# Anyons in Quantum Hall Interferometry

Matteo Carrega  CNR-SPIN, Via Dodecaneso 33, 16146, Genova, Italy  Luca
Chirolli  NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, Piazza
San Silvestro 12, 56127 Pisa, Italy  Department of Physics, University of
California, Berkeley, California 94720, USA  Stefan Heun  NEST, Istituto
Nanoscienze-CNR and Scuola Normale Superiore, Piazza San Silvestro 12, 56127
Pisa, Italy  Lucia Sorba  NEST, Istituto Nanoscienze-CNR and Scuola Normale
Superiore, Piazza San Silvestro 12, 56127 Pisa, Italy; *email =
<EMAIL_ADDRESS>
###### Abstract

The quantum Hall (QH) effect represents a unique playground where quantum
coherence of electrons can be exploited for various applications, from
metrology to quantum computation. In the fractional regime it also hosts
anyons, emergent quasiparticles that are neither bosons nor fermions and
possess fractional statistics. Their detection and manipulation represent key
milestones in view of topologically protected quantum computation schemes.
Exploiting the high degree of phase coherence, edge states in the QH regime
have been investigated by designing and constructing electronic
interferometers, able to reveal the coherence and statistical properties of
the interfering constituents. Here, we review the two main geometries
developed in the QH regime, the Mach-Zehnder and the Fabry-Perot
interferometers. We present their basic working principles, fabrication
methods, and the main results obtained both in the integer and fractional QH
regime. We will also show how recent technological advances led to the direct
experimental demonstration of fractional statistics for Laughlin
quasiparticles in a Fabry-Perot interferometric setup.

## Introduction

Over the last forty years, quantum Hall (QH) physics has represented an
extremely active research topic on the condensed matter agenda [1]. Since its
discovery, the quantum Hall effect has represented a unique platform that has
become a metrological realization of electrical resistance and, at the same
time, displays a number of highly exotic phenomena. More recently, the quantum
Hall system has been recognized as the first known example of topological
quantum matter [2, 3, 4, 5]. The topological nature of QH states is at the
heart of the precise and robust quantization of resistance and of the long
coherence length of electrons that allow for ballistic transport over
remarkable distances. These interesting features have been investigated and
exploited both in the integer quantum Hall (IQH) regime and for the intriguing
fractional (FQH) case [6, 7], where many-body effects play a crucial role in
forming a strongly correlated stable system, exhibiting highly exotic emergent
quasiparticles, that are very promising for topological quantum computation
purposes [5].

The topological nature of QH systems is reflected in the coexistence of an
insulating bulk and metallic edge states, which are responsible for transport
properties. Owing to the high degree of phase coherence of electrons, these 1D
propagating channels have been viewed as the electronic analog of optical
waveguides, opening an entire new field called electron quantum optics [8, 9,
10, 11, 12] that holds promise for quantum computation with flying qubits [13,
14, 15, 16, 17, 18]. In this setting, interferometry in the QH regime has
emerged as a unique playground to study and exploit the coherence of
propagating quasiparticles. Several electronic interferometer setups have been
put forward and successfully realized in recent years, both constituting
useful novel metrological applications and, more importantly, revealing
information on Coulomb interactions and the nature of emergent quasiparticles.
Since quasiparticle statistics can be inferred by moving one entity around
another, electronic analogues of wave interferometers are ideal platforms to
investigate these peculiar features, especially in the FQH regime, whose
excitations are predicted to be anyons [5, 4, 19, 20], instead of common
fermions or bosons.

The scope of this Technical Review is to give a survey of the main and most
recent results achieved in QH interferometry. Of course, it is not possible to
cover all aspects investigated in the vast literature present, where already
excellent reviews have been published on different topics [4, 5, 21]. Here, we
describe and compare the two main geometries realized so far, namely Mach-
Zehnder and Fabry-Perot electronic interferometers. We discuss their basic
working principles, fabrication methods, drawbacks, and the main results
obtained both in the integer and fractional QH regime. We also discuss a
recent landmark experiment in Fabry-Perot interferometry that allowed direct
demonstration of fractional statistics for Laughlin quasiparticles [22]. We
emphasize that another recent experiment [23] achieved the same milestone,
employing a beam-splitter setup with noise detection. These impressive results
open the way to new experiments involving more complex FQH states, with the
aim of demonstrating the emergence of quasiparticles with non-Abelian
statistics, a key milestone for topological quantum computation purposes [5].

## Anyons and Exchange Statistics

Quantum statistics is at the heart of quantum mechanics, reflecting the
symmetry properties of a many-body wave function under the exchange of two
identical particles. In three dimensions, only two kinds of particles can
exist, bosons or fermions, which are symmetric or anti-symmetric under
particle exchange, respectively. In two spatial dimensions (2D), the situation
turns out to be not so restricted and much richer, as realized over 40 years
ago [19] [20] — an adiabatic process in which two particles are exchanged
twice is equivalent to a process where one particle forms a closed loop around
the other particle. Indeed, in 2D, a closed trajectory encircling a particle
cannot be deformed into a point without ”crossing” the encircled particle and
the whole system does not necessarily come back to the same state [19, 20, 5,
4]. This idea defines the concept of non-trivial winding. After two
consecutive particle interchanges (or one particle moved around the other in a
closed loop) the two-particle wave function, $\psi$, can acquire a non-trivial
geometric phase, $\theta$,

$\psi({\bf r}_{1},{\bf r}_{2})\to e^{i\theta/2}\psi({\bf r}_{2},{\bf
r}_{1})\to e^{i\theta}\psi({\bf r}_{1},{\bf r}_{2})~{},$ (1)

with $\theta=0$ for bosons and $\theta=2\pi$ for fermions, where ${\bf r}_{1}$
and ${\bf r}_{2}$ define the position of each particle. In 2D, other generic
$\theta$ values are allowed, and quasiparticles obeying such generalized
statistics are called anyons [19, 20, 24, 4].

More generally, in a system of $n$ quasiparticles (with non-degenerate ground
state), an adiabatic process consisting of swapping some of the quasiparticle
positions results in a phase factor, which is composed of a trivial dynamical
phase and a statistical contribution. The latter can be expressed as the Berry
phase [25, 5] of the relative motion and can be quantified as

$\theta=i\oint d{\bf r}\cdot\langle\psi({\bf r})|\nabla_{\bf r}|\psi({\bf
r})\rangle,$ (2)

which is independent of the microscopic details of the particle trajectories
and represents a two-particle property of the wave function. Besides, a
single-particle Berry phase can be acquired separately by each particle. A
well-known example is provided by the Aharonov-Bohm (AB) phase [26], that is
acquired by a particle of effective charge $e^{*}$ encircling a region
threaded by a magnetic flux. In presence of a vector potential ${\bf A}({\bf
r})$ describing a magnetic field of magnitude $B$, the gradient $\nabla_{\bf
r}$ is replaced by $\nabla_{\bf r}-ie^{*}{\bf A}({\bf r})/\hbar$, and the
Berry phase acquires an additional term, $\theta\to\theta+\theta_{AB}$, with

$\theta_{AB}=\frac{e^{*}}{\hbar}\oint d{\bf r}\cdot{\bf A}({\bf
r})=2\pi(e^{*}/e)B\cdot A/\phi_{0},$ (3)

where $A$ represents the area enclosed by the loop, $\phi_{0}=h/e$ is the flux
quantum, and $e$ the electron charge. The AB phase does not depend on the
details of the particular trajectory, but only on the flux threading the
enclosed area.

