on our sample probe between
$T=$18\text{\,}\mathrm{m}\mathrm{K}$$ and
$T=$639\text{\,}\mathrm{m}\mathrm{K}$$. (b) The average value of the diagonal
conductance, $G_{AVG}$, plotted versus the temperature on the probe $T$ for
two ranges in $V_{C}$ marked by the red and blue arrows respectively in panel
a. The average value reflects the tunneling rate of the two QPCs, which for a
chiral Luttinger liquid is expected to change significantly with increasing
temperature. (c) Visibility, denoted as $\delta G$ and defined as the average
difference between successive maxima and minima in the interference is plotted
versus the temperature on the probe, $T$. Interestingly, the interplay between
the chiral Luttinger liquid renormalization of the QPC transmission and the
temperature induced decoherence of the interference leads to a maximum in the
visibility that depends on the initial set point of $V_{C}$.  Figure S6: Bias
Dependence of the QPC transmission in the outer edge of the $\nu=2$ state
(device 1). (A) Hall conductance as a function of $V_{C}$ at 3T. The
interferometer is characterized in the $\nu=2$ state with the center gate
fixed at the marked point. (B, C) Diagonal conductance across the device with
each QPC pinched off individually. (D) Dependence of the transmission through
each QPC at the marked points in B,C on the applied source-drain DC bias
$V_{SD}$. No zero-bias suppression in the conductance is observed, in stark
contrast to the phenomenology observed in the FQH regime. (E) Dependence of
the interference on the source drain bias and plunger gate, with both QPCs set
to the marked points in B,C simultaneously.

###  IV.4 S3. Phase Slip Magnitude Analysis

To extract the magnitudes of the phase slips reported in Fig. 1e we start by
subtracting off the mean value of $G_{D}$ for each line-trace versus $V_{P}$
at a fixed value of B. We then take the 1D discrete Fourier transform (DFT)
along the $V_{P}$ axis of the data in Fig. S7a (the mean subtracted data from
main text Fig. 1d). The absolute value of the resulting DFT is shown in Fig.
S7b. We then extract the phase from the component of the DFT which has the
largest absolute magnitude, here around 3 cycles per volt of $V_{P}$. The
point-to-point difference in the phase, extracted in this way, as a function
of B contains a nearly constant background, as expected for a linear phase
accumulation from the Aharonov-Bohm effect. However, this constant background
is interrupted by sharp spikes at the location of each sudden phase slip.

To remove the Aharonov-Bohm effect and purely extract the value of the sudden
phase slips, we take a moving trimmed mean in a 7-point window (ignoring
outliers bigger than $|\Delta\theta|=0.4$) of the point-to-point difference of
the phase versus B. This enables us to extract a rough estimate of the
background. This rough background is fit to a line, and then further refined
with a new trimmed mean of the original point-to-point difference (with the
same window size). The refined trimmed mean excludes outliers greater than the
fit value at fixed B plus $\Delta\theta=0.14$ or less than the fit value minus
$\Delta\theta=0.17$. The integrated phase (plotted in red) including the
sudden slips, along with the integrated background (plotted in black), is
shown in Fig. S7c. Subtracting the integrated phase from the integrated
background produces the dataset shown in Fig. S7d where the phase jumps are
isolated. The outlier exclusion range at the refinement step is set to
guarantee the Fourier analysis reproduces the number of observed phase slips
in Fig. S7a.

Figure S7: Extracting Phase Slip Magnitudes (a) Same data as in main text Fig.
1d, but with the mean value of $G_{D}$ for each value of magnetic field
subtracted from the corresponding trace versus $V_{P}$. (b) Absolute magnitude
of the discrete 1D Fourier transform (along $V_{P}$) of the data in panel a
plotted versus magnetic field and inverse $V_{P}$. (c, e, g) The integrated
$\theta$ vs. B extracted from the DFT is plotted in red. The integrated
background, which we subtract off from the accumulated phase to calculate the
magnitude of the phase slips, is plotted in black. Panel c shows the
integrated phase and the integrated background extracted from the DFT by
taking the phase of the largest magnitude component; this is the analysis
method we use for all phase slip extractions in the main text. Panel e shows
the same data as panel c, but the phase is extracted by taking a weighted
average over the peak in the DFT amplitude. Panel g shows the same data as
panels c and e, but the phase is extracted by taking an unweighted average
over the peak in the DFT amplitude. (d) $\theta-\theta_{AB}$ versus magnetic
field extracted by subtracting the integrated background from the integrated
phase calculated via the method in panel c. This data is identical to the main
text Fig. 1e. (f) Same as panel d, but the phase slip magnitudes are
calculated from the data in panel e. (h) Same as panels f and d, but the phase
slip magnitudes are calculated from the data in panel g.

While it is true that the calculated phase slip magnitudes weakly depend on
the choice of the trimmed mean exclusion range, the variation is limited to
the standard deviation in the point-to-point phase difference after coarsely
removing the phase slips using the same criterion as the initial trimmed mean
(outliers with $|\Delta\theta|>0.4$ are removed). We find that
$\sigma_{\Delta\theta}=2\pi\cdot 0.012$; leading to an uncertainty of
$2\pi\cdot 0.012\cdot\sqrt{2}=2\pi\cdot 0.017$ for the magnitude of a sudden
phase slip. While this is small compared to the magnitude of the observed
jumps, it is not the dominant source of error in our measurement. As can be
seen by the the resulting DFT in Fig. S7b, the peak in the DFT amplitude
spectrum has some finite width; this is ultimately the result of the
interference signal not being perfectly periodic in $V_{P}$. Consequently,
there is some freedom in the analysis method used to determine the phase of
the primary DFT component which introduces some systematic error beyond the
statistical contribution.

