Localizing individual exciton on a quantum Hall antidot

Domain: physics
Published: 2025-11-24
Editorial outcome: accept

Abstract
Excitons are bond states of electron-hole pairs formed through Coulomb interaction. While excitonic phases have been widely studied in semiconductors and quantum Hall double-layers, prior works largely focus on bulk systems with large number of excitons, limiting their applications in quantum devices. Here, employing the approach of quantum Hall antidot with two spatially separated edge channels, we demonstrate a type of quantum Hall quasiparticle exciton which represents a quantum-coherent bound state of an electron and a hole situated on their corresponding edges coupled through intralayer tunneling and Coulomb interaction. This approach allows localization and electrical tuning of individual quantum Hall excitons. Quantum-coherent dynamics of exciton are observed in the gate-dependence of antidot conductance peaks near the electron-hole resonance, which signifies a quantum superposition of vacuum- and electron-hole pairing states. Modeling the electron-hole pair as a coupled two-level system, semi-quantitative understanding of experimental observations is achieved. This work opens avenues for creating quantum systems of multiple quantum Hall quasiparticles.

Introduction
Solid-state systems have been a major area for the development of quantum devices and circuits owing to their advantages in scalability, electrical control, and constant progress in size reduction down to the nanoscale level. This combination of characteristics allows manipulation of individual quasiparticles associated with the states of charge 1–8, spin 9–14, magnetic flux and superconducting phase 15–21, or more complex combined degrees of freedom. In this context, a two-dimensional (2D) electron quantum Hall (QH) system has been a unique platform that, in addition to the basic integer charge states, hosts the more exotic quasiparticles associated with fractional QH (FQH) abelian and non-abelian anyonic states 22–26, and excitons in double-layer QH systems 27–32 which exhibit Bose–Einstein condensation (see, e.g., ref. 33) and unusual quantum solids 34. Evidence of the anyonic exchange statistics was obtained in the edge modes-based Fabry-Perot QH interferometer 35–39 and in the anyon collisions 40.

Another experimental platform that utilizes QH edge modes is QH antidot 41–43 in which the edge modes encircle the maximum of an electrostatic potential, forming closed orbits. Quantized energy levels obtained in this way can localize individual QH quasiparticles, including those with fractional charge and statistics, and provide the basis for their manipulations. Coulomb-blockade-type correlations among the quasiparticles on the antidot, combined with the discrete antidot energy spectrum, control the number of quasiparticles and minimize the phase space for scattering, hence enhancing quantum coherence. Resonant tunneling through the antidot energy levels has successfully demonstrated the fractional charges of FQH quasiparticles 44–46. The fractional exchange statistics of the anyonic quasiparticles are also theorized to be observable through transport measurement in multi-antidot molecule systems 47. Beyond single antidot atoms, however, while there has been experimental evidence of the double-antidot transport in accidental 48 and defined 42 structures, creating controllable and quantum-coherent multiple QH quasiparticle systems remains limited and challenging.

One of the archetypal interacting multi-particle quantum systems is an exciton, a bound state of an electron-hole pair formed through Coulomb attraction. In QH double-layers, interlayer Coulomb interactions can induce excitonic states from the spatially separated electrons and holes on their corresponding layers, forming a bosonic condensate 28–32. QH quasiparticle-associated excitons offer the potential of creating quantum states combining interaction effects and non-trivial quantum statistics. However, to date, the study on QH excitons has been largely limited to the bulk systems with a large number of excitons.

In this work, employing the platform of a quantum Hall antidot with two spatially separated electron- and hole-edge channels, we demonstrate a coherent and interacting quantum state of an individual localized exciton. Quantum-coherent dynamics of excitons are observed in the gate dependence of antidot conductance peaks near the electron-hole resonance, which signifies a quantum superposition of vacuum- and electron-hole pairing states. Modeling the electron-hole pair as a coupled two-level system, semi-quantitative agreement with the experimental observations is achieved.

Results
Figure 1a illustrates the basic principle of such a QH antidot exciton in comparison with a conventional exciton. A conventional exciton forms in the free space of a semiconductor by exciting a valence electron. The resulting hole in the valence band and the excited electron form an excitonic pair through Coulomb interaction, with energy lower than the conduction band edge. A QH antidot exciton, on the other hand, forms in a pair of closely placed QH antidot edge modes hosting an electron and a hole. The quantum tunneling of an electron from the hole-edge mode to the electron edge mode creates a coherent electron-hole pair, which, through Coulomb interaction, forms an excitonic state with reduced ground state energy. In contrast to the interlayer QH excitons (Fig. 1b), a QH antidot exciton is localized individually and created via intralayer Coulomb interaction and tunneling.

While Coulomb-blockade transport of individual excitons has been studied in quantum dots 49–51 as one of the approaches to the development of “on-demand” single-photon sources, and continue to attract practical interest up to now (see, e.g., ref. 52 and references therein), the distinct element added by our antidot structure to the manipulation of the individual excitons is quantum coherence between the electron and the hole parts of the exciton. Such quantum coherence within the individual excitons is potentially useful for quantum information applications, e.g., as an interface for quantum information transfer between the photons and the antidot qubits. From the point of view of qubits, electron-hole attraction within the exciton enhances the energy gap between the qubit states beyond the typically not-so-large tunnel coupling of these states, something that would be beneficial in many different contexts, most directly, in the case of the ground state quantum computing 53.

The experimental platform studied in this work is a graphene-hexagonal boron nitride (hBN) heterostructure illustrated in Fig. 1c, with hBN encapsulated monolayer graphene tunable by two graphite gates (see Supplementary Information (SI)). The bottom graphite gate tunes the carrier density uniformly throughout the whole monolayer graphene channel. The top graphite gate is patterned with an antidot and two couplers etched out. The diameter of the antidot here is D ~ 180 nm. The couplers point toward the antidot with an edge-edge distance of ~80 nm at the closest points. Combining the top and bottom gates, carrier density and QH filling factor can be separately adjusted inside the antidot and coupler region, ν = nh/eB, and outside, ν0 = n0h/eB. Here n and n0 are the corresponding local carrier densities, h is Planck’s constant, e is the electron charge, and B is the external magnetic field. When the corresponding numbers of the filled Landau levels (f and f0, positive on the electron side and negative on the hole side) are different between the two regions, QH edge modes form the boundaries of the antidot and the couplers. As a result of quantum confinement (particle in a ring), the energy of the QH edge modes encircling the antidot becomes quantized:

εj = 2ℏvQH/DQH * j + Φ/Φ0

with energy level spacing: Δε = 2ℏvQH/DQH in a fixed magnetic field 43,46 (see SI). Here, j is an integer which indexes the quantization energy levels, vQH is the QH edge mode velocity, DQH is the effective diameter of the antidot edge mode, Φ is the magnetic flux through the antidot, and Φ0 = h/e is the magnetic flux quantum. The occupation of the energy levels can be tuned through the gate voltage, while a magnetic flux shifts the manifold of the levels, manifesting the Aharonov–Bohm effect. When the distance between the couplers’ and the antidot’s edge modes is sufficiently small, coupling between the bulk edge modes through the antidot takes place and lowers the transverse resistance RT = V+/I- (I being the electric current flowing through the I+ and I- electrodes) which can be characterized through the transport measurement in the configuration shown in Fig. 1d. In the regime when the bulk graphene is tuned to a QH plateau with quantized resistance RQH, the antidot tunneling conductance can be measured from the corresponding resistance valley: GAD = 1/RT - 1/RQH.

The basic signature of QH antidot tunneling is illustrated in Fig. 2a under the simplest configuration where the antidot and the couplers are within the f = 0 insulating gap, while the outside bulk is on the f0 = ±2 QH plateau. Here, consistent with the prior studies on physically etched QH antidots 43,46, the emergence of quasi-periodic resistance valleys on a QH plateau, associated with tunneling through discrete antidot levels, is observed in our gate-defined QH antidot device. Each resistance valley corresponds to adding a single hole charge onto the antidot 43. Figure 2b goes beyond the linear conductance and demonstrates the top gate voltage and bias voltage dependencies of resistance (charge stability diagram). The dotted lines highlight the discernible Coulomb diamonds and excited states. We note that compared to typical quantum dots, the Coulomb diamond features are less pronounced in the QH antidot. This may be due to (1) break-down of the quantized plateau resistance under finite bias voltages which adds a bias-dependent background; (2) small Coulomb energy due to the relatively large antidot size and screening from the hBN dielectric layers; (3) asymmetric tunneling amplitude between the antidot and the two coupler, which results in nearly invisible resonant tunneling traces at the positive-sloped boundaries of the Coulomb diamonds. From this diagram, the on-site Coulomb repulsion energy and quantization energy spacing on a f0 = ±2 QH plateau in a 1T magnetic field can be estimated to be U ~ 1 meV and Δε ~ 0.7 meV, respectively. More examples of charge stability diagrams and Coulomb diamonds showing a similar energy scale can be found in the SI. Based on the quantization energy, the QH edge mode velocity of vQH = Δε * DQH / 2ℏ ~ 10^5 m/s can be estimated, which is in qualitative agreement with the previous reports on edge mode velocity in QH interferometers 54,55. The measured on-site Coulomb energy of one edge state can be understood considering a localized ring of charge density with thickness on the order of the magnetic length lB = √ℏ/eB, and partially screened by the nearby gate electrodes: U = 2πϵ0ϵBN * e^2 / (ϵBN * L) * ln(1 + 4t/lB) * (1 + 2t/lB)^-1/2 (see SI). Here, ϵBN ~ 3.5 ± 0.2 is the dielectric constant of hBN, and t ~ 10 nm is the effective hBN thickness for the screening effect from both graphite gates. The observed Coulomb energy of U ~ 1 meV suggests an antidot mode circumference of L ~ 600 nm, i.e., D ~ 190 ± 10 nm. This is consistent with the similar but slightly smaller diameter of the antidot in the top graphite gate (D ~ 180 nm).

The gate-control of the antidot tunneling is illustrated in Fig. 2c. The main panel shows the tunneling resistance versus the filling inside (ν) and outside (ν0) the antidot/couplers regions. Sharp bright dots (resistance valleys) signify the antidot tunneling on the QH plateau of RQH = h/2e^2 (f0 = ±2), coarsely sampled over a wider range of filling factor ν in the antidot/couplers regions that are tuned by the back gate. Four characteristic (f, f0) combinations are highlighted with their corresponding Landau level energy diagrams. The spatial profiles of the Landau level energy (in the presence of the back gate potential) are shifted by the chemical potential throughout the whole channel. The QH edge modes are located where the Landau levels cross the chemical potential μ. The edge mode tunneling strength depends sensitively on the spatial separation between the coupler and antidot edge modes, adjusted by the dual gates. To enable coupler-antidot tunneling, the tunneling distance between the coupler edge mode and the antidot edge mode needs to be near its minimum when the Landau Level associated with these edge modes is tuned slightly across the chemical potential. Directly relevant to the discussions below and for f, f0 ≤ 0, this corresponds to the lower carrier density end of a QH plateau. A more detailed discussion of the edge mode tunneling in these four regimes is given in the SI.