In the FQH regime, the competition between the high magnetic field, the low
electronic density and the Coulomb interaction, together with the two-
dimensional character of the system, promotes a strongly correlated ground
state wave function characterized by non-trivial exchange properties. In this
regime, quasiparticle excitations are believed to be emergent anyons [27, 5,
4]. The simplest instance of FQH state supporting anyons is provided by the
Laughlin state, that describes the series with fractional filling factor
$\nu=1/(2m+1)$, with $m$ an integer. In this case, evaluation of the Berry
phase leads to a statistical angle $\theta_{\nu}=2\pi/(2m+1)$ and an effective
fractional charge $e^{*}=e/(2m+1)$ [7, 28]. Indeed, the fractional
quasiparticle exchange properties follow from the statistical (two-particle)
part of the Berry phase [28, 5], while the fractional effective charge is
associated with the single-particle Aharonov-Bohm effect via the flux acquired
along a closed path.

The existence of fractional charges has been observed and confirmed with
several techniques, such as the measurement of the Fano factor in shot noise
experiments [29, 30]. Detection of fractional statistics is more subtle, and
after decades of intense research efforts, a first direct proof of fractional
statistics in the FQH regime has been reported very recently in anyon
collision experiments [23] and soon after confirmed also in interferometry
experiments [22]. The former study realizes collision experiments at a beam-
splitter based on the Hanbury-Brown-Twiss or Hong-Ou-Mandel effect [8, 9, 10,
11, 12, 23], that can probe the statistics of colliding particles via two-
particle interference. The latter is based on interference among different
paths, and several proposals that aim at probing the fractional statistics
have been engineered for FQH states. Below we focus on two of the most common
electronic interferometers that have been proposed and realized in the last
years for this purpose (see Box I for their optical analog), and we describe
the recent results obtained in this field.

## Mach-Zehnder Interferometer

Figure 1: The electronic Mach Zehnder interferometer.(a) Scheme of the
electronic Mach-Zehnder interferometer (MZI) in a Hall-bar in the Corbino
geometry, featuring a hole at the center. For filling factor $\nu=2$, the
inner edge channel (not shown) is completely backscattered at the quantum
point contacts (QPC)s. The outer channel coming out from source S (represented
by a black line) is partially transmitted at QPC1 and QPC2 and can therefore
mix with the outer channel that flows around the hole at the center of the
mesa. The outgoing signal is collected at drains D1 and D2. Modulation gates,
MG1 and MG2, allow for modulation of the area. (b) Scanning electron
micrograph of an electronic MZI. A central ohmic contact, serving as a
detector (D2), is connected through a metallic air bridge. QPC1 and QPC2
obtained via metallic air bridges act as beam-splitters, and five metallic
gates are used as MGs. (c) Aharanov-Bohm oscillations as a function of
magnetic field and modulation gate voltage, showing the typical pyjama
pattern. Color code: maxima (red) and minima (blue). (d) Visibility of the
Aharanov-Bohm oscillation as a function of QPC1 transmission $T_{1}$, together
with a fit (dashed line) with the expectation $\eta\sqrt{T_{1}(1-T_{1})}$,
giving $\eta=0.62$. All panels adapted from Ref. [31].

### Geometry and Basic Principles

An electronic Mach-Zehnder interferometer can be realized in the QH regime by
exploiting the edge states as electronic channels, in analogy with photon
waveguides (Box I). The high degree of phase coherence in the QH regime allows
for the construction of an interferometer on a scale of tens of micrometers.
Due to the chiral nature of the edge modes, dictated by the magnetic field, a
MZI in the QH regime is usually realized in a Corbino geometry that features a
hole at the center of the Hall-bar, as shown in Fig. 1a-b. Beam-splitters are
realized through quantum point contacts (QPCs) (Box II) that bring the edge
channels in close proximity, allowing for tunneling and (partial)
backscattering. Depending on the structure of the edge channels that
characterize an integer filling factor $\nu$, typically only one edge channel
is partially transmitted at the QPCs, and the other channels are completely
reflected or transmitted. The schematic in Fig. 1a highlights only the
scattering channels that coherently mix at the QPCs and allows for a clear
comparison with the optical analog (see Box I). Propagation along the two
paths introduces a relative phase, which comprises a dynamical phase
$\theta_{L}$ and the Aharonov-Bohm phase $\theta_{AB}$. The dynamical phase
depends on the energy via the momentum $k=\epsilon/(\hbar v_{\epsilon})$, with
$v_{\epsilon}$ the velocity at energy $\epsilon$. The Aharonov-Bohm phase is
given by $\theta_{AB}=2\pi\phi/\phi_{0}$, with $\phi=B\cdot A$ the flux of the
magnetic field $B$ through the area $A$ enclosed by the two paths.
Interference fringes can be observed in the output currents measured at the
detectors D1 and D2, that are expected to show out-of-phase sinusoidal
oscillations.

The electronic MZI was originally realized in the IQH regime in 2003 [31]. The
device was fabricated in a high mobility two-dimensional electron gas (2DEG)
embedded in a GaAs-AlGaAs heterojunction, by shaping the mesa in a 3 $\mu$m
wide ring (Fig. 1b). An air bridge was constructed to make an Ohmic contact
with the drain D2. The AB phase is modulated by changing the magnetic field
$B$ or by changing the area $A$ through different modulation gates (MG) placed
along one path, as shown in Fig. 1a-b. The interference pattern was measured
at filling factor $\nu=2$, with the inner edge channel completely reflected by
the QPCs. The current $I$ as a function of the modulation gate voltage and the
magnetic field shows a neat interference pattern in the form of a
characteristic striped color plot, the so-called “pyjama pattern”, as reported
in Fig. 1c.

### Implementation and Results

Figure 2: Visibility and coherence of Aharonov-Bohm (AB) oscillations. (a)
Visibility of the Aharanov-Bohm oscillations of a Mach-Zehnder interferometer
(MZI) in the $\nu=2$ integer quantum Hall (IQH) regime as a function of the
source-drain bias, showing a lobe structure (top panel) and phase rigidity
with $\pi$ jumps (bottom panel). (b) Log of the visibility, $\cal V$, of the
AB oscillations as a function of temperature for MZIs of different size.
Inset: slope of the curves versus size of the devices. An exponential decay
with temperature and size is extracted. (c) Schematics of a MZI in the $\nu=2$
IQH regime showing the interfering outer channel (black solid/dashed line),
sourced by S2, and the inner channel (gray solid/dashed line). The latter is
independently biased, through contact S3, and partitioned, through quantum
point contact (QPC0), to induce a controlled dephasing. Currents are collected
in drains D1 and D2. The interfering area is varied through a modulation gate
MG. (d) Resulting visibility as a function of the QPC0 transmission $T_{0}$
for $V_{\rm det}=V_{S3}$. Fits show the predictions for dephasing induced by
Gaussian (dotted lines) and non-Gaussian noise (black line, see equation in
the text). Panel a adapted from Ref. [35]. Panel b adapted from Ref. [42].
Panels c and d adapted from Ref. [36].

The degree of coherence of the MZI is assessed by studying the visibility,
$\cal V$, of the oscillations given by ${\cal V}=(I_{\rm max}-I_{\rm
min})/(I_{\rm max}+I_{\rm min})$. Setting the transmission of QPC2 to
$T_{2}=0.5$, the visibility as a function of transmission $T_{1}$ of QPC1 is
expected to be ${\cal V}=2\eta\sqrt{T_{1}(1-T_{1})}$. The parameter $0<\eta<1$
accounts for a reduction of the visibility. A value $\eta=0.62$ was reported
in Ref. [31], as shown in Fig. 1d. Deviations from visibility with $\eta=1$
are ascribed to phase averaging and dephasing. Phase averaging typically takes
place when independent electrons contributing to the detected signal acquire a
slightly randomized phase, for example due to slightly different velocities
appearing in the dynamical phase. Dephasing typically occurs due to inelastic
scattering with the environment and originates mainly from Coulomb
interactions. An analysis of the temperature- and bias-dependence of ${\cal
V}$ shows a monotonic decay over a common energy scale, and a study of the
shot noise points to a dominant role of phase averaging, as opposed to
dephasing [32, 33, 34].