To roughly quantify the size of this systematic uncertainty, we chose two
additional analysis methods and checked the processed phase slip magnitudes
against the method used in the main text. The second method we used was to
take a weighted average of the phase, weighted by the normalized absolute
magnitude of the DFT signal, around the peak in the DFT between
$f_{P}=$2\text{\,}\mathrm{V}^{-1}$$ and $f_{P}=$5\text{\,}\mathrm{V}^{-1}$$.
An identical set of panels to Fig. S7c-d where the phase jumps are extracted
using the weighted average (method 2), are shown in Fig. S7e-f. The third
method we used (method 3) was to take an unweighted average of the DFT phase
across the same range in method 2. Similarly to Fig. S7c-f, the integrated
phase and background, as well as the extracted phase slips, are plotted in
Fig. S7g-h. We find that for each phase slip, the typical deviation between
the three methods is $|\delta(\Delta\theta)|<2\pi\cdot 0.01$. However, for
some phase slips the deviation can be as large as $\pm 2\pi\cdot 0.037$
between methods. Since we cannot check every possible analysis method, we will
take our global systematic uncertainty to be the largest observed deviation
(across all phase slips in Fig. S7a) from the average across the three
methods; we find this to be $\pm 2\pi\cdot 0.04$.

We use the same method outlined in the beginning of this section (method 1) to
extract the sudden phase slip magnitudes for all other relevant datasets. It’s
important to note that while the statistical uncertainty from the background
subtraction is not relevant to other datasets, such as Fig. 3 and Fig. 4,
where no Aharanov-Bohm component needs to be subtracted off, we expect the
systematic uncertainty from the Fourier analysis persists. Consequently, for
datasets where we analyze phase shifts with no background component, we still
assume our uncertainty is $\sigma_{\Delta\theta}\approx\pm 2\pi\cdot 0.04$.

###  IV.5 S4. Bulk-edge and exterior-edge coupling at $\nu=\frac{1}{3}$

_The edge “capacitance.”_ In the main text we reported the common-mode
periodicity $\Delta V_{CM}=$234\text{\,}\mu\mathrm{V}$$, which we relate to
the capacitance of the edge as $e=\Delta V_{CM}C_{I}$. This measurement is
taken in the regime where the bulk is completely incompressible (no phase
slips). But we should note that in the notation of Halperin et al. Halperin
_et al._ (2011), this measurement is more properly understood as a
characterization of $K_{I}=$234\text{\,}\mu\mathrm{e}\mathrm{V}$$. When the
bulk is entirely incompressible, the relation between the two is trivial:
$K_{I}=e^{2}/C_{I}$. However, if extended to the regime when the bulk is
compressible (for example at a phase slip), and with finite-bulk edge coupling
$K_{IL}$, the parameters are related as a matrix inverse $K=e^{2}C^{-1}$, so
that $C_{I}=\frac{e^{2}}{K_{I}-K^{2}_{IL}/K_{L}}$. We do not expect $K_{I}$ to
depend too much on the bulk filling, while $C_{I}$ will depend on the filling
via quantum capacitance effects in $K_{L}$. Thus when we refer to $C_{I}$ as
“the capacitance of the edge,” strictly speaking this should be understood as
$e^{2}/K_{I}$. In any case the capacitance matrix is not so relevant to our
experiments because the bulk and edge are out of equilibirium.

_Bulk-edge coupling of the anyonic phase._ We elaborate on the derivation of
Eq.(2) of the main text, which gives the Coulomb coupling between an anyonic
impurity and the edge. Assuming the radius of curvature of the edge is large
compared with the impurity distance, we can treat the edge as a line with
capacitance per unit length $C_{I}/L$. The induced charge is then $\delta
Q_{I}=\frac{C_{I}}{L}\frac{e}{3}\int G(x,R)dx$ where $G$ is the double-gated
Green’s function with a charge $e/3$ at $(x,y)=(0,R)$. Making use of the
effective translation variance along $x$, we can then use a conformal
transformation to find that a 1D line charge $\rho$ produces a potential
$\phi(y)=\frac{\rho}{2\pi\sqrt{\epsilon_{xy}\epsilon_{z}}}\log(\tanh(\pi
y\sqrt{\epsilon_{z}/\epsilon_{xy}}/4d))\approx\frac{1}{\pi\sqrt{\epsilon_{xy}\epsilon_{z}}}e^{-\pi
y\sqrt{\epsilon_{z}/\epsilon_{xy}}/2d}$. This gives the estimate $\delta
Q_{I}=\frac{e}{3}\frac{C_{I}}{\pi\sqrt{\epsilon_{xy}\epsilon_{z}}L}e^{-\pi
y\sqrt{\epsilon_{z}/\epsilon_{xy}}/2d}$. This also provides an estimate of the
_average_ bulk-edge coupling, $K_{IL}=\frac{e^{2}}{A}\int_{y>0}\int
dxG(x,y)dxdy=\frac{e^{2}}{2A}\frac{d}{2\epsilon_{z}}=\frac{e^{2}}{2Ac_{g}}$,
where $c_{g}$ is the geometric bulk capacitance per unit area.

This discussion also points out the need to generalize the analysis of
Ref.Halperin  _et al._ (2011) in order to describe bulk-edge coupling in our
experiments. In that work, the bulk was assumed to be in electro-chemical
equilibrium and that it could be treated as a single lumped element with
charging energy $K_{L}$. In our experiment, however, we are able to resolve
charging of single impurities. To explain bulk-edge coupling in this regime
requires a more fine-grained model keeping the charge state of each impurity
$a$ and an impurity-edge coupling $K_{L,a}$, which is precisely the quantity
analyzed above.