Having demonstrated the basic principle of QH antidot operation, we next focus on the symmetry-breaking f0 = ±1 state in a magnetic field B = 3.3 T. Figure 3a shows the top gate voltage dependence of the transverse resistance RT, where quasi-periodic resistance valleys associated with coupler-antidot tunneling of the hole side of the zeroth Landau level edge modes emerge on the plateau background of RQH = h/e^2. Following Eq. (1), the gate voltage positions of these valleys shift with changing magnetic field at a rate of one resistance valley per magnetic flux quantum Φ0 through the effective area of the antidot: ΔVTG = ΔΦ0 * (4π * e * DQH^2) / (cTG * Φ0) = 4π * e * DQH^2 / (cTG * Φ0) * ΔB/B. Here, ΔVTG is the gate voltage period of the resistance valleys, ΔB and ΔΦ are the changes of magnetic field and flux, respectively. From the magnetic shift of the resistance valleys, DQH ~ 230 nm can be estimated, suggesting the edge mode to situate ~25 nm away from the physical antidot edge of the top graphite gate. With this effective diameter, we can also estimate the number of charges added to the antidot per gate period: ΔN = cTG * π * DQH^2 / (4 * e * ΔVTG) = 1.04 ± 0.06 (cTG = ϵ0 * ϵBN / dBN is the area capacitance of the top gate with dBN = 18 nm being the thickness of the hBN layer between the graphene can the top graphite gate), consistent with the addition of single fundamental charges in the integer QH regime 46. All these observations further confirm the antidot nature of the resistance valleys.

The main finding of this work is shown in Fig. 3b, which illustrates the evolution of the antidot tunneling conductance oscillations on the lower density side of the f0 = ±1 plateau over a range of filling factor ν in the antidot/coupler regions. For clarity, the horizontal axis is set to VTG + α * VBG, which directly depicts the electrostatic doping at the outside vicinity of the antidot, where α = 0.53 is the effective capacitance ratio between the bottom and the top gates for antidot charging (see SI). The antidot tunneling conductance peaks appear as near-vertical stripes with abrupt jumps. Most interestingly, anti-crossing-like features are observed, highlighted by the black dotted curves in Fig. 3b. These features are generally present under the f, f0 = (1, ±1) configuration, although the regularity of their locations is disturbed by a few random slides occurring along the yellow dotted straight lines in Fig. 3b. The source of these random slides is identified as quantum dot-like charge trapping centers located near the antidot edge on the top graphite gate (see SI), which is not the focus of this work. In the discussions below, we focus on the anti-crossing feature highlighted by the yellow box in Fig. 3b, for which the individual curves of the antidot conductance are shown in Fig. 3c. As will be argued below, after developing a quantitative model, the anti-crossing-like features observed here originate not from atomic-scale traps, but from the formation of a gate-defined pair of electron-hole edges coupled both by tunneling amplitude and Coulomb attraction.

Anti-crossing features are typically associated with level coupling. In high-quality graphene samples, symmetry-breaking interaction effects lift the 4-fold degeneracy of the zeroth Landau level. With a well-developed f0 = ±1 plateau and insulating behavior at the charge neutrality, the inner electron and the outer hole-edge modes are expected to coexist and be spatially separated by a gapped region. Increasing the magnitude of the top gate voltage, the antidot-defining potential profile becomes increasingly sharp, resulting in antidot edge modes with larger edge mode velocities (vQH ∝ dV), hence larger quantization energy level spacing (Δε = 2ℏvQH/DQH). As a result, both the outer hole- and the inner electron-antidot edge modes have their quantization energy levels fan out with increasing top gate voltage. Since the manifolds of hole- and electron-quantization energy levels are offset by the charge neutrality gap, the energy levels of the outer- and inner- modes with different indices will have different shifting rates under changing top gate voltage, and will cross each other as illustrated by Fig. 3d. As a result, just from the top gate voltage, some of the hole- and electron levels can be tuned to align with and cross each other, creating tuning and detuning of an electron-hole resonance. With the back gate tuning the chemical potential throughout the channel, one can ensure tunneling through the antidot under the conditions of the electron-hole resonance. A more analytical discussion of the gate voltage dependence of the resonance can be found in the SI.

To understand the energy spectrum of the coupled electron and hole-edge modes, we consider a basic model of graphene QH antidot 43 extended to two coupled co-centric edge modes, shown in Fig. 4a. Here, Γ is the tunneling rate between the couplers and the outer hole edge mode. The energy levels of the two edge modes are quantized, and therefore, in the main approximation, the tunnel coupling between the two modes is coherent. Besides tunneling, electrons and holes also experience Coulomb interaction, which can be described in the constant-interaction approximation for each individual mode: U(nh, ne) = 1/2 * Uh * nh(nh - 1) + 1/2 * Ue * ne(ne - 1) - u * nh * ne (2). Here, nh and ne are the numbers of particles in the hole and electron mode, respectively, U are the corresponding interaction constants for the two modes, and u > 0 is the magnitude of the Coulomb electron-hole attraction between the two edges. Similar to the basic Coulomb-blockade model for tunneling through the quantum dots, Eq. (2) leads to a periodic response of the edges to the two gate voltages that control nh and ne. For the outer edge, this translates into a quasi-periodic sequence of the resonant tunneling peaks of the antidot conductance GAD = dI/dVbias | Vbias=0, with each peak occurring at the bias conditions at which the number of holes in the edge changes, nh -> nh + 1. On the other hand, the relations among the parameters in Eq. (2): nearly equal repulsion energies Uh and Ue, and a strong attraction constant u close to U, combined with the choice of the non-vanishing tunneling terms, make this model very specific to the system we consider: concentric electron-hole edges encircling one antidot. In this case, the inner edge does not affect the conductance directly, but the number ne of electrons in it also changes periodically by 1 as a sequence of resonances.

To describe the antidot transport properties in the vicinity of one combined electron-hole resonance (where both the electron and hole antidot energy levels align with the chemical potential), we need to consider four charge states, each labeled by nh and ne: |nh, ne>. On the basis |00>, |11>, |01>, |10> of these four states, the Hamiltonian of the antidot is: H = [0, Δ*, 0, t+; Δ, ξ, t, 0; 0, t+, -δe, 0; t, 0, 0, δh] (3). Here, δh is the gate-induced energy deviation of the outer edge state from the resonance point, δe is the same for the inner edge, ξ = δh - δe - u is the energy of the electron-hole pair occupying the inner edge state. We include t, t+ terms to indicate how the states of the electron-hole system are coupled to the external charge reservoirs and produce the incoherent tunneling rates Γ illustrated in Fig. 4a. However, under the limit of weak antidot-reservoir coupling, we do not calculate these terms. Note that the electron-hole tunnel amplitude Δ that couples |00> and |11> states has a somewhat unusual form of simultaneously creating or destroying two particles, an electron and a hole, which form an exciton. The reason for this is that it describes the coupling of the two different edge states, the electron and the hole-edge states. If one reformulates the description of both edges in terms of the same-charge particles, e.g., electrons, it becomes evident that the tunnel amplitude Δ corresponds physically to the usual tunneling process of transfer of one electron between the two edge states. Such a transfer either adds an electron to the electron edge, leaving a hole in the hole edge, thus creating an exciton, or, in the reverse process, takes an electron from the electron edge, filling a hole in the hole edge, thus destroying an exciton in the process.

From the Hamiltonian, δh and δe -dependent ground states can be calculated as superpositions of the basis states. Tunneling through the antidot occurs when the system is tuned across two degenerate ground states with the same electron numbers but with hole numbers differing by one. This limits the conductance-contributing tunneling processes to |00> <-> |10> (empty electron edge mode) and |11> <-> |01> (filled electron edge mode). Without electron-hole coupling, the basis states are the eigenstates of the Coulomb energy, and their charge configurations in the ground state depend directly on the energies δh, δe, as shown in Fig. 4b. |00> <-> |10> and |11> <-> |01> lead to the vertical traces of conductance peak along the corresponding phase boundaries in the (δh, δe) plane. The horizontal shift between the two vertical parts of this boundary (which is also observed experimentally, e.g., in Fig. 3b) is the direct manifestation of electron-hole attraction within the exciton. This shift represents the additional magnitude of the gate voltage that is needed to overcome the Coulomb attraction energy u between the individual electron and hole in the two respective edge states of the antidot, when the gate voltages evolved past the resonance point.

Near electron-hole resonance, the tunnel amplitude Δ mixes the states |00> and |11> along ξ = 0, forming a ground state which is the quantum superposition of the two states (see SI). Tunneling conductance peak emerges at energy determined by the condition of degeneracy between the resonance state and the adjacent ground states of |01> and |10> (see SI): δh = |Δ|^2 / (δe - u), δh < 0; δh = δe - u + |Δ|^2 / δe, δh > 0 (4). Here, the hyperbolic δh, δe relations (which, in the absence of electron-hole tunneling, correspond to the straight boundaries between the states |11> and |01> and between the states |10> and |00> in Fig. 4b) give rise to the anti-crossing feature in the conductance peak trace.

Near the resonance, the amplitude of the tunneling conductance from the |00> <-> |10> and the |11> <-> |01> processes is determined by the composition of the |00> and |11> states in the superposition, respectively. The tunneling probability p0 for the |00> <-> |10> process rapidly decays as the superposition evolves into pure |11> state under negative ξ; and the tunneling probability p1 for the |11> <-> |01> process rapidly decays as the superposition evolves into pure |00> state under positive ξ (see SI): p0,1 = 1/2 * (1 ± ξ / √ξ^2 + |Δ|^2) (5). Hence, deviation from the resonance condition drives the system out of the electron-hole superposition and results in the termination of the curvy anti-crossing feature.

Applying the model above to our experimental system, we simulate the doping dependence of the antidot tunneling conductance as shown in Fig. 4c. Here, we assume Lorentzian-shaped tunneling conductance peaks 46 with an energy broadening of 0.2 meV to match the broadening of the measured conductance peaks. The energy (δh, δe) dependences of the peak position and peak amplitude are calculated from Eqs. (4) and (5). The gate voltage-tuning of energy at ~0.5 meV/mV is obtained from the ratio between the top gate voltage period and the corresponding charging energy measured in Fig. 2a, b. Semi-quantitative agreement with the experimental observations (Fig. 3b, c) is achieved with an electron-hole Coulomb energy of u = 0.8 meV and tunneling amplitude of Δ = 0.3 meV, reproducing the anti-crossing-like gate-dependent peak position as well as the tunneling conductance peak amplitude. In Fig. 4c, the arrows highlight tunneling processes that give rise to conductance peaks, both far off-resonance (i and iv) and near resonance (ii and iii). The corresponding electron and hole energy level configurations of the tunneling states are depicted in Fig. 4e. In processes ii and iii, exciton states are the superposition of the |00> and |11> states.