Subsequent works by different groups confirmed the functionality of the MZI in
the Corbino geometry and pointed out several non-trivial aspects [35, 36, 37,
38, 39, 40, 41, 40, 42, 43, 44, 45, 46, 47, 48]. Unexpectedly, the visibility
showed a multiple lobe structure as a function of the DC source-drain bias.
The measurement shows that the phase of the AB oscillations stay constant,
expect for abrupt changes by $\pi$ when there are minima in the visibility
(Fig. 2a) [35, 36, 37, 41, 48]. This behaviour reflects strong dephasing in
the system and is not caused by phase averaging. The lobe structure appears to
be non-universal, and structures with a single side lobe have been reported,
as well [39, 44]. For filling factor $\nu=2$, the origin of the lobe structure
in the visibility has been ascribed to interchannel Coulomb interactions [49,
50, 51, 52]. At long wavelengths, chiral excitations at the edge behave as
bosonic waves (plasmons), with a dispersion determined by the intrachannel
Coulomb interaction [53]. Interchannel Coulomb interaction mixes the two edge
plasmon modes and promotes a fast charge mode and a slow dipole mode as
eigenmodes [49, 50, 51, 52]. A tunneling event at QPC1 excites both
eigenmodes, which propagate at different velocities toward QPC2. At QPC2,
their recombination results in beatings, and the minima of the visibility are
associated with destructive interference at an energy scale set by the
velocity of the slow mode.

In order to further characterize the degree of coherence of the MZI, a
measurement of the coherence length as a function of temperature and
interferometer arm length was reported [42], showing in both cases an
exponential dependence (Fig. 2b). In an attempt to isolate the relevant
interfering channel, the upper arm of the MZI has been divided into disjoint
regions where the inner channel forms micron-scale loops [54]. This loop-
closing strategy combined with the freezing of internal degrees of freedom
provide an essential isolation from the environment, with a successful
increment of coherence length up to 0.1 mm [55].

Other implementations of an electronic MZI have also been proposed and
realized besides the Corbino paradigm. In an architecture for multichannel
interferometry between co-propagating edge states in the IQH regime [16], two
channels are mixed before and after a region in which a top gate separates
them and introduces a flux sensitive area. Beam mixing of the spin-resolved
edge states (at $\nu=2$) has been realized through a comb of magnetic fingers,
which provides a spin-flip mechanism and the correct momentum transfer through
the comb periodicity [16, 56, 57, 58]. An alternative approach has been put
forward, based on non-equilibrium population [59, 60] of compressible and
incompressible strips [61, 62]. In this context, details of edge equilibration
[63] and edge reconstruction [64] have been studied with lateral resolution by
scanning gate microscopy.

In graphene, a MZI has been realized by exploiting spin- and valley-polarized
edge channels at the boundary of a p-n junction [65, 66]. Here, conduction and
valence band electrons have opposite chirality, and for samples encapsulated
in h-BN, copropagating edge states at the two sides of the p-n junction, even
with the same spin, do not equilibrate [67, 68]. Beam mixing is achieved at
the physical boundaries of the junction, where an abrupt potential variation
introduces scattering among same-spin states. AB oscillations with 98%
visibility have been reported, showing strong robustness to dephasing [65,
66].

### Interaction Effects and Dephasing

Interferometry in an electronic MZI has proven to be demanding already at
integer filling factors. In the $\nu=1$ case, electrostatic interactions may
give rise to dephasing and to contributions of higher harmonics [53]. For the
$\nu=2$ case, several experiments were performed to study the reaction of the
system to controlled dephasing. If the inner channel is independently biased
and partitioned by a QPC0 with transmission $T_{0}$ placed before QPC1 (Fig.
2c), the injected electrons (which are fully reflected by QPC1) act as
effective detectors and dephase the interfering electrons in the outer channel
via a which-path detection [36]. A which-path detection is a determination of
which of the two possible paths the electron takes, with consequent collapse
of the wave function due to loss of indistinguishability of the interfering
paths. The visibility as a function of $T_{0}$ is shown to evolve from a
parabolic dependence at small detector bias $V_{\rm det}$ to a V-shaped
dependence at higher detector bias (Fig. 2d). The presence or absence (with
probability $T_{0}$ and $1-T_{0}$, respectively) of one detector electron
gives an extra phase $\delta\theta\propto V_{\rm det}$ to the interfering
electrons, yielding visibility ${\cal
V}\propto\left|T_{0}+(1-T_{0})e^{i\delta\theta}\right|$ that provides a good
fit to the data (Fig. 2d), suggesting an evolution from Gaussian noise at
small bias to non-Gaussian noise at higher bias [36, 43]. A qualitative change
in the lobe structure of the visibility as a function of $T_{0}$ was predicted
to occur in the presence of strongly non-Gaussian noise, for which a structure
with one lobe at low $T_{0}<1/2$ evolves to a structure with multiple lobes
for $T_{0}>1/2$ [50], yielding a noise-induced phase transition, that was
recently confirmed experimentally [52].

The use of Ohmic contacts embedded in one arm of the MZI has enabled an
alternative characterization of the coherence of the system. When an Ohmic
contact acts as a voltage probe, loss of coherence results from a which-path
detection [45]. At the same time, the charging energy of a micrometer-sized
Ohmic contact may prevent full suppression of coherence for low bias and
temperature [69], as no phase space is left for inelastic scattering, with the
visibility saturating at its maximum value, which has been experimentally
confirmed [70].

The evolution of visibility as a function of magnetic field has emphasized the
appearance of three energy scales, associated with the temperature and bias
dependence of the visibility. These scales are the inverse coherence length,
the distance between the visibility minima between lobes, and the Gaussian
decay scale of the visibility envelope with bias. All three scales change in a
similar way with filling factor [41]. Recent measurements of the evolution of
the visibility from $\nu=3.5$ to $\nu=1$ show a maximum at $\nu=1.5$, a
minimum at $\nu=2.5$, and a reduction to zero when approaching $\nu=1$ (Fig.
3a) [71]. The first minimum in the lobe structure of the visibility (at bias
voltage $V_{0}$) allows us to extract a characteristic velocity $v$ as
$eV_{0}=hv/2L$, that may be associated with a slow dipole mode. A clear
correlation appears between the velocity $v$ and the visibility, with a fading
of oscillations as the filling factor approaches $\nu=1$ [72]. The precise
mechanism responsible for the loss of coherence at filling factor $\nu=1$
still remains an open issue.

Figure 3: Impact of neutral modes on the interference signal. (a) Visibility
of the Aharanov-Bohm oscillations as a function of filling factor $\nu$. Upper
panel, velocity $v$ extracted from the first minimum of the visibility lobe at
bias $V_{0}$ (cf. Fig. 2a) via $eV_{0}=hv/2L$. Lower panel, amplitude of the
visibility at zero-bias. The two quantities show the same behavior with
filling factor $\nu$. Full and open symbols refer to different devices. (b)
Schematics of the Mach-Zehnder interferometer obtained in a Hall-bar in the
Corbino geometry, that features a hole in the center of the mesa. Quantum
point contacts (QPCs) are used to mix the beams and the interfering area is
varied through modulation gates (MGs). Carriers are injected via sources (S’s)
and collected in drains (D’s). Grounded contacts are labeled as G’s. Zoom in
of the part of the devices allowing for independent assessment of the neutral
modes propagating from source $S_{N}$ to drain D1. (c-d) Transmission as a
function of the voltage QPC0 featuring a plateau at $\nu_{\rm QPC}=1/3$ in the
case of bulk filling factor (c) $\nu_{B}=2/3$ and (d) $\nu_{B}=1$. Insets:
shot noise versus applied current showing a quasiparticle charge of (b)
$e^{*}=(2/3)e$ and (c) $e^{*}=e$. Panel a adapted from Ref. [71]. Panels b-d
adapted from Ref. [72].

