Observation of positronium diffraction

Domain: physics
Published: 2025-12-23
Editorial outcome: major_revision

Abstract
Diffraction of matter waves is a fundamental consequence of quantum mechanics, directly illustrating the core principles of wave-particle duality, quantum superposition, and quantum interference. De Broglie’s proposal that particles exhibit wave-like properties has been experimentally confirmed for electrons, neutrons, and composite systems such as helium atoms, molecules and clusters. Here, we report the observation of positronium diffraction using a high-quality, energy-tunable positronium beam transmitted through graphene. Time-of-flight selection and spatially resolved detection reveal a distinct 1st-order diffraction peak at a position consistent with the prediction from matter-wave considerations for positronium. This work provides the direct and definitive evidence of quantum interference in positronium beams and confirms that it behaves as a single quantum entity rather than two independent interfering particles. This groundbreaking experimental milestone marks a major advance in fundamental physics, not only demonstrating positronium’s wave nature as a bound lepton-antilepton system but also opening pathways for precision measurements involving positronium.

Introduction
Positronium (Ps) is the purely leptonic hydrogen-like bound state of an electron and a positron, making it a unique test bed for fundamental physics such as quantum electrodynamics, antimatter interaction and gravity1. Ps is the simplest atom composed of equal-mass constituents and, until it self-annihilates (142 ns lifetime for the triplet state), it behaves as a neutral atom in vacuum. Ps can also exist within insulators and may exhibit unique behaviour governed by quantum mechanics. For example, Ps in quartz (α-SiO2), as detected using angular correlation of annihilation radiation, can form Bloch states2,3 at room temperature, while in some alkali halides, this characteristic has been observed at low temperatures4.

To date, however, there has been no direct observation of Ps wave-like phenomena in free space. While positron matter-wave behaviour has been observed5,6, its manifestation in exotic composite systems such as Ps has remained largely unexplored. A diffraction or interference experiment with Ps would directly demonstrate its quantum superposition. Such an experiment also addresses whether Ps interferes as a single quantum object or whether its electron and positron constituents act as separate matter waves from observation of the position of the diffraction peak. The distinction is clear from the de Broglie wavelength relation, λ = h/p, where h is Planck’s constant and p is projectile momentum. Because Ps momentum is equal to the sum of the electron and positron momentum, its wavelength is half that of an isolated electron or positron. If the two constituents diffracted separately, the resulting diffraction pattern would exhibit an angular spread approximately twice as large as that expected for a bound quantum system due to the longer de Broglie wavelengths of the individual particles.

Diffraction is a powerful tool in crystal structure analysis. Notably, electrons and, more recently, positrons are used for surface structure determination via reflection high-energy diffraction7,8. Since positrons experience a repulsive potential inside a crystal, they undergo total external reflection at grazing incidence, allowing one to probe only the topmost atomic layers. Ps diffraction could similarly be used for non-destructive surface analysis9. While previous experimental studies using Ps beams, produced via neutralising slow positron beams in gaseous targets, explored mainly gas phase scattering10 and specular reflection of Ps from a LiF surface11, the use of Ps for solid surface diffraction will be an application. Since Ps is electrically neutral, its trajectory is not affected upon incidence at insulator surfaces, which may be charged or generate magnetic fields12—a unique advantage over charged particle beams. Though neutron and X-ray beams are similarly insensitive to external fields, Ps offers the added benefit of surface sensitivity, enabling the extraction of information specific to the outermost atomic layers of solids9. Moreover, positrons produced in β decay are spin-polarised; Ps beams generated using these positrons should inherit this polarisation13, opening opportunities for spin-resolved studies of magnetic surfaces. Beyond confirming its quantum properties, the observation of Ps interference opens avenues for fundamental measurements. Owing to its sensitivity, Ps interferometry has been proposed as a method for measuring the gravitational acceleration of antimatter and of purely leptonic systems14–16. Though of fundamental interest, no direct gravitational measurement has ever been performed on leptons—even electrons—as gravitational effects are many orders of magnitude weaker than the dominant Coulomb interaction. The matter-wave coherence of dense Ps is also a prerequisite for creating a Bose-Einstein condensate (BEC) of Ps17,18. Recent advances in Ps laser cooling19,20 begin to make it feasible to produce ultracold Ps, an important step towards Ps BEC.