For the Coulomb interaction between the electron and the hole QH antidot edges, we estimate u = 2πϵ0ϵBN * e^2 / (ϵBN * L) * ln(1 + 4deh/lB) * (1 + 2deh/lB)^-1/2 (see SI). Here, deh is the radial separation between the electron and hole edges, which should roughly be equal to deh ~ 35 nm to produce u = 0.8 meV, in agreement with the observations. A theoretical estimation of deh requires detailed information on the ν = 0 energy gap and the gate-dependent edge mode velocity, and is beyond the scope of this study. Our QH antidot exciton estimates, nevertheless, are consistent with the most natural model of concentric electron and hole-edge modes with similar diameters and significant tunneling between them. The order-of-magnitude of these estimates for u and Δ are consistent with the assumed geometry of our exciton model with tunnel-coupled, roughly concentric electron and hole edges. Indeed, the inter-mode Coulomb attraction energy u is close to and slightly smaller than the on-site Coulomb energy measured from the charge stability diagrams, consistent with the fact that the effective distance between the edges is slightly larger than the characteristic distance between the charges on the same edge. At the same time, the tunnel coupling Δ is significant, which implies closely spaced electron and hole edges. If an electron were localized on an atomic-scale impurity trap at such a small distance from the hole edge, it would produce more complex scattering of the hole instead of the observed coherent tunneling with amplitude Δ.

Reducing the inter-mode coupling Δ, the model suggests that the hyperbola part of the anti-crossing feature around the resonance is reduced (Fig. 4d). This is consistent with the observations at smaller filling factor ν, as shown in Fig. 3b (more discussion can be found in the SI). At small ν, the inner electron edge mode just starts to emerge near the center of the antidot and hence is physically distant from the outer hole-edge mode. As a result, the tunneling amplitude between the two modes is small, and the hyperbola feature in the gate dependence of the conductance peaks is largely reduced.

Similar to the conventional exciton, the localized QH exciton state is stabilized by reducing the ground state energy of an electron-hole pair. Compared to the non-interacting (u = 0) and isolated (Δ = 0) electron-hole states, the excitonic state has its ground state energy lowered by finite interaction and finite inter-mode tunneling: Eg(u, Δ) - Eg(0, 0) < 0. This gate-dependent energy reduction can be considered as the equivalent of the exciton binding energy. In particular, for the neutrally biased electron-hole state (δh = δe), we have Eg(0, 0) - Eg(u, Δ) = u/2 + √u^2/4 + Δ^2. Interestingly, and different from conventional excitons, the binding energy of the localized QH exciton is not only affected by the Coulomb energy but also by the tunneling amplitude between the electron and the hole modes. Finite inter-mode tunneling leads to a ground state energy reduction in comparison to the decoupled interacting electron-hole pair: Eg(u, Δ) - Eg(u, 0) = u/2 - √u^2/4 + Δ^2, with maximum reduction of Δ occurring at ξ = 0 (see Fig. 4f).

We note that a unique characteristic of the electron-hole system studied here is the combination of Coulomb repulsion on-site and Coulomb attraction between the two edges, which is different from the more familiar multi-quantum dot systems with Coulomb repulsion only. The electron-hole attraction u in the exciton breaks one continuous trace of the external hole-edge mode tunneling peak when the inner electron edge changes from being empty (|10> <-> |00>) to filled (|11> <-> |01>). Since the inter-edge Coulomb attraction effectively reduces the on-site Coulomb blockade, the shift of the off-resonance tunneling conductance peaks is toward a smaller magnitude of gate voltages with increasing number of electrons on the inner antidot edge. Classically, the two segments of the conductance peak would be connected and smoothly shifted by an energy of u, reflecting the gradual increase of the average occupation of the electron state from 0 to 1 with the gate voltages. The quantum state |00> and |11> are mixed by the tunnel amplitude Δ, creating a coherent quantum superposition of the two states. Hole tunneling through the vacuum-pair superposition, as discussed through our model, leads to the anti-crossing-like gate voltage dependence of the conductance peaks with diminishing amplitude away from the electron-hole resonance.

In conclusion, we have demonstrated a localization of individual quantum-coherent QH excitons in a graphene QH antidot. Compared to the conventional excitons on semiconductors, the QH exciton allows electrical tuning of the electron and hole states, and shows a ground state energy, which is determined both by Coulomb interaction and inter-mode tunneling. This work paves the way to a series of scientific tasks and technical developments. For example, excitonic states may be studied in more complex and versatile many-antidot systems for excitonic molecules. Time-dependent measurements may be performed on the excitonic state to explicitly study the quantum coherence. The QH antidot excitons scheme can potentially be extended to study the individual anyonic excitons in the FQH regime by tuning the bulk filling inside and outside the antidot to the fractional fillings of the zeroth Landau level. This implements a coupled state of the two fractional quasiparticles with opposite charges. The interaction energy of such fractional excitons, together with the tunnel coupling between the edge states of the quasiparticles, would be significantly smaller than in the integer regime, requiring much lower temperatures to observe the excitonic state. In the case of a single antidot, which can accommodate at most one exciton at a time, fractional exchange statistics may manifest itself indirectly, via the shape of the tunneling conductance peaks through the antidot. Beyond a single antidot, potential direct observation of the exchange statistics may be possible through exciton transport in the inter-coupled multi-antidot structures.

Methods
Sample fabrication
Graphite-hBN encapsulated graphene stack was assembled using standard van der Waals dry pickup technique, where top hBN, top graphite gate, middle hBN, monolayer graphene, bottom hBN, and bottom graphite gate are picked up sequentially following Fig. 1b with a polycarbonate (PC) stamp. All flakes were mechanically exfoliated onto precleaned (sonication in isopropanol followed by UV ozone cleaning for 2 min and 350 °C baking for 1 min) SiO2 substrates and were scanned using an atomic force microscope (AFM) to ensure contamination-free and atomically flat surfaces. The stamp base was a droplet-shaped polydimethylsiloxane (PDMS) to take advantage of its natural curvature. The PC solution at 6% was then spin-coated onto the droplet-like PDMS base at 1000 rpm and left to dry on a hot plate at 100 °C for a total of 20 min.

The assembly process started with picking up the top hBN layer. The substrate carrying the hBN flake, preheated to 70 °C, was carefully raised until touching the lowest point of the PC stamp. The substrate was then further raised manually until the touch area expanded to the close vicinity of the destination hBN flake. The final contact was made using thermal expansion by heating the stage slowly to 110 °C at ~3 °C/min. The following separation process was done by cooling the stage/substrate back to 65 °C, at which the hBN flake got picked up from the substrate with the thermal contraction of the substrate stage as well as the PC stamp. The subsequent layers of graphite gates, hBN dielectric layers, and graphene were picked up in a similar manner. With all the layers designed in order and orientation on the PC stamp, the stack was then pressed onto a precleaned SiO2 substrate at 160 °C and heated up to 180 °C. Upon separation, the PC was detached from the PDMS stamp and transferred onto the SiO2 substrate together with the heterostructure stack. As the final step, we soaked the stack in chloroform for at least 10 h to dissolve PC. The stack was then transferred to isopropanol to remove chloroform before being blow-dried with nitrogen gas. The sample was then thermally annealed in forming gas (5% hydrogen in argon) at 400 °C for 2 h to thoroughly remove polymer residue on the top surface. The completed stack consisted of, from top to bottom, hBN (8 nm), top graphite (~2 nm), middle hBN (18 nm), monolayer graphene, bottom hBN (36 nm), and back graphite (~2 nm).

After assembly, the heterostack was scanned with an AFM for surface topography, and a bubble-free area was identified for the active area of the sample. The overall device outline was patterned into van der Pauw geometry with standard e-beam lithography and etched into shape with reactive ion etch (RIE) in CHF3/O2 plasma (40 sccm/4 sccm, 30 mTorr, 60 W). The four edge-contacts to the encapsulated graphene channel were made using the same etching method followed by thermal evaporation of 5 nm Cr and 50 nm Au. Similar edge-contact was fabricated on the top graphite gate. To prevent shorting top graphite to the graphene channel underneath the top gate lead at the edge of the channel, a cross linked PMMA block was fabricated covering the exposed graphene edge, and the metal lead toward the top graphite gate contact was deposited over this block. In the final step, the two couplers and the antidot were etched out of the top graphite layer, together with four additional cuts made just outside the four side contacts to isolate the top graphite gate from the graphene channel. A scanning electron microscopy image of the device is shown in the SI.

Sample measurement
Transport measurement of the device was performed inside an Oxford VTI with a 12 Tesla superconducting magnet and He-III cryogenic insert with a base temperature of 300 mK. All electrical cables that enter the sample chamber go through low-pass pi-filters at room temperature and RC filters at 2 K, to minimize the high-frequency noise and the sample’s electron temperature. The high quality of the sample is confirmed by the quantum Hall (QH) plateaus, which start to form in very low magnetic fields (see S1).

All electrical measurements were carried out with the standard lock-in technique. Current excitation of 5–10 nA was provided by a Keithley 6221 current source at a frequency of 23 Hz. In the stability diagram measurements, the current course also provided a controllable DC bias. The AC component of the voltage response was measured by a Stanford Research SR830 lock-in amplifier, while the DC voltage bias was measured by a Keithley 2182A nanovoltmeter. The top graphite gate, bottom graphite gate, and Si gate voltages were provided by a Keithley 2450, a Keithley 2400, and a Keithley 6487, respectively.

Several segments of the patterned monolayer graphene acting as a contact to the channel are not covered by the graphite gates. When these segments are near charge neutral and under a strong magnetic field, large contact resistance can emerge and result in poor measurement results. To solve this problem, a Si back gate voltage was applied to electrostatically dope the segment with the same polarity as the graphene channel to avoid the highly resistive zero-filling gap at a p-n junction in a strong magnetic field.

Data availability
The data represented in Figs. 2–4 are available at the Figshare database: https://doi.org/10.6084/m9.figshare.29042864. All other data that support the findings of this study are available from the corresponding authors upon request. Any questions regarding the data can be addressed to the corresponding authors.