### Fractional Case

An anyonic MZI for quasiparticles in the FQH regime can be designed in a
similar way to that for the IQH regime, which exploits edge modes in a Corbino
geometry. The expected interference signal depends on the structure of the
edge, and the simplest case is provided by Laughlin fractions $\nu=1/(2m+1)$.
For weakly closed QPCs (Box II), edge channels propagating on opposite sides
of the Hall bar get closer and quasiparticles tunnel from one edge to the
other traversing the Hall bulk. Let us assume that $n_{q}$ quasiparticles are
present in the bulk of the MZI. Each time a quasiparticle tunnels from one
edge to the other at a QPC, the value of $n_{q}$ changes by one. At zero
temperature, the flow is unidirectional and determined by the chemical
potential difference between the two edges. At a given stage, after $j$
quasiparticles have tunneled, the rate for transition from a state with $j$
quasiparticles to a state with $j+1$ quasiparticles depends on $j~{}{\rm
mod}(1/\nu)$. In the limit of weak tunneling at the QPCs, the rate of
quasiparticle tunneling, ${\tau_{j}}^{-1}$, through the MZI can be written as
[73, 74]

$\frac{1}{\tau_{j}}=c_{1}(|\Gamma_{1}|^{2}+|\Gamma_{2}|^{2})+c_{2}[\Gamma^{*}_{1}\Gamma_{2}\exp\left(2\pi
i\nu\phi/\phi_{0}+2\pi i\nu(n_{q}+j)\right)+c.c.],$ (4)

where $\Gamma_{1}$ and $\Gamma_{2}$ are the tunneling amplitudes at the first
and second QPC, and $c_{1}$ and $c_{2}$ are coefficients that depend on
temperature and bias voltage. A current $I=e/\tau$ is obtained when one
electron is collected at the detector, after time
$\tau=\sum_{j=1}^{1/\nu}\tau_{j}$ in which a number $1/\nu$ of quasiparticles
have tunneled [73, 74]. Assuming $\Gamma_{2}\ll\Gamma_{1}$, to first order in
$\Gamma_{2}$, destructive interference takes place in the current among the
$1/\nu$ different rates, and the leading flux-dependent contribution scales as

$I_{\Phi}\sim\Gamma_{2}^{1/\nu}\exp(2\pi i\phi/\phi_{0})+{\rm c.c.},$ (5)

so that the AB oscillations in devices in the Corbino geometry show a
periodicity with $\phi_{0}$, in agreement with the Byers-Yang theorem [75,
76].

In addition, signatures of fractional statistics can be extracted from the
Fano factor, $F=S/(2eI)$. Here, $S$ is the current noise that is sensitive to
charge fluctuations and carries information of the discrete nature of the
quasiparticles involved in the tunneling process. In a non-interferometric
setup (single QPC) the Fano factor is typically used to extract information on
the quasiparticle effective charge. In the MZI, the Fano factor for electrons
for which $\nu=1$, is flux independent despite a flux dependence of current
and noise. For fractional quasiparticles, a flux-dependent Fano factor is
expected [77, 74].

To date, despite tremendous experimental efforts, AB oscillations in a MZI in
Corbino geometry have never been reported for the FQH regime. The principal
reason for that is believed to be dephasing due to the appearance of neutral
upstream modes that may act as a which-path detector and destroy the
indistinguishability of the interfering paths. Neutral modes are predicted to
occur in hole-conjugate states that feature fundamental counter-propagating
edge modes (see Box III). Due to the Coulomb interaction, the confining
potential and disorder, edge reconstruction is predicted to occur, resulting
in downstream charged modes and upstream neutral modes [78, 79]. In the
prototypical example of $\nu=2/3$ (see Box refBox3), a downstream $\nu=1$ mode
is accompanied by an upstream $\nu=1/3$ mode [80, 81, 82]. Analogously to the
discussed $\nu=2$ case, the interchannel Coulomb interaction promotes charge
and (neutral) dipole eigenmodes, the latter propagating upstream with respect
to the direction dictated by the magnetic field [78, 79]. As a consequence,
quasiparticles are no longer fundamental edge excitations, and tunneling at a
QPC excites both charge and neutral modes. In the MZI schematically depicted
in Fig. 3b, a quasiparticle originating from S2 and collected at D2 may either
tunnel at QPC1 and follow the path around the hole or keep propagating along
the mesa edge and tunnel at QPC2. The two events may excite distinct jets of
upstream neutral mode wave packets, which are spatially separated and time
delayed, as it takes the quasiparticle some time to propagate from QPC1 to
QPC2. If the separation is large enough so that the neutral mode wave packets
do not overlap, this excitation of neutral jets breaks the
indistinguishability of the interfering paths and introduces dephasing via
which-path detection. Their overlap decreases with the overall size of the MZI
and can be thus recovered only by reducing the size of the system, the working
temperature, or the bias voltage, posing quite severe restriction in practice
[83].

In addition to the composite nature of edge modes, an unexpected appearance of
a $\nu_{\rm QPC}=1/3$ conductance plateau in the QPCs accompanied by shot
noise has been reported, which suggests the presence of additional neutral
modes [71, 72]. For the device sketched in Fig. 3b, this is shown in Fig. 3c-d
both for the case of bulk integer and fractional filling [72]. A conductance
plateau typically indicates full transmission (or full reflection) of certain
edge modes. Since no partitioning takes place, the transmitted current is
expected to be noiseless. An anomalous “noise on plateau” was identified by
independent assessment to result from excitation of upstream neutral modes at
the QPC.

The reported phenomenology can be set in a more general framework of edge
reconstruction and renormalization among multiple channels. In hole-conjugate
fractions, a more complex structure featuring additional counter-propagating
channels may arise possibly due to a smooth confining potential [84]. Edge
reconstruction and equilibration over long distances wash away the detailed
structure and result in a conductance signal featuring no noise on the QPC
plateaus. When probed on shorter length scales, equilibration may not be
effective, and the complex edge structure may result in noisy conductance
plateaus [85]. In general, currents and noise may be affected by the presence
or absence of symmetries among the edge channels [86].

Possible signatures of AB oscillations in a MZI in the FQH regime have been
reported for the $\nu=1/3$ case [60]. The geometry of this device does not
contain an etched region inside the interference loop. Aharonov-Bohm
oscillations with $\phi^{*}=(e/e^{*})\phi_{0}=\phi_{0}/\nu$ have been reported
and were ascribed to interference of fractional charge quasiparticles.