Here we report on the direct observation of Ps diffraction obtained by passing Ps beam at energies of several keV through graphene. This success is due to the fact that the Ps beam used in this study, which is generated by photodetachment of accelerated Ps−, possesses sufficient coherence for diffraction observation.

Results
Production of a high-quality, energy-tunable Ps beam and preparation of graphene samples
A key requirement for success in Ps diffraction experimentation is the quality of the Ps beam. Here we employed a high-quality, energy-tunable Ps source, developed previously using laser photodetachment of Ps− ions21. Compared to Ps beams generated by the neutralisation of slow positrons in gas, this source delivers higher energies (up to 3.3 keV), a narrower energy spread—determined by photoelectron recoil to ~2%—and a reduced angular divergence of less than 0.3 degrees. The fact that Ps beams can be obtained in ultra-high vacuum is also an important feature to keep the graphene surface clean so that diffraction can be clearly observed. These characteristics were essential for enabling the observation of Ps diffraction. This beam has already been instrumental in a number of precision measurements, namely: hyperfine structure frequency by motion-induced transition22; electron affinity of Ps−23; anisotropic photodetachment of Ps−24; and measurements of Ps transmittance through graphene25.

In this study, positrons emitted from a 22Na source capsule were moderated by a solid neon moderator and collected in a Surko-type trap21. The positrons were then extracted from the trap as a pulsed slow positron beam and incident upon a tungsten film, where a fraction was emitted as Ps− ions from the Na-coated downstream surface. The experimental setup from this point is shown in Fig. 1. The ions were then accelerated to the required energies and photodetached by light pulses from a Nd:YAG laser synchronised with the Ps− pulses. To observe Ps diffraction, the resulting Ps atoms were directed onto a graphene sample. With a lattice constant of 246 pm, graphene is a suitable material with which to perform diffraction experiments in this study, where the Ps de Broglie wavelength is approximately 0.87 nm / E(eV). The sample, purchased from EM Japan Co. Ltd., consisted of nominally 2–3 layers of graphene of diameter 2.6 mm, supported by a lacey carbon film on a 300 mesh transmission electron microscopy (TEM) grid to ensure flatness and mechanical rigidity. This was fixed in a stainless steel holder and mounted on a transfer rod. In order to maintain the graphene in optimal condition, it was heated every few hours (see Methods). The component of the Ps beam transmitted through the graphene was detected using a position sensitive detector of active diameter 83 mm consisting of a microchannel plate (MCP) with a delay-line anode (Roentdek, DLD-80). The detector’s spatial calibration is described in Methods. The Ps count rates were ~0.02 s−1 at 3.3 keV and ~0.01 s−1 at 2.3 keV. The low rates arise from cumulative losses due to: (i) initial collimation by a 2.0 mm-diameter aperture immediately downstream of the source; (ii) further limitation by the 2.6 mm aperture in the graphene sample holder; (iii) a combined geometric transmittance of ~40% for the lacey carbon support and 300-mesh Cu grid; and (iv) an additional ~50% reduction in flux through the 2–3-layer graphene sample25.

In the present study, we analysed the data from the MCP detector, which included Ps atoms that had undergone diffraction by the graphene. However, to ensure the integrity of the diffraction signal, contributions from other particles and background noise had to be systematically eliminated. A fraction of the Ps atoms dissociated upon interaction with the graphene target, travelling towards the MCP as electrons and positrons each of ~1.65 keV when the incident Ps energy is 3.3 keV. Electrons thus produced were repelled by the application of a −2 kV electric potential to the front of the MCP, while a mesh plate positioned prior to the MCP, biased at +2 kV, removed the positrons. Additionally, significant background γ-ray emission arose from multiple sources. Positrons incident on the tungsten film could annihilate without forming Ps, generating prompt γ-rays. Further γ-ray contributions originated from self-annihilation of Ps atoms emitted from the tungsten surface, as well as pickoff annihilation occurring upon collisions with chamber electrodes. Although these high-energy photons interacted inefficiently with the MCP, their substantial numbers posed a risk of spurious signal contamination. Therefore, time-of-flight (TOF) selection, with the timing of the laser irradiation used as the start signal, was employed to discriminate γ-ray events from the Ps signals. To achieve the statistical precision required for clear identification of diffraction features, data acquisition times were 210 h for the 3.3 keV Ps and 370 h for the subsequent 2.3 keV measurement. Several operational constraints contributed to the overall duration of the experiment: the photodetachment laser system required overnight cooldown; the graphene sample accumulated surface adsorbates, necessitating in-situ cleaning every few hours; and the solid Ne moderator periodically degraded, requiring regeneration every few days.