_Bulk-edge coupling of the AB phase._ In Fig. 1 of the main text we found that
in the regime of Aharonov-Bohm oscillations in which $N_{qp}$ is kept fixed,
the field period $\Delta B$ varies from $15\text{\,}\mathrm{m}\mathrm{T}$ to
$18\text{\,}\mathrm{m}\mathrm{T}$ as the plunger gate voltage $V_{P}$ is
varied from $-0.75\text{\,}\mathrm{V}$ to $-2.25\text{\,}\mathrm{V}$. Assuming
pure Aharonov-Bohm evolution, $\frac{e}{3}A\Delta B=h$, this gives a range of
areas $A\sim 0.69-$0.83\text{\,}\mu\mathrm{m}^{2}$$ in good agreement with the
nominal geometric area of the device. Here we discuss two qualifications to
this analysis, and their possible resolution:

  * •

The measured variation in $\Delta B$ is larger than can be explained by the
$V_{P}$-dependence of $A$.

  * •

In the standard theory of bulk-edge coupling,Halperin  _et al._ (2011) the
field period should receive a correction
$\frac{e}{3}\bar{A}(1-K_{IL}/K_{I})\Delta B=h$, and from theoretical estimates
of the $K_{IL}$ it is surprising the field period is in such good agreement
with the nominal area.

We propose that the resolution to both arises from a novel feature of our
graphene Fabry-Perot interferometer geometry, which has not been considered in
the literature focused on GaAs devices. Specifically, the filling outside the
interferometer is of opposite sign to the interior ($\nu_{\textrm{ext}}\ll 0$
for positive filling of the interferometer), giving rise to coupling between
the edge state and this exterior region we term “exterior-edge” coupling.
Exterior-edge coupling counteracts the effect of the typical bulk-edge
coupling between the edge state and the interior of the interferometer.

The variation in $\Delta B$ would at first seem to be intuitive, as
$\theta=\frac{e}{3}A_{I}(V_{P},B)B/\hbar$ and $V_{P}$ increases the area
$A_{I}$. However, the observation of only 4 interference periods over this
range of $V_{P}$ implies $A_{I}$ has changed by an area of $4\cdot
2\pi\ell_{B}^{2}/\nu\sim${10}^{-3}\text{\,}\mu\mathrm{m}^{2}$$, which is
negligible compared to the observed $20\%$ variation in $\Delta B$. We
conjecture the $V_{P}$ dependence instead occurs through the $B$-dependence of
$A_{I}$,
$\partial_{B}\theta=\frac{e}{3}A_{I}(1+BA_{I}^{-1}\partial_{B}A_{I})/\hbar$.
This dependence is one manifestation of bulk-edge Coulomb coupling: as $B$ is
increased at fixed $N_{qp}$ (and hence $\nu$), the electron density $n=\nu
eB/\hbar$ in the bulk increases, which in turn tends to shrink the edge due to
Coulomb coupling, reducing $A_{I}$. The standard theory Halperin  _et al._
(2011); von Keyserlingk  _et al._ (2015) of bulk-edge coupling predicts

$\displaystyle\theta=(1-\frac{K_{IL}}{K_{I}}\frac{\nu_{\textrm{in}}}{\Delta\nu})e^{\ast}\bar{A}B/\hbar$
(1)

where in the present case $\nu_{in}=\Delta\nu=1/3,e^{\ast}=e/3$. Here
$K_{I}\sim e^{2}/C_{I}\approx$234\text{\,}\mu\mathrm{e}\mathrm{V}$$ describes
the compressibility of the edge and $K_{IL}$ is a phenomenological bulk-edge
Coulomb coupling. The change in $\Delta B$ could then arise from a $V_{P}$
dependence of $K_{I}$ or $K_{IL}$. Based on the electrostatics of a double-
gated device, we expect bulk edge coupling of
$K_{IL}=e^{2}/2c_{g}A\approx$86\text{\,}\mu\mathrm{e}\mathrm{V}$$, where
$c_{g}=2\epsilon_{z}/d$ is the capacitance to the gates. Furthermore, we have
measured $K_{I}\sim$234\text{\,}\mu\mathrm{e}\mathrm{V}$$ and find it changes
by only about 15% as $V_{P}$ is varied over an even larger range of $V_{P}$
than displayed in Fig 1. The prediction of the standard theory then raises a
puzzle, as the resulting correction by $(1-86/234)\approx 0.63$ would imply
that the actual area is in fact $\sim 60\%$ larger than previously inferred,
$\bar{A}\sim 1.1-$1.3\text{\,}\mu\mathrm{m}^{2}$$. This is at odds with the
nominal size of the interferometer $0.74-$0.83\text{\,}\mu\mathrm{m}^{2}$$.