## Fabry Perot Interferometer

### Geometry and Basic Principles

A complementary geometry is the so-called electronic Fabry-Perot
interferometer (FPI) [87]. Two constrictions are patterned on a Hall-bar to
define two QPCs placed at a distance $d$, which play the electronic analog of
two semi-transparent mirrors (Fig. 4a). On a QH plateau, current carrying
states counter-propagate on the two opposite edges of the Hall-bar (kept at
different chemical potentials) with a quantized two-terminal Hall conductance
$G_{H}=\nu\frac{e^{2}}{h}$. Interedge tunneling is induced by the two QPCs,
which leads to a finite backscattered current (Fig. 4b). Backscattering
amplitudes $t_{1}$, $t_{2}$ at the QPCs, experimentally controlled by two
split-gates, produce interference patterns in the backscattered current
$I\sim|t_{1}+t_{2}e^{i\theta}|^{2}$ in the same way as partial waves interfere
in the famous two slit experiment. Interference fringes depend on three main
parameters: the external magnetic field $B$, the effective area $A$ of the
FPI, defined by the two edges and the two QPCs regions, and the number $n_{q}$
of quasiparticles localized inside it. Quasiparticles backscattered at the
constrictions will move around localized quasiparticles inside the FPI, thus
realizing the braiding gedanken experiment in a quite natural fashion [87, 4].
Interference phase shifts are expected by changing the number $n_{q}$ of
quasiparticles inside the interferometer, constituting a direct hallmark of
anyonic statistics. For small tunneling amplitudes $t_{i}$, the backscattering
current response has an oscillatory behavior, where the phase $\theta$ is a
combination of the phase acquired by a quasiparticle encircling an Aharonov-
Bohm flux, scaled with the effective quasiparticle charge $e^{*}$, and the
anyonic statistical angle $\theta_{\nu}$ [88]:

$\theta=2\pi\frac{e^{*}}{e}\frac{B\cdot A}{\phi_{0}}+n_{q}\theta_{\nu}~{}.$
(6)

For the Laughlin sequence, the statistical angle is predicted to be
$\theta_{\nu}=2\pi/(2m+1)$, with $m$ an integer. Equation (6) holds also for
other FQH states with multiple internal edge structures [89, 90, 91] or more
exotic filling factors like $\nu=5/2$ [5, 4], with the proper statistical
angle $\theta_{\nu}$ associated to the underlying quasiparticles. Abrupt
changes in the phase $\theta$ upon variation of $n_{q}$ inside the
interferometer are thus a consequence of the anyonic statistics. The total
current backscattered by the interferometer will depend on $\cos(\theta)$.
Therefore, the interference phase can be probed by measuring the two-terminal
conductance across the device. In the strong backscattering regime with almost
closed QPCs, the FPI resembles a quantum dot with a well-defined number of
excitations. Here, rather than a sinusoidal behaviour with magnetic field or
interferometer area, one expects resonance lines to appear in correspondence
to degeneracy points of addition energies (Coulomb blockade) [92, 88, 93]. It
is worth stressing that the predicted (and also observed) periodicities do not
depend on transmission amplitudes.

Figure 4: The electronic Fabry-Perot interferometer. (a) Representative SEM
image of a Fabry-Perot interferometer (FPI) where the bright regions are the
metallic gates. (b) Schematic showing an interference path with multiple edge
states in which the outermost mode is fully transmitted ($f_{t}=1$), the
innermost mode is fully backscattered, and the middle mode is partially
transmitted by both quantum point contacts (QPCs); in this configuration only
the middle mode is interfered. (c) Oscillations in diagonal resistance,
$R_{D}$, as a function of magnetic field $B$ for a FPI device. Panel 1 shows
$R_{D}$ as a function of $B$, with well-quantized integer quantum Hall
plateaus. Different colored backgrounds indicate different numbers of fully
occupied Landau levels ($\nu$) through the device. Panels 2 and 3 report
zoom–ins of the data in panel 1, at filling factors 1 and 2, respectively,
showing oscillations in $R_{D}$, and their $B$ periods $\Delta B$. (d) $\delta
R_{D}$, i.e. $R_{D}$ with a smooth background subtracted, as a function of $B$
and modulation–gate voltage $V_{g}$, for a 2.0 $\mu$m2 device. This plot shows
a clear Coulomb-dominated behavior. (e) Same as in (d), but for a 18 $\mu$m2
device. This plot shows a clear Aharonov-Bohm behavior. Panel a adapted from
Ref. [104]. Panel b adapted from Ref. [109]. Panel c, d and e from Ref. [99].

### Implementation and Results

Owing to this physical picture, great experimental efforts have been made to
realize FPIs both in the integer and fractional QH regime [94, 95, 96, 97, 98,
99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 22], in order to
reveal non-trivial anyonic statistics. Early realizations of FPIs, effectively
realizing dots and anti-dots in the fractional quantum Hall phases [111, 112,
113, 114, 115], reported oscillations that turned out to be manifestations of
fractional charge [116, 117, 118, 119]. A typical experimental setup involves
the simultaneous measurement of quantized Hall resistance $R_{H}$ and the
diagonal resistance of the FPI $R_{D}$. The latter is proportional to the two-
terminal Hall conductance and hence to the backscattered current signal, in a
four-terminal configuration, performed at low temperature (of the order of
tens of mK). Interference patterns can be inspected by varying the magnetic
field $B$ or by applying a voltage $V_{g}$ on a plunger (modulation) gate
which effectively shrinks and modifies the area $A(B,V_{g})$ (Fig. 4c).
Unexpectedly, by analyzing the obtained periodicities as a function of these
external knobs, a richer panorama, compared to the naive picture exposed
above, has been observed, even at integer filling $\nu$ where quasiparticles
are electrons with trivial statistical angle $\theta_{\nu}$ [95, 96, 99, 100,
101]. It turned out that Coulomb interactions play a critical role,
determining different operating regimes of FPIs. In a first generation of
devices, two main regimes have been identified, namely a Coulomb-dominated and
an Aharonov-Bohm dominated regime [92, 99, 104, 88].

### Coulomb Blockade limits Visibility

Distinct behaviors have been reported analyzing different geometrical FPI
sizes. A seminal work [94] demonstrated resistance oscillations as a function
of magnetic field in an electronic FPI, opening the way to a large amount of
experiments which showed rich physics even in the (naively thought simple)
case of the IQH regime [95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106,
107, 108].

Indeed, for a small area, of the order of a few $\mu$m2, strong bulk-edge
capacitive coupling characterizes the FPI operation, and a change in the bulk
density determines a change of the area enclosed by the edge. Therefore, in
this Coulomb-dominated regime, changing $n_{q}$ amounts to a continuous
variation of the area of the interference loop and the phase jump becomes an
unobservable integer multiple of $2\pi$ both for integer and fractional QH
[92, 88]. Conversely, in the extreme AB regime, where the bulk and the
interfering edge are not coupled, the area of the interference loop does not
vary with $B$, or by changing $n_{q}$ [104, 107]. Therefore, it is only in
this latter regime that anyonic statistics can be probed [88]. Measurements of
$R_{D}$ revealed different oscillating behaviors as a function of both
magnetic field and side gate $V_{g}$. Looking at color maps of lines of
constant phase in a $B-V_{g}$ plane, the Coulomb-dominated regime is
characterized by a pyjama pattern with lines of positive slope (an increase in
$B$ is accompanied by an increase in $V_{g}$, in order to keep the phase
constant [99, 104, 107] (Fig. 4d).