Figure 2 presents the TOF spectrum recorded for a Ps beam energy of 3.3 keV. Two prominent peaks were observed at ~202 ns and 217 ns. When the Nd:YAG laser pulse was deactivated, the latter peak disappeared21, confirming that the first and second peaks correspond to γ-ray and Ps events, respectively.

Experimental results of Ps diffraction
Figure 3a shows the measured event density distribution ρ(r) as a function of radial distance r from the Ps beam centre, for a Ps energy of 3.3 keV, which corresponds to a de Broglie wavelength of 15 pm. The black dots represent events detected by the MCP within the TOF selection window of 215.5–219.5 ns, chosen to isolate Ps events as determined in Fig. 2. A distinct peak is observed near r ~ 8 mm. Arrows at 8.1 mm, 14.0 mm and 16.2 mm indicate the expected positions of the 1st-, 2nd- and 3rd-order Ps diffraction maxima, respectively, calculated using the de Broglie wavelength of Ps and the lattice spacing of graphene. The arrow at 16.2 mm also corresponds to the expected position of the 1st-order diffraction peak that would arise if the electron and positron comprising Ps diffracted independently. Thus, the presence of a prominent peak at 8.1 mm serves as direct evidence for Ps behaving as a single quantum entity, producing interference as a composite particle. In contrast, within the limits of statistical uncertainty in this study, no peaks are seen at either 14.0 mm or 16.2 mm. The absence of a peak at 14.0 mm suggests that 2nd-order Ps diffraction is not observed under the current conditions—unlike the case for electron diffraction, where higher-order peaks typically appear with similar relative intensities (see Methods). Furthermore, the absence of a peak at 16.2 mm confirms that the observed interference pattern is not due to individual diffraction of electrons or positrons, but rather from the coherent diffraction of the bound Ps system.

The Ps diffraction signal is superimposed on a background that includes contributions from γ-ray events originating upstream in the beamline. These γ-ray events were estimated independently using TOF windows of 210–212 ns and 222–224 ns, and are shown as blue triangles in Fig. 3a. For radial distances r > 4 mm, the γ-ray event distribution remains relatively uniform, indicating that it is not significantly dependent on position. Additional background may arise from Ps scattering off C atoms in the lace frame and from reflection and diffraction at the sample holder.

In order to isolate the Ps diffracted component, we fit the following equation, a sum of a Gaussian function representing the 1st-order diffracted Ps and a polynomial representing a smooth background, to the experimental data:
ρ(r) = A / sqrt(2πσ^2) * exp(-(r-μ)^2 / 2σ^2) + c1 / |r-r0|^n + c2*r + c3
with parameters of intensity A, peak position μ, peak width σ and parameters for the background c1, n, c2 and c3 applied over the radial range from 4 mm to 18 mm, where r0 = 18 mm. The thick red line in Fig. 3a shows the fitted curve. The reduced chi-squared value of the fit was χ2/ν = 11.5/14. The peak position was determined to be μ = 8.4 ± 0.4 mm, which is consistent with the calculated position of 1st order Ps diffraction r = 8.1 mm. The fitted peak width, σ = 0.74 ± 0.38 mm, corresponds to a full-width at half-maximum (FWHM) of 1.7 ± 0.9 mm, which is consistent with the measured Ps beam diameter of ~2.0 mm (FWHM). As the reduced χ2/ν gave a value slightly less than 1, the possibility of overfitting was considered. To test robustness against the background parameterisation, we repeated the fit with the background parameter n fixed to values about ± 25% from that at the best fit of n = 7.9, such that χ2/ν ~ 1. The inferred peak position changed by <4%, and the remaining parameters stayed within their uncertainties, indicating that the result is insensitive to the precise background model. The Ps diffraction efficiency at 3.3 keV, εPs_3.3keV, was determined to be 0.0032 ± 0.0018, with the uncertainty calculated from the fitting result. This value is approximately an order of magnitude lower than the electron diffraction efficiency εe of 0.045, measured at 1.6 keV under the equiveloctiy condition (see Methods).