We conjecture this discrepancy arises from an effect not previously considered
in the context of GaAs FPI. In the standard theory of bulk-edge coupling, it
is implicitly assumed that $\nu\to 0$ within a short distance of the edge. In
our graphene FPI, however, $\nu\sim-5$ in the exterior of the interferometer.
As a result we expect additional Coulomb coupling between the charge in the
exterior of the FPI and the edge. When the exterior is operated in a gap, the
exterior charge density will introduce a new term to the Coulomb correction:
$(1-\frac{K_{IL}}{K_{I}}\frac{\nu_{\textrm{in}}}{\Delta\nu}-\frac{K_{I,\textrm{ext}}}{K_{I}}\frac{\nu_{\textrm{ext}}}{\Delta\nu})$.
In double-gated devices, we again estimate $K_{I,\textrm{ext}}\sim K_{IL}\sim
e^{2}/2c_{g}A$ (while it may seem strange for $K_{I,\textrm{ext}}$ to depend
on $A$, this is an artifact of the factor of $A$ which has been factored out
in Eq.(1): for both the bulk and exterior coupling, the important scale is the
gate distance relative to the perimeter, $d/L$). For $\nu_{\textrm{ext}}<0$
the exterior charge will thus counteract the bulk-edge coupling. In double-
gated devices, fields are screened over a scale $d$, so the relevant
$\nu_{\textrm{ext}}$ is one averaged over a distance of $\sim d$ from the
interfering edge. Over this range we expect the density to fall from
$\nu=\frac{1}{3}\to-5$ through a series of steps whose spatial structure
depends on the details of the electrostatics and interaction energies. It is
thus difficult to quantitatively predict $\nu_{\textrm{ext}}$. Nevertheless,
this hypothesis can be be tested through a future study of the interferometer
field period as a function of $\nu_{\textrm{ext}}$, though this requires a
modified device design.

###  IV.6 S5. Thermal dephasing of the interference phase in in the FQH regime

Figure S8:  Scaling of the interference visibility with temperature,
suggesting a finite amount of thermal dephasing from $N_{qp}$ fluctuations
(device 1) (A) $G_{D}$ vs. $V_{P}$ oscillations in the $\nu=1/3$ state,
measured as a function of temperature at fixed
$V_{C}=$2.230\text{\,}\mathrm{V}$$. (B) The average value of the conductance
$\bar{G}$ as a function of temperature shows strong temperature dependence,
characteristic of a chiral Luttinger liquid edge. (C) Interference visibility,
defined by taking the absolute value of the largest Fourier component of the
oscillations at base temperature, and tracking the amplitude of that component
as a function of temperature. (D)To cancel the power-law dependence of the
backscattering, here we normalize the visibility by the average backscattering
across the interferometer, $(e^{2}/3h-\bar{G})$. The normalized visibility
exhibits exponential suppression $e^{-T/T_{0}}$ from which we extract
$T_{0}=$87\text{\,}\mathrm{m}\mathrm{K}$$

As the temperature increases, the amplitude $\delta G$ of the conductance
oscillations is observed to decrease; this is theoretically expected due to
dephasing processes. In the FQH case, quantitative analysis of this dependence
is complicated by the non-linear $T,V$ characteristic of the current: the
effective back-scattering amplitude of a _single_ quantum-point contact scales
as a power-law in $T$.Wen (1991); Kane and Fisher (1992); Chamon  _et al._
(1997) At least in the limit of perturbative backscattering, this RG flow can
be cancelled by considering the ratio of the oscillation amplitude and the
mean current, $\delta I_{B}/\bar{I}_{B}=\delta G_{D}/(e^{2}/3h-\bar{G}_{D})$
Halperin  _et al._ (2011). The result is shown in Fig. S8A, and over a range
of $T$ shows exponential dependence $e^{-T/T_{0}}$ with
$T_{0}=$87\text{\,}\mathrm{m}\mathrm{K}$$.

When accounting for thermal fluctuations of the conformal field theory on the
edge alone, Ref.Chamon  _et al._ (1997) predict $kT_{0}=\frac{\hbar v}{\pi\nu
L}$ (note their notation for the edge geometry is related to ours by $a=L/2$).
Relating this to the measured edge capacitance $\frac{hv}{\nu
L}=\frac{e^{2}}{C_{I}}\approx$234\text{\,}\mu\mathrm{e}\mathrm{V}$$, we would
then predict
$kT_{0}=\frac{e^{2}}{2\pi^{2}C_{I}}=$137\text{\,}\mathrm{m}\mathrm{K}$$, in
rather poor agreement with our experiment. Here we show the discrepancy with
the CFT prediction can be accounted by for by including fluctuations in the
bulk quasiparticle number, Halperin  _et al._ (2011) an effect also observed
in Ref.Nakamura  _et al._ (2020).

Noting that the interference phase is proportional to the total electron
charge $\theta=2\pi N$, the thermal dephasing of $\langle
e^{i\theta}\rangle_{T}$ arises from fluctuations in $N$. The result can then
be estimated given the capacitance of the edge $C_{I}$ and bulk
$C_{L}$.Halperin  _et al._ (2011) Focusing on an operating point where
$\langle N_{qp}\rangle_{T}=0$, and, for simplicity, ignoring bulk-edge
coupling, we obtain

$\displaystyle\frac{\langle e^{i\theta}\rangle_{T}}{\langle
e^{i\theta}\rangle_{0}}$ $\displaystyle=\left(\frac{1}{Z_{I}}\int e^{2\pi
iN-\frac{e^{2}N^{2}}{2C_{I}kT}}dN\right)\left(\frac{1}{Z_{B}}\sum_{N_{qp}}e^{i2\pi
i(N_{qp}/3)-\frac{(e/3)^{2}N^{2}_{qp}}{2C_{L}kT}}\right)$ (2)
$\displaystyle\approx e^{-2\pi^{2}kT(C_{I}+C_{L})/e^{2}}=e^{-T/T_{0}}$ (3)

To obtain the exponential approximation we have taken the high-$T$ limit where
fluctuations in $N_{qp}$ are large; for $kT<\frac{(e/3)^{2}}{C_{L}}$, $N_{qp}$
fluctuations will be frozen out (or un-observable when the equilibriation time
becomes long). Below we obtain the estimate
$\frac{(e/3)^{2}}{C_{L}}\sim$170\text{\,}\mathrm{m}\mathrm{K}$$, and
experimentally we indeed find good exponential behaviour above this scale. As
discussed, using the measured value of
$\frac{e^{2}}{C_{I}}=$234\text{\,}\mu\mathrm{e}\mathrm{V}$$, we find that the
edge alone contributes
$kT_{0}=e^{2}/2\pi^{2}C_{I}\sim$137\text{\,}\mathrm{m}\mathrm{K}$$.