Similar interference fringes have been also reported considering several
fractional filling factors [96, 120, 99, 100, 105]. Importantly, it has been
recognized that, together with the interfering edge, the number of fully
transmitted channels $f_{t}$, belonging to the lowest Landau levels, comes
into play in determining the periodicity of the FPI oscillations [92, 99, 88].
Devices with front gates deposited on etch trenches allowed a fine and
independent control of the filling factor, the number of edge channels in the
constrictions, and the FPI area with respect to the bulk filling factor [96].
In the Coulomb-dominated regime, magnetic field periodicities turn out to be
$\Delta B=\phi_{0}/(Af_{t})$ (no oscillations when $f_{t}=0$); similarly, gate
voltage periodicity $\Delta V_{g}$ depends solely on $f_{t}$. These
periodicities can be understood assuming that the relevant area of the
interferometer changes as the magnetic field is varied, with a threaded flux
$\phi=B\cdot A\left(B,V_{g}\right)$. Therefore, the phase $\theta$ responds to
small variations in magnetic field or gate voltage as

$\displaystyle d\theta$ $\displaystyle\propto$
$\displaystyle\frac{\partial(A\cdot B)}{\partial B}dB+\frac{\partial(A\cdot
B)}{\partial V_{g}}dV_{g}$ (7) $\displaystyle=$
$\displaystyle\left(A+B\frac{\partial A}{\partial B}\right)dB+B\frac{\partial
A}{\partial V_{g}}dV_{g},$

For the number of (outer) fully transmitted channels $f_{t}=0$, increasing $B$
is followed by a decrease in the area $A$ in such a way that the flux, and
hence the number of occupied states, is kept constant, but the charge density
at the center of the FPI area grows. A spatial imbalance between electrons and
ionized donors takes place. Increasing the magnetic field further must
eventually lead to a relaxation of that imbalance by expulsion of an electron
(creation of a hole) within the interfering Landau level. The area thus
”breathes” with magnetic field [107], and it decreases monotonically and
increases abruptly while keeping the average area constant. The same happens
for $f_{t}>0$, only with a faster decrease with magnetic field related to the
possible transfer of charges from the interfering edge to the lower Landau
levels.

Conversely, larger FPI devices (of the order of $\sim 20$ $\mu$m2) showed an
opposite behaviour with lines of constant phase with a negative slope [99,
104], consistent with the AB regime (Fig. 4e) [107]. Here, reported
periodicities are no longer independent of magnetic field, and in particular
$\Delta V_{g}\propto 1/B$, while $\Delta B\propto 1/A$. Although coherent
Aharonov-Bohm and Coulomb-dominated regimes seem to be of different nature,
the former being interference of independent quasiparticles and the latter
being associated with periodic charging (and discharging) of the FPI with
single electrons, they reflect two sides of the same coin, that can occur on
the same device [88].

A unified picture for both Coulomb-dominated and Aharonov-Bohm regimes based
on energetic considerations has been developed in Ref. [88]. Physically, this
model consists of a capacitance network, taking into account self-capacitance
of the interfering edge, the capacitance of the localized quasiparticles, the
capacitance between the bulk quasiparticles and the edge, and the capacitive
coupling of the gate to the interferometer. By minimizing the total energy of
the system, one can deduce the expected periodicities of the interfering
fringes, recovering the Coulomb-dominated regime with a strong capacitive
bulk-edge coupling and the opposite AB regime, where this mutual capacitance
is negligible.

Figure 5: Coulomb-dominated versus Aharonov-Bohm regime. (a, b) Aharonov-Bohm
interference in the $h/e$ and $h/2e$ regimes measured with a 2 $\mu$m2 Fabry-
Perot interferometer (FPI) with grounded center-Ohmic contact. Conductance
$G=1/R_{D}$ of the FPI versus both magnetic field $B$ and modulation–gate
voltage $V_{g}$ (a) in the $h/e$ regime and (b) in the $h/2e$ regime, measured
at $B=5.2$ T (bulk filling factor $\nu_{B}\sim 2$) and $B=3$ T ($\nu_{B}\sim
3$), respectively. (c) Differential conductance at $\nu_{bulk}=1$ interfering
the $\nu=1$ mode as a function of side-gate voltage $\delta V_{\rm gate}$
(relative to $-1.4$ V) and source-drain voltage $V_{\rm SD}$. (d) Edge-state
velocities extracted from the differential conductance oscillations for
different edge modes for the $N=0$, 1, and 2 Landau levels. Panels a and b
adapted from Ref. [106]. Panels c and d adapted from Ref. [109].

### Screening unwanted Coulomb Interactions

Several technological advances have been made in order to suppress
electron–electron interactions and bulk-edge coupling, achieving the AB regime
in a controlled fashion, even in rather small FPIs. A central top gate
covering the whole interferometer has been employed to reduce the charging
energy [99]; alternatively, low temperature illumination has been used to
enhance screening by the doping layer [100]. A more effective screening has
been obtained using a grounded Ohmic contact of the size of $\sim 0.5$ $\mu$m2
inside the incompressible bulk of the FPI [104, 106, 107]. Placing such an
Ohmic contact in the vicinity of the FPI, instead of placing it within the FPI
center, results in slightly less effective screening, allowing to inspect also
the intermediate regime between AB and Coulomb. This leads to the observation
of a lattice structure (checkerboard pattern) in the $B-V_{g}$ plane with more
than one characteristic frequency [107]. However, these methods lower the
charging energy of the FPI as a whole, suppressing Coulomb-dominated features,
without much effect on the interchannel capacitance. Residual Coulomb
interactions between different edge channels can give rise to interesting
behavior even for devices operating at IQH filling in the AB regime.

Measurements in the AB regime for FPIs of different sizes ($\sim 2-12$
$\mu$m2) with a center Ohmic contact have been reported, for integer filling
in the range $\nu=2$ to $\nu=4$ [106, 108]. Monitoring the flux dependence of
the conductance at $\nu>2$, an unexpected halving of the oscillation periods
both in $B$ field and $V_{g}$, without any sign of the fundamental periodicity
$h/e$ was found (Fig. 5a-b) [106]. To explain this unexpected doubling of AB
frequencies, different scenarios have been tested. For instance, the
possibility that only an even number of interfering windings can occur, has
been excluded by comparing the $h/(2e)$ observed visibility and the one
estimated based on the reported dephasing length [106]. The doubling of
frequence has been attributed to strong interedge coupling, with an effective
electron pairing among them that is consistent with quantum shot noise
measurements reporting an effective charge $e^{*}=2e$ [106, 108]. These
intriguing results, reminiscent of effective attractive (Cooper-like) pairing
among electrons, have been confirmed in subsequent experiments [108, 109] and
explained [121, 108, 122, 123] in terms of mutual edge capacitances and the
emergence of neutral edge modes whose exchange can mediate such
counterintuitive effective attractive interactions.

Figure 6: Detection of anyonic statistics. (a) Conduction band edge (red) and
electron density (blue) versus growth direction (z–axis) calculated using a
self–consistent Schrödinger-Poisson method. The sheet density in each well is
indicated. (b) Conductance oscillations $\delta G$ versus magnetic field $B$
and variation in side gate voltage $\delta V_{g}$. The predominant behavior
shows negatively sloped Aharonov-Bhom interference fringes, with a small
number of discrete phase jumps (highlighted by dashed lines). Least-squares
fits of $\delta G$ are shown with highlighted stripes, and the extracted
changes in phase $\frac{\Delta\theta}{2\pi}$ are indicated for each discrete
jump. The latter are directly related to the statistical angle of
quasiparticles. Panel a adapted from Ref. [109]. Panel b adapted from Ref.
[22].