Code availability
The code underlying this study is available at at Figshare database: https://doi.org/10.6084/m9.figshare.29042864. Any questions regarding the code can be addressed to the corresponding authors.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript by Rui Pu et al. introduces a new material platform based on graphene antidots, engineered to host a novel\nquantum Hall (QH) quasiparticle exciton. In contrast to conventional excitons, here the electron and hole comprising the\nexciton are spatially separated at the inner and outer edges of the antidot, respectively. Through the measurement of\ntunneling resistance across the sample the authors have demonstrated the detection of a quantum-coherent QH exciton,\nwhich is individually localized and created via intralayer Coulomb interaction and tunneling within a single\nantidot structure. The theoretical framework developed aligns well with the\nexperimental results. Overall, my impression is positive since the work explores a compelling and largely uncharted area. However, a few aspects of the explanation lack clarity and require\nrevision before the manuscript can be considered for publication.\n1. A primary concern is the limited visibility of the Coulomb diamond features in Fig. 2b, where the observed signatures\nappear rather faint. The authors attribute this to screening from hBN layers. It is recommended that the authors consider\nreproducing similar measurements using a separate device where screening effects are reduced. This could potentially\nenhance the visibility of the features and provide clearer evidence for the reported effects, while also serving as a\nreproducibility test for the observed features in general.\n2. In figure 1, it is recommended that the authors also add a schematic, highlighting excitons in QH double layers, as\nreported in works by Eisenstein et al. in GaAs, and Philip Kim & Cory Dean groups in graphene.\n3. For fig.2, the selection of f = 0, f’ = -2 is not very intuitive. A more detailed justification for focusing on this regime will help.\nAdditionally, it would be valuable to include data from other carrier densities especially on the electron-doped side as this\ncould offer further insight. Such results could be appropriately included in the supplementary material.\n4. The discussion for Fig. 2 should be more elaborate, particularly Fig. 2c. While the supplementary materials provide\nadditional context, the authors should expand on the explanation of the simultaneous dual gate tuning of antidot tunneling\nwithin the main manuscript to enhance clarity and accessibility for the reader.\n5. Fig. 3 illustrates the central findings of the paper. It is not clear to me why the pronounced excitonic state emerges\nspecifically around ν≈0.8. Also, the explanation provided for Fig. 3d is insufficient, particularly given the presence of multiple\ncrossings. The authors may better explain this figure in the text. Including data from an independent device would\nsignificantly strengthen the claims in this part.\n6. While the avoided crossing within the yellow box in Fig. 3b is clear, I am not so sure about the remaining regions. Though\nthere are clear bright spots, they don’t appear to behave as avoided crossings.\n7. It is quite evident how the density is tuned by two separate gates inside and outside of the quantum dot. However, it’s not\nclear how the quantization energy levels at the inner and outer edges of the antidot are changed by the top gate alone.\nCan the authors add more explanation to this part?\n8. The theoretical part in the main text may be reduced. Rather than presenting extensive technical details, the authors may\nfocus on highlighting the key intuitive results that directly support the experimental findings and move additional material to\nSI. Additionally, Fig. 4e is neither referenced nor discussed in the text in the current version of the manuscript. This figure\nshould be properly integrated and explained to ensure completeness and clarity. Also, while the modelling does explain the\nfeatures, it would be beneficial to include an intuitive explanation of why the avoided crossing indicates the presence of an\n‘exciton.’\nOther than these major comments, authors should be careful while preparing the manuscript about smaller details and Fig.\nreferences. Few minor comments below:\n1. Line 157: The text mentions Fig. 1(b) but it is Fig. 2 (b) which is being referred to.\n2. Line 164: ‘f’=-2’ instead of ‘ν’=-2’?\n3. Color scale bar is missing in fig. 3b", "reviewer_2": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature\nCommunications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career\nResearchers who co-review manuscripts.", "reviewer_3": "In this work, the authors demonstrate a new type of bound states formed around an antidot in mono-layer graphene in the\nquantum Hall regime. They control carrier density inside and outside an antidot to form a hole bound state outside the\nantidot, and an electron bound state inside the antidot, separated by a short distance. In the tunnelling conductance through\nthe outer (hole) state, they observe a signature of coupled states between the two quasiparticles, for which they use a term\n“exciton”. This should not be confused with conventional exciton, which is a pair of a conduction-band electron and a\nvalence-band hole separated by a bandgap. The quasiparticle pair in the exciton in this work has energy close to each\nother, and their relative energy can be tuned by gate voltages. There are many similarities to a double-quantum-dot system.\nThey see a similar anti-crossing conductance peaks with a honeycomb-like structure, often observed in a double-dot system.\nHowever, there is a difference in the topology. One state is enclosed by the other. Tunnelling mainly occurs through the\nouter state only.\nThis is a unique system that may be difficult to produce by another method. For example, a fabry-perot type interferometer\nexists, but it would be difficult to produce such a coupled states of electron-hole pair. These excitonic states could be useful\nfor further studies, such as anionic systems. I would like to recommend publication in Nature Communications, however, the\nmanuscript is quite technical, and it is difficult to follow in some places. I would like the authors to clarify the following points\nbefore publication.\n1) Hole or electron\nThe authors have chosen to model the system with outer hole state and inner electron state. This is understandable given\ntheir hole-like and electron-like band structures. However, it is also possible to model the system from a different\nperspective. For example, the inner state could be modelled as a hole state (such as in conduction-band holes). This would\nchange the notation |n_h n_e> to be |n_ho n_hi> (o for outer and i for inner). For example, |00> and |11> mentioned in the\nline 339 would change to |01> and |10>, and it becomes straightforward that these two states are simply an exchange of a\nhole (or an electron). Readers familiar with double-quantum dot system may be more comfortable with this notation. Could\nauthors clarify/comment on their thoughts on modelling the system with two quasiparticles of the same type (such as hole or\nelectron)? I think this is important as they state in the lines 400-402 that their unique characteristic of their system is the\ncombination of on-site Coulomb repulsion and inter-edge Coulomb attraction, but obviously, with two holes (or electrons),\ninter-edge interaction would become repulsion.\n2) Coherent bound state.\nIn the abstract, the authors mention “quasiparticle exciton which represent a quantum-coherent bound state”, but I am not\nsure how this coherence has been demonstrated in their exciton state. Could author clarify this?", "reviewer_4": "The present manuscript demonstrates the localization and coherent control of individual excitons in a graphene quantum\nHall antidot system. Using dual-gated graphene/hBN heterostructures, the authors engineer closely spaced, concentric\nelectron and hole edge modes in the integer quantum Hall regime, whose coupling via tunneling and Coulomb interaction\ncreates a bound electron-hole exciton state that has been studied in several bilayer Graphene samples recently.\nComplementary to the drag measurements that have been employed recently to probe interlayer excitons, the present work\ncombines transport measurements like AB oscillations and Coulomb blockade with a theoretical model capturing e-h\ntunneling and interaction effects. The authors extract key energy scales and demonstrate gate tunable coupling and anti-\ncrossing behavior of electron and hole modes.\nThis work is novel in that it presents the first experimental realization of individual localized quantum Hall excitons within a\ngraphene antidot device. It further demonstrates gate-controlled coupling and coherent superposition between electron\nand hole edge modes, which allows the authors to propose these QH excitons as potential qubits for future quantum\ntechnologies. The discussion is well contextualized within the broader field, and the transport measurements clearly reveal\ndiscrete tunneling resonances, AB oscillations, and anti-crossing features consistent with the proposed model.\nBefore making a recommendation, I would like to see the authors response to the following:\n1. The manuscript interprets the observed anti-crossing like features and conductance peak behaviors as evidence of\ncoherent coupling between the electron and hole edge modes. However, no direct time-domain or phase sensitive\nmeasurements of coherence (like interference visibility, coherence times, or dephasing behavior) are presented. Could the\nauthors clarify whether such measurements are feasible in this setup, and comment on how the coherence of the excitonic\nstate might be quantitatively assessed in future experiments?\n2. The conductance maps in Fig. 3(b) exhibit random shifts attributed to quantum dot like charge traps near the antidot.\nWhile the authors note this issue and focus on clean regions of the data, a more systematic characterization of the impact of such\ntraps on the exciton resonances would strengthen the conclusions. Could the authors comment on the device stability over\ntime, hysteresis behavior, or temperature dependence of these random features? Do they anticipate that these fluctuations\ncould affect the coherence of the localized exciton?\n3. The paper provides estimates for the coulomb interaction and tunneling amplitudes, but the exciton binding energy\nrelative to the thermal energy scale at the measurement temperature is not explicitly discussed. Could the authors comment on\nthe expected binding energy of the quantum Hall exciton in this system, and how robust the bound state is against\nthermal fluctuations? Additionally it would be helpful to understand whether thermal activation might impact the visibility or\ncoherence of the excitonic tunneling features.\n4. The manuscript discusses an interesting possibility of realizing excitonic states in the fractional quantum Hall regime and\nexploring anyonic statistics within such systems. This is a particularly exciting direction. Indeed very recently two papers\ndemonstrated the existence and control of such anyon excitonic states in bilayer Graphene samples in Corbino geometry.\nCould the authors comment in more detail on the experimental feasibility of achieving this in their current platform? For\ninstance, in FQH regime, there are several bilayer constructions possible for fractional excitons. What differences do the\nauthors expect in the experimental feasibility for such fractional excitons?"}, "author_response": "Reply to Reviewer #1\nThe manuscript by Rui Pu et al. introduces a new material platform based on graphene\nantidots, engineered to host a novel quantum Hall (QH) quasiparticle exciton. In\ncontrast to conventional excitons, here the electron and hole comprising the exciton\nare spatially separated at the inner and outer edges of the antidot, respectively.\nThrough the measurement of tunneling resistance across the sample the authors have\ndemonstrated the detection of a quantum-coherent QH exciton, which is individually\nlocalized and created via intralayer Coulomb interaction and tunneling within a single\nantidot structure. The theoretical framework developed aligns well with the\nexperimental results. Overall, my impression is positive since the work explores a\ncompelling and largely uncharted area. However, a few aspects of the explanation\nlack clarity and require revision before the manuscript can be considered for\npublication.\nWe thank reviewer#1 for the overall positive opinion and comments, and the constructive\nquestions. Below we answer the reviewer’s questions point-by-point.