Additional fluctuations from $N_{qp}$ are then needed to explain the measured
value, which requires knowledge of the bulk capacitance $C_{L}$. The
capacitance of the bulk arises from taking the geometric and quantum
capacitance in series, $C^{-1}_{J}=C_{g}^{-1}+C_{q}^{-1}$. The quantum
capacitance can in turn be related to the electronic compressibility of the
device, $A\frac{e^{2}}{C_{q}}=\frac{\partial\mu}{\partial n}$. Fortunately the
compressibility of the MLG 1/3-state was measured by our group in Ref.Yang
_et al._ (2021), where we obtained $\frac{\partial\mu}{\partial
n}\approx$0.15\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\mu\mathrm{m}^{2}$$ at
$B=$18\text{\,}\mathrm{T}$$ (this can be obtained from the published data by
extracting the in-gap slope of
$\frac{1}{2\pi\ell_{B}^{2}}\frac{\partial\mu}{\partial
n}=\frac{\partial\mu}{\partial\nu}\sim$646\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$).
It remains to convert this to an estimate of the compressibility at the
$B=$9\text{\,}\mathrm{T}$$ used here. Proceeding phenomenologically, it is
reasonable to assume the in-gap density of states arises from quasiparticles
pinned to impurity sites. Assuming the spatial density of these states is set
by the impurity density, and is hence independent of $B$,Assouline  _et al._
(2024) with energies equally distributed over the FQH gap
$\Delta_{1/3}\propto\sqrt{B}$, we conclude the compressibility will scale as
$1/\sqrt{B}$. We thus scale the compressibility by $\sqrt{9/18}$ to obtain
$\frac{\partial\mu}{\partial
n}\approx$0.1\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\mu\mathrm{m}^{2}$$. The
estimated quantum capacitance is then
$\frac{e^{2}}{C_{q}}\sim$0.13\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$ using
the estimated size $A\sim$0.8\text{\,}\mu\mathrm{m}^{2}$$ of the
interferometer. The quantum capacitance then adds in series with doubly-gated
geometric capacitance
$\frac{e^{2}}{C_{g}}=\frac{e^{2}\epsilon_{z}d}{2A}\sim$0.17\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$:
all together, we thus estimate
$\frac{e^{2}}{C_{L}}=$0.3\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$.

Taking the bulk and edge capacitances in parallel, we obtain
$e^{2}/C_{t}\sim(1/0.3+1/0.234)^{-1}$\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$\sim$0.131\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$,
giving a dephasing temperature $T_{0}=$77\text{\,}\mathrm{m}\mathrm{K}$$,
quite close to the measured value of
$T_{0}=$87\text{\,}\mathrm{m}\mathrm{K}$$. This may be partially a matter of
good fortune given the estimates involved in $C_{t}$, but the scales are
clearly consistent with a dephasing temperature set by a combination of bulk
and edge contributions. In the future this analysis could be made quantitative
using devices capable of measuring the capacitance of the center gate defining
the interferometer.

###  IV.7 S6. Quasiparticle dynamics and bulk-edge coupling at $\nu=-1$

To further explore the evolution of charge dynamics, and to characterize the
role of bulk-edge coupling in the absence of contributions from the anyon
phase, we study integer filling of $\nu=-1$ at range magnetic fields near B =
4T in a second device, where the single chiral edge mode of $\nu=-1$ avoids
the complexity inherent in compound edges in higher filling factors
Werkmeister  _et al._ (2023). Because no anyon phase slips are expected at
$\nu=-1$, we can attribute the observed phase jumps entirely to changes in
interferometer area via the bulk-edge coupling Halperin  _et al._ (2011).

Figure S9:  Continuous evolution of the quasiparticle charging time at
$\nu=-1$ (device 2). (A) The transmitted current across the interferometer as
a function of the plunger gate voltage, $V_{P}$, and magnetic field B,
measured at a fixed electron density, with $V_{B}=0.3$V and $V_{C}=-0.580$V
corresponding to the $\nu=-1$ plateau near
$B\approx$3.96\text{\,}\mathrm{T}$$. The interference is Aharonov-Bohm
dominated throughout. The phase slip behavior evolves from continuous in this
field range to (B) noisy near $B\approx$4.2\text{\,}\mathrm{T}$$ to (C)
discrete near $B=$4.47\text{\,}\mathrm{T}$$. (D) Dependence of the
differential diagonal conductance on the applied DC source-drain bias,
$V_{SD}$, applied to the source contact, and a common-mode voltage $V_{CM}$,
applied to both the source and drain contacts (E) A representative trace of
the transmission vs. plunger gate voltage $V_{P}$ at the point circled in
panel B. Clear switching behavior between two phase-offset curves is observed.
(F) The phase $\theta$ as a function of B, extracted from the Fourier
Transform of panel C. The median phase difference between successive traces is
subtracted to remove the smooth evolution of the Aharonov-Bohm phase. The
remaining discrete jumps have a characteristic value of
$\Delta\theta/2\pi=-0.092\pm 0.008$. (G) Measurement of $R_{xy}$ on the W side
of the device at $V_{C}=-0.580V$ and $V_{B}=0.3$V. The grey boxes mark the
field ranges I, II, and III in which interference data of panels A, B, and C
were taken, respectively. (H) Evolution of the quasiparticle switching rate as
a function of magnetic field inside the “noisy” regime. The error bar plotted
is the value of the asymmetry of the 1-to-2 and 2-to-1 switching rates,
$\Delta\tau=|\tau_{12}-\tau_{21}|$.