By applying a finite source-drain voltage $V_{SD}$ [103, 109], electrons with
different energies (and thus different momentum) are injected in the
interferometer, inducing an additional phase shift,

$\delta\theta=\delta\epsilon\frac{\partial
k}{\partial\epsilon}L=\frac{\delta\epsilon L}{v_{edge}},$ (8)

with $L$ the FPI path length and $v_{edge}$ the edge mode velocity. With this
non-equilibrium configuration, it is possible to infer the speed of edge
states by looking at the conductance oscillations as a function of both
$V_{SD}$ and gate voltage $V_{g}$. Such measurements show a characteristic
checkerboard lattice with nodes in the interference pattern (Fig. 5c)
occurring at $V_{SD}=\pm{\pi v_{edge}}/{eL}$, from which one can extract
$v_{edge}={eLV_{SD}}/{2\pi}$. Measurements of edge velocity for different
Landau levels were reported as a function of filling factor $\nu$, (Fig. 5d)
[109]. The inner, higher index Landau levels were found to generally have
lower velocity and correspondingly lower coherence. The general trend observed
is an increasing edge velocity with decreasing magnetic field (larger filling
factor), consistent with the simple picture that edge currents are generated
by Hall drift. These observations recall previous work [61, 62] that
demonstrated that the confining potential is steepest at the outer edge,
resulting in a higher electric field and thus higher velocity for the outer
edge mode. For higher filling $\nu>2$, these larger velocities are also
consistent with the interedge coupling picture, where interacting edge modes
are expected to have different velocities [93, 121, 123], supporting the idea
behind the halving of periodicities described above.

These recent results have been obtained in a novel FPI geometry where the
sample contains three GaAs wells: a primary quantum well $30$ nm-wide and two
additional $13$ nm-thick wells, located on either side of the primary well and
separated by thin insulating AlGaAs spacers (Fig. 6a). The central well
contains the 2D electron gas while the two side wells act as screening layers.
These fabrication advances allowed the study of FPIs in the extreme AB regime
for small interferometer areas of the order of $\sim 1$ $\mu$m2. Structures
utilizing screening layers resulted in charging energies, and hence mutual
bulk edge coupling, lower by more than one order of magnitude with respect to
devices of the same size with conventional gating schemes [124, 125, 109, 22].
Remarkably, these new devices allowed to observe, for the first time,
oscillations with negatively sloped lines of constant phase in the $B-V_{g}$
plane, hallmark of the AB operating regime, also in the FQH regime. Indeed, by
operating the device at high magnetic field, at the center of the $\nu=1/3$ QH
plateau, the conductance variation $\delta G$ of the FPI could be studied as a
function of $B$ and variation in gate voltage $\delta V_{g}$. An overall
behavior with negatively sloped stripes is clearly visible with a number of
discrete phase jumps (Fig. 6b) [22]. Although not regularly spaced, these
jumps have nearly the same slope in the $B-V_{g}$ plane and have been verified
as reproducible in the same study [22]. Impressively, these findings are fully
consistent with Equation (6): the second term on the right hand side predicts
a discrete change in phase when the number of localized quasiparticles
changes. These discrete phase jumps are directly reflected in the behaviour of
the measured conductance and constitute the first direct observation of the
effect of the anyonic phase contribution $\theta_{\nu}$. Moreover, the
observation of a positive slope strongly suggests that these discrete phase
jumps are associated with changes in localized quasiparticle number:
increasing $B$ is expected to remove quasiparticles from the bulk [87, 4],
while increasing gate voltage would make it electrostatically favorable to
increase the number of localized quasiparticles. The authors of Ref. [22]
performed a least-squares fit in the regions between the phase jumps, fitting
the conductance data to the form $\delta G=\delta
G_{0}\cos\left(2\pi\frac{1}{3}\frac{B\cdot A}{\phi_{0}}+\theta_{0}\right)$, to
determine the value of the change in phase associated with each phase jump.
Analyzing $\theta_{0}$ across adjacent regions, and under the assumption that
each phase jump corresponds to the removal of a localized quasiparticle
($\delta n_{q}=1$), they have extracted the value
$\theta_{\nu}=2\pi\times\left(0.31\pm 0.04\right)$, which well agrees with the
predicted value of $1/3$ for the Laughlin state at $\nu=1/3$ [4, 88]. To
further corroborate their findings, additional measurements varying magnetic
field $B$ in a wider range have been performed. A variation of the $3\phi_{0}$
superperiodicity in $\Delta B$ was reported while moving away from the center
of the $\nu=1/3$ QH plateau, consistent with the recent prediction of Ref.
[126]. These impressive measurements provide experimental confirmation for the
prediction of fractional braiding statistics at the $\nu=1/3$ quantum Hall
state, opening the way for the study of more complex FQH states with composite
edge physics beyond the Laughlin series.

## Towards non-Abelian Statistics

The described recent advances inevitably fuel and revitalize the search for
signatures of non-Abelian anyons [5, 4] in more exotic filling factors such as
$\nu=5/2$. Indeed, when the ground state of a many-body system is degenerate,
the effect of a closed trajectory is not necessarily a mere phase factor. The
system can start and end in different ground states belonging to the same
degenerate subspace. Such adiabatic processes may be expressed in terms of a
generic unitary transformations within this subspace. Therefore, the
topological phase determined by Eq. (2) is generalized to a matrix rather than
a simple phase factor. Braiding quasiparticles within such degenerate subspace
allows to build unitary operations which are topologically stable and immune
to external perturbations and decoherence [5]. Non-Abelian exchange statistics
have been predicted for quasiparticles of FQH states such as $\nu=5/2$ or more
exotic (and fragile) ones [127, 128, 129, 130, 5]. Indeed, the most promising
candidates to describe the even-denominator FQH state at $\nu=5/2$ are the
Pfaffian (or Moore-Read) [131, 132] or its conjugate [133, 134, 4, 135, 136,
137], which would host non-Abelian anyons. Such quasiparticles have been
predicted to possess $e^{*}=e/4$ fractional charge, and some indications in
this direction have been reported by shot noise transport measurements [133,
136]. Several models have been put forward [131, 138, 139, 134, 135, 140] for
the $\nu=5/2$ state, including some without non-Abelian excitations, and from
such measurements it is not possible to conclusively determine the non-Abelian
nature of this FQH state. It is worth mentioning that measurements seem to
confirm the non-Abelian nature of the $\nu=5/2$ FQH state [141, 142].

Interferometric setups like Fabry-Perot configurations [100, 143] have been
realized at filling $\nu=5/2$. The reported results, mostly in the Coulomb-
dominated regime, have been subject of a long debate [138, 100, 140, 88, 144,
143]. Several interpretations of the reported observations have been proposed,
but a conclusive answer towards a unique model unequivocally supporting non-
Abelian anyons is still lacking. We believe that recent technological
advances, such as the screening layer mechanism [109, 22] and the anyon
collider platform [8, 9, 10, 11, 12, 23], would be beneficial for the
inspection of $\nu=5/2$ or other exotic filling factors, allowing to clearly
enter the AB regime, where evidence of statistical angle should be unambiguous
and could constitute a smoking gun for such longstanding detection of non-
Abelian quasiparticles and their ensuing manipulation.

## Discussion

The two interferometric setups described in the previous sections clearly
share differences and similarities, complementarities and incompatibilities,
pros and cons. Due to its open geometry, one of the advantages of the MZI
scheme is the absence of Coulomb blockade physics that strongly influences the
FPI. The success of the FPI in demonstrating AB oscillations and clear
signatures of the fractional statistics in no way takes away importance from
the studies performed on MZI. Moreover, preliminary results on setups that
avoid the subtleties of the Corbino geometry [60] suggest possible alternative
routes for the design and construction of a MZI able to show signatures of
fractional statistics. Further investigations may reveal the key to a MZI
working in the FQH regime and allow full exploitation of a setup that is free
from Coulomb blockade effects, whose control have been proven crucial for the
success of the FPI.

Very recently, novel heterostructures based on high-quality graphene devices
have been demonstrated to be an interesting and alternative platform for
electronic interferometry [65, 145, 146, 66]. By properly patterning several
split-gates, it has been possible to realize FPIs on graphene devices free of
charging effects, observing clear evidence of AB oscillations in the integer
QH regime. These promising results, and the reported long coherence length,
will allow for a new generation of electronic interferometers with 2D
materials, in which several FPIs can be easily joined, in order to realize
possible braiding schemes both at integer and fractional filling factors.