When the Ps beam energy was decreased to 2.3 keV, the diffraction peak shifted outward, as shown in Fig. 3b. At this energy the de Broglie wavelength is 18 pm. Calculated positions of the 1st and 2nd-order Ps diffraction peaks are indicated by arrows at 9.7 mm and 17.0 mm, respectively. (The 3rd-peak is outside the measurement range in the current experimental setting.) The presence of the peak at 9.7 mm provides direct evidence of Ps diffraction at this Ps energy. No distinct 2nd-order diffraction peak was observed. The experimental data were fitted using the same composite function as described previously, yielding a reduced chi-squared value of χ2/ν = 9.0/14, indicating good agreement with the model. The fitted peak position was determined to be 10.0 ± 0.2 mm, consistent with the predicted 1st-order Ps diffraction position at 9.7 mm. Again χ2/ν was less than 1, therefore, we repeated the test for robustness by refitting with background parameter n fixed to values ± 35% from the best fit of n = 8.3, chosen to bring χ2/ν ~ 1. The peak position shifted by <3%, indicating insensitivity to the precise background parameterisation. The Ps diffraction efficiency at 2.3 keV, εPs_2.3keV, was determined to be 0.0029 ± 0.0005, which is of the same order as εPs_3.3keV. The obtained intensities of Ps diffraction are significantly smaller than that of electrons in the few keV energy region. In single-scattering theories, the diffracted signal scales with the elastic scattering amplitude. For positronium, however, the 1st Born elastic amplitude in a static potential cancels because the electron and positron have equal masses and opposite charges, so the leading non-zero elastic contribution arises only at higher order26. By contrast, inelastic channels such as internal excitation, breakup, or pickoff annihilation appear already at first order. These features naturally suppress the coherent elastic signal and provide a consistent explanation for the small Ps diffraction efficiency observed here. While recent studies have revealed similarities between electron and Ps scattering from molecules at comparable velocities in the range below a few 100 eV27, these findings have prompted ongoing theoretical investigations28. Further theoretical work is needed to interpret the reduced Ps diffraction efficiency observed here and to develop refined models of Ps-graphene interactions in the keV energy regime.

The present measurement was conducted using apparatus similar to that used in the work of Mikami et al.25. Whereas the previous work25 required auxiliary measurements to correct for a slight decrease in count rate in order to determine the absolute Ps transmittance, the objective here was a long-duration run that required no such correction. Thus, the data were acquired independently. In this study, we used a graphene thin film consisting of 2–3 layers rather than a monolayer. The diffraction intensity from graphene depends on the number of layers in the sample; consequently, using a monolayer leads to a corresponding reduction in peak intensities. As evident from the present data, halving the diffraction intensity would render the peak comparable to the statistical scatter. To perform the experiment with monolayer graphene, it will therefore be necessary to increase the Ps beam intensity further.

Methods
Electron diffraction
Prior to conducting the Ps diffraction experiment, electron diffraction patterns were observed in order to evaluate the graphene sample used and to obtain the data for comparison with the Ps diffraction data. The graphene sample was moved from the Ps beam axis using the transfer rod and repositioned in front of the electron gun, as shown in Fig. 1. The electron diffraction patterns obtained from the sample were observed on a phosphor screen and their profiles were measured using a CCD camera.

Figure 4a shows the CCD profile for an incident electron energy of 6 keV. The characteristic ring structure30 indicative of 2–3 layer graphene was observed in the obtained pattern. The corresponding radial density distribution is shown in Fig. 4b, where distinct 1st- and 2nd-order diffraction peaks were identified at approximately r ~ 5 mm and r ~ 8.5 mm from the beam centre, respectively.