Figs. S9a-c show interference data taken in three regions of the $\nu=-1$
transport plateau, approaching the plateau center from the high filling-factor
side. Note that for $\nu=-1$, depleting the interferometer area requires
applying a positive voltage to $V_{P}$, so the negative slope of the constant-
phase lines is consistent with Aharonov-Bohm dominated interference, and the
observed field period $\Delta B=$8\text{\,}\mathrm{m}\mathrm{T}$$ implies an
effective area of approximately $A_{I}=$0.52\text{\,}\mu\mathrm{m}^{2}$$ when
ignoring bulk-edge coupling, slightly smaller than the lithographic area of
$A=$0.76\text{\,}\mu\mathrm{m}^{2}$$ (as discussed in a previous section, the
quantitative discrepancy may arise from bulk-edge coupling).

In Fig. S9a, we observe regular phase slips that are reversibly tuned by both
plunger gate and magnetic field, spaced by approximately one flux quantum.
This is consistent with the breathing of the interferometer area due to the
repeated entry of single electron quasiparticles each with a small but finite
bulk-edge coupling of approximately the same magnitude. As these data are
taken near the edge of the transport plateau, we expect the system to have a
finite compressibility, and the quasiparticle states to be relatively
delocalized.

In addition, the measurement of the interference signal as a function of the
dc common mode voltage $V_{CM}$ allows us to estimate the expected degree of
bulk-edge coupling as done in section S4. From the measured common-mode period
$\Delta V_{CM}$ at $B=4.0T$, $K_{I}=e\Delta
V_{CM}=$1.24\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$. The geometric
capacitance of the island can be approximated from the thickness of the hBN
spacers ($45\text{\,}\mathrm{n}\mathrm{m}$ each) and measured lithographic
area $A_{I}$ of the graphite top gate, leading to an estimated (following
section S4) of
$K_{IL}=e^{2}/2c_{g}A\approx$89\text{\,}\mu\mathrm{e}\mathrm{V}$$. Thus, the
expected phase shift from the bulk edge coupling is
$\delta\theta_{BE}/2\pi=K_{IL}/K_{I}=0.072$. This is in agreement with the
size of the observed phase slips within our systematic uncertainty.

As we increase the magnetic field and move closer to the center of the
plateau, we observe a striking qualitative change in the interference signal
near the phase slip lines. In Fig. S9B, while the pattern of phase slips
remains regular, telegraph-noise develops in the transmitted current near the
charge degeneracy line associated with each phase slip. This is illustrated in
Figs. S9E, which shows a trace of the transmitted current as a function of
$V_{P}$, acquired slowly over the course of 36 seconds across the marked
charge-degeneracy point. Evidently, the transmission through the
interferometer switches between two distinct patterns offset by a small phase
shift on a few-100ms timescale. As we increase the magnetic field further
towards the center of the $\nu=-1$ plateau, the switching behavior becomes
less apparent, and we recover sudden and irreversible phase slips as described
in the fractional quantum Hall regime in the main text: interference data show
lines of continuously evolving AB phase interrupted by sharp, irregularly
spaced phase slips that are instantaneous and irreversible (Fig. S9C). In
contrast to the fractional quantum Hall regime, however, the magnitude of
these phase slips is much smaller than $2\pi/3$, with
$\Delta\theta/2\pi\approx-0.092\pm 0.04$ for the slips shown in Fig. S9C -
slightly greater than the estimated equilibrium value but consistent within
our relatively large systematic uncertainty. However, a larger effect from
bulk-edge coupling is not unexpected as some bulk states may be physically
closer to the edge than others contributing to a larger observed change in
$A_{I}$ upon loading.

Evidently, the charge equilibration time $\tau$ varies strongly over a small
magnetic field range, going from faster than $10\text{\,}\mathrm{m}\mathrm{s}$
(the time to acquire a single pixel in Figs. S9A-C) to slower than 7s (the
time to acquire a complete line in Fig. S9A-C) over a range of 500 mT. This
increase can be quantified precisely in the regime of Fig. S9B, where the
switching time $\tau\approx 1s$. The plotted values of the characteristic time
$\tau$ are plotted in Fig. S9H, illustrating a sharp and monotonic increase
over a very narrow range of $45\text{\,}\mathrm{m}\mathrm{T}$.

To extract the data-points in Fig. S9H, we analyze the telegraph noise
quantitatively using a two-state switching model where our readout is the
interference signal at distinct charge-degenerate points for several values of
B. Fig. S10a shows a measurement of the transmitted current illustrating
roughly the location of one of the measured charge-degenerate lines in the
switching-noise regime. Precisely at the charge degeneracy point, we expect
the tunneling rate to have some characteristic time $\tau$. In general,
however, we expect the ratio $P_{1\rightarrow 2}/P_{2\rightarrow
1}=e^{-\Delta/k_{b}T}$, where $P_{1\rightarrow 2}$ is the probability for an
electron to hop from state 1 to state 2, $P_{2\rightarrow 1}$ is the
probability for the electron to hop from state 2 to 1, and $\Delta$ is the
detuning away from the charge degeneracy point. As a result, the location of
the charge-degenerate point in $V_{P}-B$ plane must be determined
systematically by measuring the switching noise as a function of the detuning.