\n1. A primary concern is the limited visibility of the Coulomb diamond features in Fig.\n2b, where the observed signatures appear rather faint. The authors attribute this to\nscreening from hBN layers. It is recommended that the authors consider reproducing\nsimilar measurements using a separate device where screening effects are reduced.\nThis could potentially enhance the visibility of the features and provide clearer\nevidence for the reported effects, while also serving as a reproducibility test for the\nobserved features in general.\nWe thank the reviewer for the critical comment on the visibility of the Coulomb diamond\nfeatures in Figure 2b. Various factors can affect the apparent visibility of the Coulomb\ndiamonds. Generally, a weaker screening leads to a larger Coulomb energy, hence the large\nCoulomb diamonds which help to stand out of any broadening factors. This is most evident\nin suspended graphene-based antidots where the screening is minimized, as shown by the\nexamples from our previous lab work below.\nFigure R1-1. Coulomb diamonds measured from physically etched QH antidots in\nsuspended graphene, with minimal screening and sharp confinement.\nWith our device structure, and given that the diameter of the antidot must be much larger\nthan the magnetic length, the only practical way to significantly reduce the screening is to\nmake the hBN layer between graphene and the patterned top gate thicker, and we have\nindeed recently tested such a sample. However, we found that a thicker hBN layer leads to\nbroadened electric potential profiles, making it very difficult to achieve a proper antidot\ntunneling. In addition, the broadened electric potential also weakens the spatial\nconfinement of the edge modes, resulting in a significant level of broadening which has a\nnegative impact on both the formation of quantum states and the visibility of the Coulomb\ndiamonds. (By comparison, the figures above were measured from physically etched\ngraphene, where the confinement potential profile sharpness is maximized. However, such\nphysically etched antidots preclude the possibility of achieving the co-centric e-h double\nedge modes, which is required for this work). Generally, the sharpness of the top gate-\ndefined potential profile is critical for confining the edge modes and is more important than\nthe visibility of the Coulomb diamonds, for realizing well-defined antidot states and the\ndelicate, complex excitonic states.\nMore recently we have identified another important factor for the visibility of the Coulomb\ndiamonds (which was not discussed in our original manuscript but has been added to the\nrevised version). It is the tunneling amplitude between the antidot and both couplers.\nClarity of a Coulomb diamond (which by itself a low conductance regime in the charge\nstability diagram due to the Coulomb blockade) is often enhanced by the traces of the\nbright, high conductance resonant tunneling peaks which highlight their boundaries (see\nFigure R1-2 below). For example, the good visibility of the Coulomb diamonds in Figure R1-1\nis largely due to the bright outlines. In Figure 2b in the manuscript, such resonant tunneling\ntraces are pronounced only on the negative slope side but are nearly invisible on the\npositive slope side. This suggests that the coupling of the antidot to one of the couplers is\nmuch weaker than to the other. This also explains the rather strong tilt in the diamonds.\nFigure R1-2: The visibility of Coulomb diamonds, from Coulomb blockade (the dark\n“diamonds”) and resonant tunneling (the yellow outlines).\nWith the current device scheme and without an independent control to the antidot\ncoupling to both couplers, we have rather limited and somewhat random control for\nrealizing more ideal resonant tunneling. Our devices were always designed to have a\ncompletely symmetric coupling. Nevertheless, the actual outcome of antidot tunneling,\nwhich is highly sensitive to the tunneling distance and the local electric potential profile\nnear the barrier, is affected by a number of uncontrollable factors from the complex\nchemical and physical processes in the nanofabrication.\nBecause of the limitation in the reliable control to the tunneling, reproducing the identical\ntypes of samples does not guarantee better data for our main purpose in this study.\nCurrently we are working on developing and studying a new type of sample which has\nindependent side-gates to control the antidot tunneling. This work, however, is outside the\nscope of the current manuscript.\nDespite the limited control of the antidot tunneling to the charge reservoirs, we stress that\nthe coupling to the antidot only serves the purpose of detecting the exitonic states (not the\nformation of such states), and thus it should be kept weak in general, as stated by our\nmodel. As long as the antidot coupling is not too strong, the coupling strength does not\naffect the key result of this work. The weaker tunneling, although resulting in visually “faint”\nColumb diamonds, does not have any negative impact on the observation of the underlying\nphysics. This is because Coulomb diamonds are associated with the finite-bias, non-linear\nconductance regime and excited states. However, the measurements on the excitonic\nstate discussed in this work are only related to the ground states (as discussed in the\nmodeling section), which is manifested by the zero-bias conductance peaks clearly shown\nin Figure 2a and Figure 3a. The only purpose of measuring the Coulomb diamonds is to\nestimate the Coulomb energies and to compare/confirm the energies with the basic theory\nto be used as the parameters for our modeling.\nTo address the reviewer’s comment on the visibility of Coulomb diamond, we have made\nseveral revisions. In our previously Figure 2b the color scheme (“jet”) of black diamond in\nred background gives rather poor visual contrast. We have revised the plot using a different\ncolor scheme (“copper”), as shown below. We also include a few more data sets taken\nfrom this device in the Supplementary Information, to help clarify the estimation of the\nCoulomb and quantization energies. In particular, all Coulomb diamonds suggest a similar\nCoulomb energy of ~1meV as stated in our manuscript.\n)\n)\nV\nV (\n(\nb\nV b\nV\nV TG (V) V TG (V)\n) V ) V\nV ( b V ( b\nV (V) V (V)\nTG TG\nFigure R1-3. Updated figures for the Coulomb diamonds. Top left: (f,f’)=(0,-2). Top right\n(f,f’)=(-6,-2). Bottom left (f,f’)=(0,-1). Bottom right (f,f’)=(0,-6). The dotted lines highlight\nexamples of Coulomb diamonds.\nWe would also like to point out two interesting facts from these plots which are related to\nour discussion above: 1) In the top left plot, the negative-sloped traces of the resonant\ntunneling are bright while the positive-sloped traces are nearly invisible. The opposite can\nbe observed for the other three plots. This suggests that the tunneling distance between\nthe antidot and each one of the two couplers depends on the details of the electrostatics.\n2) In the bottom right plot, the two arrows point to resonant tunneling traces which are\nweak but still visible. Moving from the lower density end of the QH plateau towards the\nhigher density end (left to right), the resonance tunneling amplitude becomes weaker, and\neventually the weaker negative-sloped traces become invisible. Then, the diamonds lose\nhalf of the outlines and become much less visible even though the Coulomb blockade\nitself remains the same.\n2. In figure 1, it is recommended that the authors also add a schematic, highlighting\nexcitons in QH double layers, as reported in works by Eisenstein et al. in GaAs, and\nPhilip Kim & Cory Dean groups in graphene.\nWe thank the reviewer for the suggestion. Indeed, the inter-layer exciton is one of the most\nimportant exciton systems. We have revised Figure 1 to include a schematic on it.\n3. For fig.2, the selection of f = 0, f’ = -2 is not very intuitive. A more detailed\njustification for focusing on this regime will help. Additionally, it would be valuable to\ninclude data from other carrier densities especially on the electron-doped side as this\ncould offer further insight. Such results could be appropriately included in the\nsupplementary material.\nThe main reason we focus on the selection of f = 0, f’ = -2 is because the antidot-reservoir\ncoupling happened to be better on the hole side. While in principle graphene should have\nelectron-hole symmetry, in real-life devices/measurements, it is very common that the QH\nrelated signatures are not completely symmetric between electron and hole doping.\nMoreover, in such QH edge tunneling measurements, even a slight asymmetry can critically\naffect the accessibility to the observables. Below in Figure 1R-4 we show a comparison of\nthe conductance oscillation on the two sides, showing the tunneling conductance peaks\non the hole side (f=-2 plateau) but the lack of tunneling on the electron side (f=2 plateau).\nThe asymmetry is clearly due to the longer tunneling distance on the electron side related\nto the details in the electrostatics at the coupler-antidot region.\nWe believe that the asymmetric tunneling is related to some of the detailed factors in our\nfabrication processes and/or the particular design of the structure of our device. Such\nstructure serves the main goal of this presented work, which focuses on the proof-of\nprinciple demonstration of the antidot exciton state. However, we expect to observe similar\nresults in the tunneling asymmetry if another of this sample type is produced for our\npurpose. As discussed above, currently we are continuing our work focusing on a new\ndevice design with an effective gate-controllable coupling. This is our on-going work to\ntackle certain complexities we encountered in the presented work, including the observed\ntunneling asymmetry, but it is outside the scope of the manuscript.\nFigure R1-4: Comparison between antidot-reservoir tunneling on the electron (blue) and\nhole (red) sides. Conductance peaks are expected on the low-density side of the v=2\nplateaus.\nWe should stress that while the device structure reported in this work has limited tunability,\nit does not affect the key scientific claim of this work. In our measurements, we first ensure\nthat the weak tunneling is established between the couplers and the antidot, as\nmanifested by: 1) well-defined tunneling resistance dips (conductance peaks) on the low-\ndensity side of the well-defined QH plateaus, with the periodicity agreeing with the antidot\nsize (Figure 2a and Figure 3a); 2) top and back gate dependence on the amplitude of the\nconductance peaks which agrees with the understanding from the electrostatics and the\nLandau level diagrams (Figure 2c and Figure 3d); 3) quantized magnetic flux shift of\ntunneling peaks (Figure 3a). Only under these conditions did we proceed to measure the\nfeatures of the excitonic state. Satisfying all these conditions is a time-consuming trial-\nand-error process for the reported device structure. Still, once these conditions are\nmatched, the experimental system is well-defined and there is no ambiguity to any further\nobservations.\n4. The discussion for Fig. 2 should be more elaborate, particularly Fig. 2c. While the\nsupplementary materials provide additional context, the authors should expand on\nthe explanation of the simultaneous dual gate tuning of antidot tunneling within the\nmain manuscript to enhance clarity and accessibility for the reader.\nWe thank the reviewer for this suggestion. We have added more detailed explanations to\nthe dual-gate tuning and the operation of the antidot device.\n“The gate-control of the antidot tunneling is illustrated in Figure 2(c). The main panel\nshows the tunneling resistance versus the filling inside (𝜈) and outside (𝜈’) the antidot/couplers\nregions. Sharp bright dots (resistance valleys) signify the antidot tunneling on the QH plateau of\n𝑅 = ℎ (𝑓′ = −2), coarsely sampled over a wide range of filling factor 𝜈 in the\n𝑄𝐻 2𝑒2\nantidot/couplers regions that are tuned by the back gate. Four characteristic (𝑓,𝑓′) combinations\nare highlighted with their corresponding Landau level energy diagrams. The spatial profiles of\nthe Landau level energy (in relation to the chemical potential) are tuned through the patterned top\ngate, while the back gate uniformly shifts the chemical potential throughout the whole channel.\nThe QH edge modes are located where the Landau levels cross the chemical potential 𝜇. The\nedge mode tunneling strength depends sensitively on the spatial separation between the coupler\nand antidot edge modes, adjusted by the dual-gates. To enable coupler-antidot tunneling, the\ntunneling distance between the coupler edge mode and the antidot edge mode needs to be near its\nminimum, when Landau Level associated with these edge modes is tuned slightly across the\nchemical potential. Directly relevant to the discussions below and for 𝑓𝑓′ ≤ 0, this corresponds\nto the lower carrier density end of a QH plateau. A more detailed discussion of the edge mode\ntunneling in these four regimes is given in the SI.”