Finally, remarkable results have been obtained exploiting time-dependent
signals. An entirely complementary field of electron quantum optics [8, 147,
10, 11, 12] has been developed in parallel to the standard interferometry.
These platforms allow to exploit two-particle interference like the Hanbury-
Brown-Twiss [148, 149, 150, 151, 152, 153, 154] and Hong-Ou-Mandel [155, 156]
schemes, that are directly sensitive to two-particle statistics and paved the
way to the recent success of the anyon collider experiments [9, 23].
Furthermore, anyon colliders and interferometric setups could be merged and
joined together, offering the opportunity to exploit complementarities and
reciprocal advantages, and develop novel platforms for topological quantum
computing.

## Acknowledgements

This activity was partially supported by the SUPERTOP project, QUANTERA ERA-
NET Cofound in Quantum Technologies, and by the FET-OPEN project AndQC. L. C.
acknowledges funding by the EU Marie Curie Global fellowship TOPOCIRCUS-841894
- Simulations of Topological Phases in Superconducting Circuits.

##  I BOX: Optical interferometers

The Mach-Zehnder interferometer (MZI) is a two-path interferometric setup
originally aimed at determining a relative phase shift between two collimated
beams originating from a single source (panel a). The device is named after
Ludwig Mach and Ludwig Zehnder, who proposed the interferometer in 1891, and
it has many applications, from aerodynamics and plasma physics to
gravitational wave detection. An incoming beam of photons from source S is
split at the first beam-splitter BS1, and the two outgoing beams follow
different paths via mirrors M1 and M2. The beams rejoin at BS2, and
interference as a function of their relative phase $\theta$ acquired along the
two different paths is observed at the detectors D1 and D2. Photons are
generated with a precise momentum $k$, and their relative phase is due to the
dynamical contribution $\theta_{L}=k\Delta L$ that is accumulated due to
propagation along paths of different length $\Delta L$.

The Fabry-Perot interferometer (FPI) is a resonator originally constituted by
two partially reflecting optical flats placed one in front of the other, with
the reflecting surfaces facing each other (panel b). It is named after Charles
Fabry and Alfred Perot, who developed the instrument in 1899. The optical
flats act as beam-splitters. Beam-splitters are rotated by $\pi/2$ with
respect to the MZI. A photon incoming from source S impinges on the first
beam-splitter BS1, where it can be either reflected (and reach detector D1) or
transmitted. In the second case, it propagates for a given distance $L$ that
separates the first beam-splitter from the second, BS2, where it can be either
transmitted and reach detector D2, or reflected back to BS1. This sequence can
repeat an arbitrary number of times, and the outgoing signals at D1 or D2 are
the result of the coherent sum of all amplitudes for multiple bouncing between
the two beam-splitters. A sharp resonance in the transmitted signal at D2
occurs when the light exhibits constructive interference after one round trip,
at photon wavelength matching twice the distance between the mirrors,
$\lambda=2L$. FPIs are widely used as resonators, spectrometers, optical
filters, and interferometers in optics, telecommunications, astronomy, and
gravitational wave detection.

##  II BOX: Quantum Point Contacts

Starting from a 2DEG, a further restriction of carrier motion to only one
dimension can be obtained by using a split-gate geometry, where a small
channel region is defined by two gates, separated by a few hundreds of
nanometers. By applying a negative gate voltage, these gates tailor the width
of the channel by electrostatically depleting the 2DEG under and next to them.
This leaves only a small opening in the 2DEG for the electrons to pass
through, as sketched in the figure, where the part of the 2DEG depleted by the
gates is indicated in gray. If a large negative gate voltage is applied, the
gates can even completely close the channel (pinch-off). Such device is called
a quantum point contact (QPC).

When the Fermi wavelength $\lambda_{F}$ is comparable to the width of the QPC,
the transverse momentum becomes quantized, forming a series of parabolic one-
dimensional subbands [157, 158, 159, 160, 161, 162]. In a 2DEG in the QH
regime, current is carried by chiral edge channels, which at the two opposite
sides of the Hall-bar propagate in opposite directions. If a QPC is inserted
in the device, by squeezing such constriction, the counter-propagating edge
channels are forced to get closer. If their distance becomes comparable to the
magnetic length, they will start to interact and (partially) be backscattered.
The first backscattered edge states are the ones formed by the highest Landau
levels, because they travel farther from the edges. After, edge states with
lower Landau index are backscattered, until all edge channels are
backscattered (pinch-off). The figure illustrates this for $\nu=2$ in the
bulk. Far from the constriction, the two edge channels propagate on either
side of the Hall-bar, but the inner one (red) is backscattered by the QPC, so
only one (the outer) channel passes through it (blue). Therefore the filling
factor in the constriction is $\nu_{\text{QPC}}=1$.

##  III BOX: Counter-propagating edge states and neutral modes

The complexity of edge reconstruction in the FQH effect can be visualized by
considering the prototypical hole-like $\nu=2/3$ state. The latter has been
described as the formation of a $\nu=1/3$ Hall droplet by depletion of a
$\nu=1$ state in the bulk [80, 81, 82]. Edge excitations are described by the
coexistence of a $\nu=1$ forward propagating (downstream) and a $\nu=1/3$
backward (counter-propagating or upstream) edge mode (panel a). However, this
picture results in a value of the Hall conductance
$G_{H}=\left(1+\frac{1}{3}\right)e^{2}/h$ that is different from the
experimentally observed universal value of $\frac{2}{3}\,e^{2}/h$.
Furthermore, the presence of an upstream charge flow is at odds with the
absence of any experimental signature in this sense. The key observation is
that counter-propagating charge modes, besides Coulomb interactions, are
sensitive to the presence of disorder (panel b). Interchannel interactions can
be considered in a dipole scheme by introducing ”charge” ($c$) and ”neutral”
($n$) modes (panel c). However, the resulting conductance still has a
nonuniversal value

$G_{H}=\Delta\frac{2}{3}\frac{e^{2}}{h},$

where $\Delta\geq 1$ depends on the velocities of the modes and the Coulomb
interaction strength.

It has been argued that a correct inclusion of disorder-induced scattering
between the two edge modes is necessary to reconcile with the proper quantized
value of $G_{H}$ [78, 79]. The crucial point is that the counter-propagating
nature of the two charge modes allows for backscattering mediated by disorder,
which is a relevant interaction between the two channels. Physically,
subtraction of one electron from the forward channel is accompanied by the
addition of 3 quasiparticles to the backward channel, so that tunneling
between the two modes effectively involves only the neutral mode, as
emphasized in panel c. The residual interaction between charge and neutral
mode can be shown to be irrelevant in the renormalization group sense, leading
to the fixed point $\Delta\to 1$, eventually corresponding to the
experimentally observed conductance $G_{H}=\frac{2}{3}\,e^{2}/h$ [78, 163].

Notably, the $\nu=2/3$ edge state has recently been experimentally synthesized
by realizing a junction between $\nu=1$ and $\nu=1/3$ Hall droplets, bringing
the counter-propagating associated edge states into close proximity. The
experiment demonstrated that equilibration between the two states results in a
$G_{H}=\frac{2}{3}\,e^{2}/h$ Hall conductance, whereas in absence of
scattering a value of $G_{H}=\frac{4}{3}\,e^{2}/h$ is observed [164]. At the
same time, we note that the conditions needed to approach the low energy fixed
point may be demanding, and a robust universal value of 2/3 is alternatively
predicted in presence of edge equilibration involving also decoherence [165,
166].

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