Figure S10: Analysis of the quasiparticle switching rate in the switching-
noise regime in $\nu=-1$ (A) Measurement of the interference pattern in the
transmitted current across a single charge degeneracy line in the switching
noise regime of $\nu=-1$. (B) The median current in each of the two stable
configurations in the switching noise regime. The median value of $I_{T}$ in
each state, plotted in red and blue, are extracted from 20 minute measurements
of the current at each fixed $V_{P}$, by splitting the histogram of the
acquired data into two halves at the midpoint and taking the median value of
current for each half. (C) The fractional probability to occupy state 1 at
each plunger gate voltage in B, acquired by simply taking the number of data
points lying on either side of the halfway-point threshold between the two
stable values of current. The charge 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 used to extract the dwell times $\tau_{12}$ and $\tau_{21}$.
G,H,I Histograms of the dwell times in state 1, $t_{12}$, each showing an
exponential distribution with a characteristic time $\tau_{12}$ that decreases
as the charge degeneracy line is crossed from left to right. (J,K,L)
Histograms of the dwell times in state 2, $t_{21}$, showing that the
characteristic time $\tau_{21}$ increases as the charge degeneracy line is
crossed from left to right.

Fig. S10B shows the characteristic values of the transmitted current $I_{T}$
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. S10A. The charge degenerate point is determined by
plotting the fraction of time spent in state 1, $P(1)$, as function of the
plunger gate voltage, plotted in Fig. S10C.

Fig. S10G-L show histograms of the two dwell times, $\tau_{1}$ and $\tau_{2}$,
for different plunger gate voltages near a phase slip line, showing a strong
dependence of the relative rates on detuning. For a given $B$, we identify the
the value of $V_{P}$ where $\tau_{12}\approx\tau_{21}$ as the charge
degeneracy point, and plot $\tau\equiv(\tau_{12}+\tau_{21})/2$ in Fig. 3I,
with the difference $\Delta\tau\equiv|\tau_{12}-\tau_{21}|$ as the error bar.
We find that $\tau$ increases by nearly one order of magnitude over a
$45\text{\,}\mathrm{m}\mathrm{T}$ range of magnetic fields.

The dramatic slowing of charge dynamics within the plateau is not unexpected:
as the filling factor moves closer to the center of the $\nu=-1$ plateau,
quasiparticle states are increasingly localized around impurity sites.
Individual impurity orbitals will, consequently, be increasingly far from the
sample edge. The dramatic resulting difference in $L$-dependence allows $\tau$
to change dramatically while the bulk edge coupling remains nearly unchanged.
For example, reducing the radius of an impurity state centered 100 nm from the
edge will increase $\tau\sim 1/\Gamma$ by three orders of magnitude, while
only changing the bulk edge coupling by 15%.

##  V S7. Sharply Confined Charge Traps and Full Range of Data in Figure 1

Expanding the range of data presented in the main text Fig. 1d we observe two
phase slips which are continuously tuned with both B and $V_{P}$. While these
phase slips are more analogous to those observed in previous GaAs measurements
Nakamura  _et al._ (2020), it is notable that the charge degeneracy line
follows a nearly horizontal trajectory in the B-$V_{P}$ plane. This suggests a
large capacitive lever arm between the plunger gate and the charge occupation
number in the interferometer, at least for this particular charge trap. In
fact, we find these strongly coupled charge loading events to be fairly
ubiquitous in these devices. In the integer quantum Hall regime, where all
phase slips are related to bulk-edge coupling effects, we find several
isolated, reversible, slips where the slope of the charge degeneracy line
varies depending on which gate is modulating the interference. We present here
sample measurements of the interference on the inner edge of $\nu=-2$, where
the slope of the charge degeneracy line is measured for various top-gate
combinations. Taken collectively, the asymmetry in the lever arm of each top-
gate to the charge trap occupation number, firmly indicates the charge trap is
strongly localized. This further indicates that the disorder landscape in
graphene van der Waals devices consists of sharp confining potentials, likely
from Coulombic impurities in the hBN. This disorder picture, likely
sufficiently different than in GaAs, will require new theoretical treatments
of bulk edge coupling and how the spatial arrangement of charge affects future
interferometry experiments.

Figure S11: Soft-phase slips and locate-able defects. (a) Full range of data
explored for main text Fig. 1d. In addition to the sharp phase slips studied
in the main text we also observe two distinct ‘soft’ phase slips that are
continuously tuned by both $V_{P}$ and B as indicated by the inset white
arrows. We understand these to be sharp Coulombic defects that strongly
localize charge near the edge. To validate this hypothesis we perform
interference experiments in the integer quantum Hall regime, where no anyon
phase effects should be expected, near one of these defects. (b) Isolated
‘soft’ phase slip across interference of the inner edge mode in $\nu=-2$ in
the second device shown in Fig. S1c-d. $G_{D}$ is plotted versus $V_{P}$ such
that lines of constant Aharanov-Bohm phase have negative slope (plunger gate
requires positive voltage to deplete the interferometer area on the hole side
of the device). The phase slip is observed to have a nearly vertical slope in
the $B-V_{P}$ plane. The phase slip is the result of a finite amount of bulk-
edge coupling when a single hole exits the interferometer. (c) Phase across
the slip, with the Aharnov-Bohm component removed, calculated using the same
DFT method used in the main text to extract the value of the sudden phase
slips. (d) $G_{D}$ as a function of B and $V_{NW/SW}$ in the same operating
regime as panel b, with all other gates fixed, which shows the same phase
slip. Notably, the phase slip, similarly to panel b, has a nearly vertical
slope in the $B-V_{NW/SW}$ plane. (E) $G_{D}$ as a function of B and $V_{NE}$,
with all other gates fixed, again showing the same phase slip as in panel b.
Here the phase slip is noted to have a steep, but discernibly less than
vertical, slope in the $B-V_{NE}$ plane. (F) $G_{D}$ as a function of B and
$V_{SE}$, all other gates fixed, showing the same phase slip as panel b. In
this dataset the phase slip is shown to have an extremely shallow slope in the
$B-V_{SE}$ plane. This indicates a strong capacitive coupling to the charge
trap not observed when studying the slip as the interference phase is varied
using the other top gates. (g) AFM topograph, as shown in Fig. S1d, which
shows the edge state configuration for panels b-f overlaid with the gate
structure. The relative lever-arm each top gate has to the chemical potential
of the charge trap allows us to put a strong constraint on the location of
said trap. This location is marked by the red dot.