\n5. Fig. 3 illustrates the central findings of the paper. It is not clear to me why the\npronounced excitonic state emerges specifically around ν≈0.8. Also, the explanation\nprovided for Fig. 3d is insufficient, particularly given the presence of multiple\ncrossings. The authors may better explain this figure in the text. Including data from an\nindependent device would significantly strengthen the claims in this part.\n6. While the avoided crossing within the yellow box in Fig. 3b is clear, I am not so sure\nabout the remaining regions. Though there are clear bright spots, they don’t appear to\nbehave as avoided crossings.\n7. It is quite evident how the density is tuned by two separate gates inside and outside\nof the quantum dot. However, it’s not clear how the quantization energy levels at the\ninner and outer edges of the antidot are changed by the top gate alone. Can the\nauthors add more explanation to this part?\nThe reviewer’s questions 5,6,7 all relate to the understanding of the anti-crossing features.\nHere we give a combined answer to the three questions.\nThe emergence of the anti-crossing features is associated with the resonance of the hole\nand electron quantization energy levels. Qualitatively, the crossing of electron- and hole-\nenergy levels by tuning of a single top-gate can be visualized through the figure below.\nFigure R1-5. Gate-defined edge confinement and energy quantization. Top panel: for a\nsingle gate-defined antidot, the quantization energy level spacing increases with increasing\ntop gate voltage. Bottom panel: two sets of quantization energy levels originated from\nelectron and hole Landau levels, offset by the v=0 gap, intersect with each other as top gate\nvoltage increases.\nIn the top panel of Figure R1-5, the 3d surfaces depict the spatial variation of the Landau\nlevel energy near a single QH antidot. At near-zero top-gate voltage, the electric potential\nprofile is mostly flat (top left panel). The edge mode confinement is very weak with the\n𝑑𝑉\nsmall edge mode velocity (which is 𝑣~ , where 𝑉 is the confinement potential and 𝑟 is\n𝑑𝑟\nℏ𝑣\nthe radial coordinate), and the quantization energy (~ , 𝑅 being the antidot radius) is very\n𝑅\nsmall. All the quantization levels are closely packed near the bulk Landau level.\nWith an increasing top-gate voltage (top right panel), the antidot becomes strongly gate-\ndefined and the QH edge modes circulating the antidot possess a higher group velocity due\nto the higher potential profile contrast. The stronger quantum confinement results in the\nlarger level spacing.\nGenerally, at each fixed top-gate voltage, there is a set of electron edge mode levels and a\nset of hole edge mode levels, whose energies “expand out” with the increasing top gate\nvoltage. The electron and hole energy levels evolve similarly with the top-gate voltage, as\nshown in the bottom panel above as red (electron) and blue (hole) lines, which initiate from\nthe zeroth electron and hole Landau level energies in absence of the top-gate voltage.\nBecause of the finite energy gap between the zeroth and electron- and hole- Landau levels,\nthe manifolds of the electron- and hole- quantization levels are offset by  (the charge\ng\nneutrality gap). With the increasing top-gate voltage and hence the quantization energy\nlevel spacing, the electron- and hole- quantization levels will cross as shown in the figure\nabove. In particular, the larger-index (hence faster moving) hole levels will intersect with the\nsmaller-index (hence slower moving) electron levels. This is how a single top-gate tunes the\nlevel-crossing between the two antidot modes. At each crossing point, electron-hole\nresonance occurs, and an anti-crossing feature is expected to emerge. This is why multiple\nanti-crossings are observed in our data. In the simplistic illustration above, N electron and\nN hole level will lead to a maximum of (N-1)N/2 crossings. In a real-life sample, however,\nmuch depends on the details of the electrostatics and on the tunneling distance between\nthe antidot and the couplers (which is also top-gate dependent).\nThe detailed shape of the anti-crossing feature (including the anti-crossing curvature which\nthe reviewer questioned for some of the weaker-looking features), as understood from our\nmodel, is determined by two parameters: the Coulomb interaction energy and the electron-\nhole tunneling amplitude. The Coulomb interaction energy causes a shift in the energy of\nthe conductance peak when the electron- and the hole-levels cross each other; while the\nelectron-hole tunneling amplitude causes bending in the tunneling conductance trace\nwhich gives the “anti-crossing” shape.\nThe observed shapes of the anti-crossing features can be understood from the impact of\nthe gate voltages on these two parameters. This can be illustrated by the schematic below.\nFigure 1R-6: the shape of the anti-crossing feature.\nThe left panel here corresponds to when the value of antidot filling factor 𝜈 is small, and the\ninner electron edge mode (f=1) just starts to emerge as the chemical potential enters above\nthe zeroth electron Landau level (red line). The distance between the inner electron antidot\nmode (red circle) and the outer hole antidot mode (blue circle) is large, hence the electron-\nhole tunneling amplitude is small. This is why at small 𝜈, the anti-crossing features show\nlittle curvature and look more like a shift (due to the Coulomb interaction energy). With the\nchemical potential shifting upwards and 𝜈 increasing, the diameter of the electron antidot\nmode approaches that of the hole antidot mode as shown in the right panel, and the inter-\nmode tunneling amplitude becomes larger as the inter-spacing between the electron and\nhole orbits reduces. As a result, much stronger curvature was observed in the anti-crossing\nfeatures.\nThere is a finite number of accessible quantization energy levels (limited by the height of\nthe top-gate induced potential step and by the finite number of levels with a sufficient\ntunneling probability to the source/drain). As a result, the number of crossings between the\nelectron and hole levels is also finite. Within the range of our measurement, the largest 𝜈 at\nwhich we could observe the anti-crossing feature was around 0.8. There we observed the\nanti-crossing feature with the strongest curvature.\nOverall, all the anti-crossing features observed here are in good agreement with the\nquantum mechanical model and the electrostatics. This includes the features with the\nweaker curvature, and the evolvement of the curvature of the anti-crossings with the back\ngate voltage and 𝜈, and the semi-quantitative agreement with the modeling parameters of\nthe Coulomb energy. The multi-aspect agreement between the experimental and the\ntheory achieved in this work, within our current understanding, cannot be due to other\nsample-dependent random factors.\n8. The theoretical part in the main text may be reduced. Rather than presenting\nextensive technical details, the authors may focus on highlighting the key intuitive\nresults that directly support the experimental findings and move additional material to\nSI. Additionally, Fig. 4e is neither referenced nor discussed in the text in the current\nversion of the manuscript. This figure should be properly integrated and explained to\nensure completeness and clarity. Also, while the modelling does explain the features,\nit would be beneficial to include an intuitive explanation of why the avoided crossing\nindicates the presence of an ‘exciton.’\nWe thank the reviewer for this suggestion. We have revised the theory/modeling part of the\nmanuscript and focused more on the conceptual understanding of the underlying physics,\nwhile moving the theoretical derivations to the SI.\nThe main point in respect of an intuitive exciton picture is that the observed resonant\nfeature in the gate-voltage dependence of the conductance peaks is different from the\nusual anti-crossing behavior of a pair of quantum levels. As one can see from the\ntheoretical results presented in Fig. 4 b,c,d, and in the experimental Fig. 3c, in contrast to\nthe standard anti-crossing behavior, the position of the conductance peak after the anti-\ncrossing point is shifted relative to the position of this peak before the anti-crossing.\nThe simple explanation of this shift is that it represents the additional magnitude of the gate\nvoltage needed to overcome the Coulomb attraction energy 𝑢 between the individual\nelectron and hole in the two respective edge states of the antidot, after one extra electron\nis added to the internal antidot edge, when the gate voltages evolves past the resonance\npoint. So intuitively, this shift represents the most direct manifestation of the formation of\nthe exciton. The anti-crossing itself is rather the manifestation of the coherent quantum\ndynamics this exciton is participating in. We added this intuitive discussion of our\ntheoretical results to the main text while redistributing the theory part of our work between\nthe main text and the SI, as suggested by the referee.\n9. Other than these major comments, authors should be careful while preparing the\nmanuscript about smaller details and Fig. references. Few minor comments below:\n1. Line 157: The text mentions Fig. 1(b) but it is Fig. 2 (b) which is being referred to.\n2. Line 164: ‘f’=-2’ instead of ‘ν’=-2’?\n3. Color scale bar is missing in fig. 3b\nWe thank the reviewer for pointing out these errors. We carefully reviewed the manuscript\nand corrected all these and similar errors.\nReply to Reviewer #2\nI co-reviewed this manuscript with one of the reviewers who provided the listed\nreports. This is part of the Nature Communications initiative to facilitate training in\npeer review and to provide appropriate recognition for Early Career\nResearchers who co-review manuscripts.\nWe thank Reviewer #2 for co-reviewing our manuscript.\nReply to Reviewer #3\nIn this work, the authors demonstrate a new type of bound states formed around an\nantidot in mono-layer graphene in the quantum Hall regime. They control carrier\ndensity inside and outside an antidot to form a hole bound state outside the antidot,\nand an electron bound state inside the antidot, separated by a short distance. In the\ntunnelling conductance through the outer (hole) state, they observe a signature of\ncoupled states between the two quasiparticles, for which they use a term “exciton”.\nThis should not be confused with conventional exciton, which is a pair of a\nconduction-band electron and a valence-band hole separated by a bandgap. The\nquasiparticle pair in the exciton in this work has energy close to each other, and their\nrelative energy can be tuned by gate voltages. There are many similarities to a double-\nquantum-dot system. They see a similar anti-crossing conductance peaks with a\nhoneycomb-like structure, often observed in a double-dot system. However, there is a\ndifference in the topology. One state is enclosed by the other. Tunnelling mainly\noccurs through the outer state only.\nThis is a unique system that may be difficult to produce by another method. For\nexample, a fabry-perot type interferometer exists, but it would be difficult to produce\nsuch a coupled states of electron-hole pair. These excitonic states could be useful for\nfurther studies, such as anionic systems. I would like to recommend publication in\nNature Communications, however, the manuscript is quite technical, and it is difficult\nto follow in some places. I would like the authors to clarify the following points before\npublication.\nWe thank the reviewer for the positive comments and constructive questions. Below we\nanswer the questions point-by-point.\n1) Hole or electron\nThe authors have chosen to model the system with outer hole state and inner electron\nstate. This is understandable given their hole-like and electron-like band structures.\nHowever, it is also possible to model the system from a different\nperspective. For example, the inner state could be modelled as a hole state (such as in conduction-\nband holes). This would change the notation |n_h n_e> to be |n_ho n_hi> (o for outer\nand i for inner). For example, |00> and |11> mentioned in the line 339 would change to\n|01> and |10>, and it becomes straightforward that these two states are simply an\nexchange of a hole (or an electron). Readers familiar with double-quantum dot system\nmay be more comfortable with this notation. Could authors clarify/comment on their\nthoughts on modelling the system with two quasiparticles of the same type (such as\nhole or electron)? I think this is important as they state in the lines 400-402 that their\nunique characteristic of their system is the combination of on-site Coulomb repulsion\nand inter-edge Coulomb attraction, but obviously, with two holes (or electrons),\ninter-edge interaction would become repulsion.