###  V.1 S8. Anomalous State Preparation-Dependent Phase Slips

After the measurement presented in the main text Fig. 3 was performed, the
center gate raster was continued over the trajectory shown in S12A. During the
four subsequent sweeps up and down across the same range of $V_{C}$, no phase
slips at all are observed, indicating that the system remains in a highly
(meta)stable charge configuration. At 140 minutes, the $V_{C}$ range was then
made to excurse to a higher filling factor, before returning to the range
studied previously (no interference is visible in this regime, as the applied
$V_{C}$ causes the QPCs to be fully open). After this, the reverse-direction
sweep to Fig. 3A was performed once again, and 11 phase slips were observed.

These phase slips present several mysteries. First, their magnitude is
consistent, with an average value of $\Delta\theta/2\pi=-0.425$ (See Fig.
S12B). This is not particularly close to any integer multiple of $2\pi/3$,
which suggests a significant and repeatable amount of bulk-edge coupling is
relevant for every slip.

While their occurrence during an interval of decreasing of $V_{C}$ leads one
to expect they are associated with an overall removal of quasiparticles
(addition of quasiholes), their magnitude is larger than $-\theta_{a}$. Due to
the fact that bulk-edge coupling should only serve to decrease the magnitude
of the phase slips, this leads one to the conclusion that each slip is
associated with the removal of more than one quasiparticle every time.
Confusingly, the magnitude of each slip is consistent, also suggesting that
the same number of quasiparticles is removed with every event.

An alternative possibility to account for the consistent magnitudes would be
that the system is merely switching back and forth between two distinct charge
configurations. This should be considered, given that the phase slip magnitude
is nearly $\Delta\theta=\pm\pi$, almost to within our expected systematic
uncertainty. However, examination of the “charge fingerprint” of sets of
curves which are separated by two phase slips (several of which are shown in
Fig. S12D) indicates a strong degree of dissimilarity. This suggests that each
charge configuration is in reality distinct.

While not definitive evidence, these facts taken together raise the
possibility that after loading a higher density of quasiparticles, the
unloading of these quasiparticles takes place in a correlated fashion, with
each discretely-resolved event corresponding to the (at least nearly)
simultaneous removal of more than one quasiparticle. This highlights the need
for further experiments to probe the potentially complicated anyon dynamics in
this system.

Figure S12:  Anomalous phase jumps for a different state preparation. (A) The
trajectory of the center voltage of the $V_{C}$ sweep. The data in main text
Fig. 3A are taken from the indicated range. Subsequently 0 phase slips are
seen as the gate voltage range is slowly swept back and forth over the same
range. After an excursion to higher $V_{C}$ at
$\sim$140\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$$, the 11 slips analyzed here
and shown in Panel B occur. (B) Analog of Fig. 4 in the main text, taken in
the marked range of panel A, shortly after the gate voltage excursion to
higher filling factor. (C) The extracted phase (in red) and magnitude of each
discrete phase jump (in blue) of the data in panel B. (D) Overlapping traces
from several classes that each are separated by two discrete jumps. The subtle
differences in curve shape (“charge fingerprint”) indicate that, while the
phase difference is nearly $2\pi$, the classes seem to correspond to distinct
configurations of charge in the interferometer, ruling out that the observed
phase slips are merely back-and-forth switching between two distinct charge
states.

###  V.2 S9. Additional data referenced in main text

Figure S13A-D show four repetitions of identically-measured interference data
in the same regime as in the main text Fig. 1D. The exact locations of the
individual phase slips differ between measurements. However, there is a clear
increase in frequency of the slips in the high-field region of the plot (above
$8.94\text{\,}\mathrm{T}$). The extracted phase from each panel in Fig. S13E
shows that in this range, while the exact location of the slips is random,
their average spacing ensures that the phase remains roughly constant, with
approximately one phase slip per additional flux quantum.

Figure S13: Repeated measurement of interference in the regime of Fig. 1 (A-D)
Repeated measurements of $G_{D}$ as a function of magnetic field and plunger
gate voltage, corresponding to a subset of the range over which data in Fig.
1d were taken. The field is swept from low to high every time. (E) The
extracted phase $\theta$ as a function of B. No background is subtracted, but
the line-by-line change in $\theta$ is assumed to lie within the interval
$(-\pi,\pi)$. The phase evolves smoothly due to the Aharonov-Bohm effect, and
is punctuated by sharp slips corresponding to changes in the quasiparticle
content. The spacing between these slips is stochastic, as the charging time
is long — however, above 8.94T the slips happen approximately once per
additional flux quantum, causing the phase to remain constant on average,
despite large time-dependent deviations. Figure S14: Difference in charge
fingerprint between all pairs of phase classes separated by 3 phase slips from
Fig. 3 of the main text. Figure S15: Stability of phase over time while B
field is swept. Repeated $V_{C}$ ramps as a function of time, in a similar
regime of parameters as those shown in Fig. 4. Here no phase slips are
observed.