\nWe agree with the referee that in principle, it is always possible to use the electron/hole\ntransformation of the electron operators to express the system dynamics in terms of either\nelectrons or holes. The important question to be answered in this regard is what\ndescription is more natural and convenient. In our system, similarly to the usual\nsemiconductor physics, we think it is natural to follow the band structure (as is also\nacknowledged by the referee) and use the description in terms of electrons for the\nconduction band and holes for the valence bands. In the case of graphene in the magnetic\nfield, the quantum Hall effect states are different from the conventional energy bands, but\nthe position of the electron and hole states in terms of the gate voltages is clearly indicated\nby the position of electron and hole quantum Hall conductance plateaus. In this way, we\nidentify the electron and hole states, correspondingly, inside and outside of the antidot.\nAs the referee points out, there was one aspect of our problem that would look more\nunderstandable if one used the “same-charge” notations (either both holes or both\nelectrons) for the edge states inside and outside of the antidot, and the tunnel coupling\nbetween these two edges. In the electron-hole language, the tunneling term in the antidot\nHamiltonian (Eq. 3 of the manuscript) looks like it creates or destroys simultaneously two\nparticles, electron and a hole, which form an exciton. We agree with this point and actually\nmade an attempt already in the original version of the manuscript to explain this form of the\ntunneling term by mentioning (right next to Eq. 3) that these exciton creation/annihilation\nprocesses described by the tunneling “correspond physically to a transfer of one electron\nbetween the electron and the hole edge modes”.\nGenerally, changing the description of the antidot completely to the same-charge edge\nmodes, while converting the tunneling term into a more transparent form, would make\nmany other aspects of the problem more difficult to follow. For example, a change of\nnotation involves not only the change of sign of interaction energy but also the shift of the\nsingle particle energies. As a result, we prefer to stick with the “band-structure” supported\nelectron-hole description.\nWe would like to stress that even with the same-charge notation, the Coulomb attraction\nstill presents to form the excitonic state. This can be understood from the following intuitive\npicture. Unlike two similar antidot states charged by a single gate, in our system the inner\nand the outer antidot energy levels shift at different rates with changing top-gate voltage,\nleading to a level crossing. Near the resonance, the charge-neutral reference, which\ncorresponds to both levels being infinitesimally close to the chemical potential, has the\ninner level empty of electron and outer level empty of hole (Figure R3-1). This is the |01>\nstate in the hole-hole notation suggested by the reviewer. Starting from the charge-neutral\nreference of |01>, when a hole tunnels from the inner orbit to the outer orbit, it leaves the\ninner orbit negatively charged (effectively due to charge imbalance between the inner\nantidot and the back gate) which creates the Coulomb attraction. This process is similar to\nthe formation of a conventional exciton when a particle is excited and interacts with the\nbackground behind which is oppositely charged (i.e., an antiparticle).\nFigure R3-1. An outer antidot level and an inner antidot level are about to cross each other.\nThe dotted line is the chemical potential, which is fixed at the resonance energy by tuning\nthe bottom-gate voltage. The arrows point to the direction each level shifts in relation to the\nchemical potential, under increasing top-gate voltage. The charge-neutral reference shown\nhere corresponds to |00> in the electron-hole notation and |01> in the hole-hole notation.\nTo make the tunneling term more understandable, we expanded our previous brief\nexplanation:\n“Note that the electron-hole tunnel amplitude Δ that couples |00⟩ and |11⟩ states has somewhat\nunusual form of simultaneously creating or destroying two particles, electron and hole which\nform an exciton. The reason for this is that it describes the coupling of the two different edges\nstates, the electron and the hole edge states. If one reformulates the description of both edges in\nterms of the same-charge particles, e.g., electrons, it becomes evident that the tunnel amplitude Δ\ncorresponds physically to the usual tunneling process of transfer of one electron between the two\nedge states. Such a transfer either adds an electron to the electron edge leaving a hole in the hole\nedge, thus creating an exciton, or, in the reverse process, takes electron from the electron edge\nfilling a hole in the hole edge, thus destroying an exciton in the process.”\n2) Coherent bound state.\nIn the abstract, the authors mention “quasiparticle exciton which represent a\nquantum-coherent bound state”, but I am not sure how this coherence has been\ndemonstrated in their exciton state. Could author clarify this?\nAs seen from the antidot Hamiltonian, the tunnel coupling between the electron and the\nhole edge modes produces the quantum-coherent evolution that leads to the creation and\nthe annihilation of the individual excitons. Quantitatively, this evolution is described in\nterms of a two-state quantum system in the basis of |00> and |11> states (in the notations\nused in the manuscript) and accounts for both the tunneling and the electron-hole\nattraction within the exciton. In the main approximation, this two-state system is\ncompletely coherent, and its quantum description predicts several qualitative features of\nthe wavefunctions of the two energy eigenstates of the system and the energy difference\nbetween them, which both are reflected in the position and the magnitude of the\nconductance peaks for tunneling through the antidot; how the conductance peaks on the\ntwo sides of the resonant point are shifted relative to each other by electron-hole\nattraction, and on which branches of the resonance the conductance peaks are decreasing\nin amplitude due to the structure of the ground state wavefunction of the exciton two-state\nsystem. The predictions of this quantum-coherent model are in agreement with the\nobserved behavior of the conductance peak in the vicinity of the resonance point shown in\nFig. 3 of the manuscript. This agreement allows us to claim the observation of the\nquantum-coherent dynamics of the individual excitons on the antidot.\nWe added to the abstract a brief statement reflecting this explanation:\n… quasiparticle exciton which represents a quantum-coherent bound state of an electron and a\nhole situated on their corresponding edges coupled through intralayer tunneling and Coulomb\ninteraction. Quantum-coherent dynamics of the exciton is reflected in the observed evolution of\nthe position and magnitude of the antidot conductance peaks around the electron-hole resonance.\nReply to Reviewer #4\nThe present manuscript demonstrates the localization and coherent control of\nindividual excitons in a graphene quantum Hall antidot system. Using dual-gated\ngraphene/hBN heterostructures, the authors engineer closely spaced, concentric\nelectron and hole edge modes in the integer quantum Hall regime, whose coupling via\ntunneling and Coulomb interaction creates a bound electron-hole exciton state that\nhas been studied in several bilayer Graphene samples recently. Complementary to\nthe drag measurements that have been employed recently to probe interlayer\nexcitons, the present work combines transport measurements like AB oscillations and\nCoulomb blockade with a theoretical model capturing e-h tunneling and interaction\neffects. The authors extract key energy scales and demonstrate gate tunable coupling\nand anti-crossing behavior of electron and hole modes.\nThis work is novel in that it presents the first experimental realization of individual\nlocalized quantum Hall excitons within a graphene antidot device. It further\ndemonstrates gate-controlled coupling and coherent superposition between electron\nand hole edge modes, which allows the authors to propose these QH excitons as\npotential qubits for future quantum technologies. The discussion is well\ncontextualized within the broader field, and the transport measurements clearly\nreveal discrete tunneling resonances, AB oscillations, and anti-crossing features\nconsistent with the proposed model.\nWe thank reviewer#4 for the positive comments and constructive questions. Below we\nanswer the questions point-by-point.\n1. The manuscript interprets the observed anti-crossing like features and\nconductance peak behaviors as evidence of coherent coupling between the electron\nand hole edge modes. However, no direct time-domain or phase sensitive\nmeasurements of coherence (like interference visibility, coherence times, or\ndephasing behavior) are presented. Could the authors clarify whether such\nmeasurements are feasible in this setup, and comment on how the coherence of the\nexcitonic state might be quantitatively assessed in future experiments?\nWe thank the reviewer for this insightful question. The DC charge transport measurements\nperformed in this work are essentially the time-averaged measurements. Our\nmeasurement setup is limited to a very narrow bandwidth (<100Hz) in order to filter out the\nhigh frequency noise and to minimize the electron temperature. So, with our measurement\nsetup, direct time-domain measurements are not applicable. In our measurements, the\nobservation of the anti-crossing-like behavior at the resonances is a direct manifestation of\nthe coherent quantum mechanical nature of the excitonic states. As seen from the antidot\nHamiltonian, the tunnel coupling between the electron and the hole edge modes produces\nthe quantum-coherent evolution that leads to the creation and the annihilation of the\nindividual excitons. Quantitatively, this evolution is described in terms of a two-state\nquantum system in the basis of |00> and |11> states (in the notations used in the\nmanuscript) and accounts for both the tunneling and the electron-hole attraction within\nthe exciton. In the main approximation, this two-state system is completely coherent, and\nits quantum description predicts several qualitative features of the wavefunctions of the\ntwo energy eigenstates of the system and the energy difference between them, which both\nare reflected in the position and the magnitude of the conductance peaks for tunneling\nthrough the antidot; how the conductance peaks on the two sides of the resonant point are\nshifted relative to each other by electron-hole attraction, and on which branches of the\nresonance the conductance peaks are decreasing in amplitude due to the structure of the\nground state wavefunction of the exciton two-state system. The predictions of this\nquantum-coherent model are in agreement with the observed behavior of the conductance\npeak in the vicinity of the resonance point shown in Fig. 3 of the manuscript. This\nagreement allows us to claim the observation of the quantum-coherent dynamics of the\nindividual excitons on the antidot.\nIn principle, through the DC-like measurements there are features that can be modeled\nand provide estimates to the phase coherence, such as relating the width of the tunneling\nconductance peaks to the quasiparticle lifetime. In our antidot system, phase coherence\nis directly related to the formation of the self-interfering standing waves and quantization\nlevels. Experimentally, such coherence is partly reflected by the narrow width of the\ntunneling conductance peaks. However, several extrinsic factors can affect the peak width,\nincluding the finite tunneling probability to the charge reservoir and the finite electron\ntemperature which is often affected by hard-to-filter high frequency noises, etc.\nNevertheless, a rough estimate on the coherence may be obtained by modeling the\ntemperature -dependent tunneling conductance peak width.\nIt would be interesting to involve photons in the characterization of the QH antidot\nexcitons. Additionally, the pump-probe style measurements may provide time-domain\ninformation on the coherence of the excitonic states. The energy scale of the QH exciton is\nin the THz regime, and probing a single exciton may require a significantly low photon flux,\nand hence very sensitive photon detectors. So, this would be very challenging.\nNevertheless, timed-domain measurements are indeed very interesting and important\nfuture\n2. The conductance maps in Fig. 3(b) exhibit random shifts attributed to quantum dot\nlike charge traps near the antidot. While the authors note this issue and focus on clean\nregions of the data, a more systematic characterization of the impact of such traps on\nthe exciton resonances would strengthen the conclusions. Could the authors\ncomment on the device stability over time, hysteresis behavior, or temperature\ndependence of these random features? Do they anticipate that these fluctuations\ncould affect the coherence of the localized exciton?\nThe charge trap-associated shift originates from the Coulomb interaction between the\nremote charges and the antidot state. An intuitive way to see the impact of a charge trap is\nto treat them as a gate which abruptly changes its voltage whenever the charge occupation\nof the trap changes. Its sole impact is to shift the entire gate dependence of the antidot\ntunneling peaks by a small bit. This can be clearly seen from the fact that one can\n“manually” recover the continuity of the tunneling conductance peak traces by shifting the\ngating curves backward at each trap-induced jumps.\nFigure R4: “cut-and-slide” along the dotted lines where charge trapping occurs at fixed top\ngate values allows recovering the continuity of the conductance peak traces.\nSo, as long as the charging of the trap does not happen at the exciton resonance, it does\nnot affect the exciton state.\nAll the measurements were carried out at 0.3K, and throughout our experiment the sample\nwas kept below 10K. Over such temperatures we did not observe any time-dependent or\nhysteric behavior. At those temperatures, we expect the charged traps to be largely static\nand elastic. Hence, we do not expect their presence to significantly affect the coherence of\nthe exciton states.\nWe have added more detailed discussion to the SI to clarify the impact of the charge\ntrapping-induced shifts.\n3. The paper provides estimates for the coulomb interaction and tunneling\namplitudes, but the exciton binding energy relative to the thermal energy scale at the\nmeasurement temperature is not explicitly discussed. Could the authors comment on\nthe expected binding energy of the quantum Hall exciton in this system, and how\nrobust the bound state is against thermal fluctuations? Additionally it would be helpful\nto understand whether thermal activation might impact the visibility or\ncoherence of the excitonic tunneling features.\nThe binding energy of the QH antidot exciton can be defined as the energy required to\nseparate the electron and the hole in an exciton, so they no longer interact. From this\ndefinition we can calculate the exciton binding energy as the difference in the ground state\nenergies between non-interacting and interacting (including Coulomb attraction and\ntunneling) e-h pair: 𝐸 = 𝐸 (0,0)−𝐸 (𝑢,∆). This binding energy depends on the energy\n𝑏 𝑔 𝑔\nlevels of the electron and hole edge modes and thus is gate-tunable. It reaches maximum\nmagnitude when the two levels are aligned (in resonance) with 𝐸𝑚𝑎𝑥 = 𝑢 +√ 𝑢2 +Δ2 ,\n𝑏\n2 4\nwhere (as defined in the manuscript) 𝑢 is the Coulomb energy and ∆ is the tunneling energy\nbetween the electron and the hole edge states. From this definition, the magnitude of the\nbinding energy of the exciton demonstrated here is ~1.5meV. Thermal fluctuations and\nexcitations impact on the exciton state in several ways, depending on the antidot\nparameters; 1) Directly smearing the quantum coherent electron-hole resonance\ncharacterized by the level anti-crossing. The excitonic state observed in our work is a\nquantum superposition of the |00> and |11> state, with the energy lowered into the ground\nstate. Temperature 𝑇 larger than the tunnel coupling Δ would drive the system also into the\nexcited state destroying the quantum-coherent anti-crossing of the electron-hole levels. If\nthe temperature is still smaller than the electron-hole attraction energy 𝑢, the conductance\npeaks should still exhibit the excitonic shift before and after the resonance. 2) Destroying\nthe excitonic shift of the conductance peaks. Since a sufficiently large temperature\ndetermines the width of the conductance peaks, large temperature 𝑇 > 𝑢 should make the\nexcitonic shift of the conductance peaks invisible. 3) If 𝑇 is larger than the energy\nseparation ∆𝜀 of the original energy levels of the electron and hole edge modes, it would\nsmear these levels, hence destroying their coherence which builds the exciton. Without\ncoherence of the two edges, the quantum mechanical model used in this work would no\nlonger apply.\nWe have added part of this discussion on the possible effects of thermal excitations to the\nSupplementary Information.\n4. The manuscript discusses an interesting possibility of realizing excitonic states in\nthe fractional quantum Hall regime and exploring anyonic statistics within such\nsystems. This is a particularly exciting direction. Indeed very recently two papers\ndemonstrated the existence and control of such anyon excitonic states in bilayer\nGraphene samples in Corbino geometry. Could the authors comment in more detail on\nthe experimental feasibility of achieving this in their current platform? For instance, in\nFQH regime, there are several bilayer constructions possible for fractional excitons.\nWhat differences do the authors expect in the experimental feasibility for such\nfractional excitons?\nThe QH antidot-based excitons are very different from the bilayer excitons. In the bilayer\nexciton setup, excitons are formed by the inter-layer Coulomb interaction between positive\nand negative QH quasiparticles on the two separated QH layers, when the total filling of the\nlayers adds up to that of a QH ground state. The excitons are free on the bulk layers. Also,\nthe two layers are electrically insulated from each other so interlayer tunneling is\nprohibited. In the QH antidot-based exciton setup, both the positive and the negative QH\nquasiparticles are pre-confined on their antidot orbits. The pairing state of the two\nquasiparticles are formed via the intra-layer Coulomb interaction and the inter-orbit\ntunneling, creating a two-level-like quantum system. In FQHE, both quasiparticles would\nbe fractionally charged (as previously demonstrated by our group in suspended graphene,\nand in various experimental work in GaAs-based 2DES). It would be interesting to study the\nimpact of the fractional charge and the anyon exchange statics on the energy spectrum of\nsuch exciton states. This can be explored, for example, by tuning to the bulk filling inside\nand outside the antidot to the fractional fillings of the zeroth Landau levels.\nReply to Reviewer #1\nThe authors have satisfactorily answered all my questions. I have only one minor\ncomment – the responses to the queries have been very detailed and I found\nsome of the figures in the response letter to be illustrative and useful. Therefore, I\nsuggest that the authors add Fig. R1.5 and Fig. R1.6 in the SI, in a suitable\nmanner. The manuscript can be accepted for publication.\nWe thank the reviewer for the positive comments. We also thank the reviewer for the suggestion\non adding part of our reply to the SI. We have followed the reviewer’s suggestion and added Fig.\nR1.5 and Fig. R1.6 from our previous reply letter and corresponding discussions to the SI.\nReply to Reviewer #2\nI co-reviewed this manuscript with one of the reviewers who provided the listed\nreports. This is part of the Nature Communications initiative to facilitate training\nin peer review and to provide appropriate recognition for Early Career\nResearchers who co-review manuscripts.\nWe thank the reviewer for reviewing our manuscript.\nReply to Reviewer #3\nIn my previous report, I asked authors two questions. The first one was about the\nrepresentations of pair particles, whether they should use exciton (electron-hole)\npairs, or same-charge pairs (electron-electron or hole-hole). In their rebuttal letter,\nthe authors have made a convincing argument that the exciton representation\nfollowing the band structure is the most natural option. The authors have also\nmade clarification in the text with regard to the tunnelling term. My second\nquestion was about the demonstration of coherence. Towards this, the authors\nclaim that the observation of anti-crossing behaviour in Fig. 3 due to the tunnel\ncoupling between electron and hole states indicates quantum-coherent\ndynamics. The authors have added a new sentence to emphasise this point in the\nabstract. This modification has cleared my confusion (it sounded like the\nauthors claimed to have observed coherent oscillations in the original manuscript). In\nsummary, I am satisfied by the authors’ response and by their modification to the\nproposal.\nI recommend the publication of this manuscript in Nature Communications for the\nfollowing reasons. The authors’ device represents a unique system in which they\ncan tune the tunnelling coupling between electron-hole states. The device\nstructure is inventive and seemed to have been carefully designed. More\nrefinement in device quality and controllability would lead to further study of the\nexcitonic structure and their applications. The revision by the authors have made\nconsiderable improvement in the readability from the original manuscript. The\nsupplementary information includes sufficient information to support the\nconclusion of the main manuscript.\nWe thank the reviewer for the positive comments.\nReply to Reviewer #4\n1. Direct coherent measurements:\nThis is reasonable. The authors are transparent that coherence is inferred rather\nthan directly measured, but they tie the observed anti-crossings tightly to their\nmodel. I do not think the direct phase-sensitive measurements are essential for\npublication, and I find their explanations sufficient.\n2. Charge traps/device stability\nI thank the authors for the clarification - It removes my concerns that disorder\nfundamentally alters their conclusions. I am happy with their explanation.\n3. Exciton binding energy / thermal robustness\nGiven that 1.5 meV >> kT at 0.3 K, the excitons are indeed robust. No additional\nfollow-up clarifications are needed.\n4. Extension to the FQH regime\nI thank the authors for their convincing response. I had expected a more explicit\ndiscussion of the differences among various bilayer-like constructions of\nfractional excitons, beyond the contrast with antidot excitons. My only\nsuggestion is that it may benefit the reader if the authors elaborate further on\nhow factors such as statistics, confinement, or inter-edge spacing could\nconcretely influence excitonic properties.\nMy overall impression is that the authors' rebuttal is thorough and sound. They\nhave addressed my core criticisms, and I find their responses convincing. I\nbelieve the manuscript is now ready for publication, and I am happy to\nrecommend it for acceptance.\nWe thank the reviewer for the positive comments and the suggestions. Regarding extensionin the\nFQH regime, detailed theoretical understanding of the individual fractional excitons in the\nantidot structures similar to the one studied in our present work would require a dedicated theory\neffort. To respond to the referee’s request, we added to the manuscript a more detailed brief\nqualitative discussion of what the basic requirements are to observe such fractional excitons and\nwhat one can expect for the properties of these particles:\nThe QH antidot exciton scheme can potentially be extended to study the individual anyonic\nexcitons in the FQH regime by tuning the bulk filling inside and outside the antidot to the\nfractional fillings of the zeroth Landau level. This implements a coupled state of the two\nfractional quasiparticles with opposite charges. The interaction energy of such fractional excitons\ntogether with the tunnel coupling between the edge states of these quasiparticles would be\nsignificantly smaller than in the integer regime, requiring much lower temperatures to observe\nthe excitonic state. In the case of single antidot, which can accommodate at most one exciton at a\ntime, fractional exchange statistics may manifest itself indirectly, via the shape of the tunneling\nconductance peaks through the antidot. Beyond single antidot, potential direct observation of the\nexchange statistics may be possible through exciton transport in the inter-coupled multi-antidot\nstructures."}]

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