To evaluate the condition of the graphene sample, we defined the electron diffraction efficiency as εe = Ie1 / Ie0, where Ie1 and Ie0 represent the intensities of the 1st-order diffraction peak and the total transmitted electron beam, respectively, and was calculated in the same manner as the Ps diffraction efficiency εPs. The red line in Fig. 4b is a fit to the data at the 1st order diffraction peak position using a Gaussian function with a smooth background of a quadratic polynomial. To determine Ie0, additional measurements were performed with a reduced CCD exposure time as the central portion of the diffraction pattern (r < 1 mm) exhibited saturation in Fig. 4b. The data were obtained by taking into account the CCD exposure times and the phosphor screen efficiency.

For incident electron of energy 6 keV, the measured efficiency, εe, for the 2–3 layer graphene sample was on the order of 1%, which was a factor of two higher than that observed for a single layer of graphene.

To further optimise the diffraction quality, a heat treatment was applied to the graphene sample using a CW laser with a wavelength of 976 nm and a power density of 70 mW/cm2. The laser was activated for 2 min to remove surface adsorbates while the graphene sample was rotated by 90 degrees using the transfer rod. Following this process, εe was noted to increase by a factor of 1.2–1.4 compared to its pre-treatment value. However, it was observed that after ~4 h εe returned to its original level, indicating a gradual reaccumulation of surface contaminants. To maintain optimal diffraction conditions, the heat treatment procedure was repeated every 3–4 hours throughout the Ps diffraction experiment.

Data for comparison with the results of Ps diffraction were measured soon after the heat treatment. For the electrons of energy 1.6 keV, the diffraction efficiency of the 1st-order diffraction peak was 0.045, and the intensity of the 2nd-order peak was of similar magnitude.

Position calibration of the MCP delay-line detector
The spatial calibration of the MCP delay-line detector was performed using a precision-engineered triangular-lattice array mask mounted directly in front of the MCP. The mask consisted of circular holes with a diameter of 0.25 mm, arranged in a regular triangular grid with a centre-to-centre spacing of 1.09 mm.

For the calibration, positrons from the Surko-type trap were injected directly into the MCP through the mask, producing a well-defined beam profile that replicated the array pattern of the mask. The resulting detected positions were used to determine the MCP detector’s spatial response. The x- and y-coordinate scales of the detector were calibrated using these reference data, with a position resolution of better than 0.2 mm. The mask was removed for the Ps beam experiments.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "This manuscript presents an experimental demonstration concerning\nan old and well studied topic, the diffraction of matter waves.\nThe actual specific interest of this current research is that the\nsystem under analysis is the positronium (Ps), the atom composed by\na positron and an electron. It is very important that the demonstration\nof a quantum interference is performed on Ps atoms, which are in fact\nunstable composite particles. On my knowledge, this is the first time\nthat quantum interference experiments concern unstable combinations\nof leptons and antileptons elementary particles, affected by an\nannihilation behavior. That this is a groundbreaking experimental\nadvance is correctly claimed in the title and in the abstract of the\nmanuscript.\nComing now to discuss about the core of the manuscript, it is\nto be noted that the writing is well conducted and extremely clear,\nstarting with a wide discussion on the usefulness of the diffraction\nad interference phenomena in various research fields, with hint also\non recent and future advances, as exposed in literature.\nThe description of the experimental apparatus is vivid and exhaustive,\nshowing how this difficult experiment is based on a suitable\nconstruction of a coherent Ps beam, a technology developed by this\nresearch group in recent years.\nThe analysis of the diffraction signal is performed by separating the\nbackground contribution of gamma ray events from the radial density\ndistribution of Ps events, with a suitable fitting model.\nHence the first order diffraction peak results well evident.\nMoreover, the radial position of the diffraction peak also demonstrates\nthat the quantum interference involves the whole atom and not the\nsingle components, as usefully discussed in the manuscript.\nJust a small comment: the diffraction experiment uses graphene samples\nwith 2 or 3 layers as gratings. For the purpose of helping the reader,\nit can be useful to write explicitly the lattice spacing of the\ngraphene sample, and add some information over the particular samples\nemployed and if it is possible to conduct similar experiments with\nsingle layer graphene.\nConcluding, in my opinion this manuscript reports about fundamental\naspects of quantum wavelike properties on a particular instable\nsystem, never analyzed before. Publication is warmly recommended.", "reviewer_2": "Review Report - Recommendation: Accept after minor corrections.\nThis manuscript reports the first demonstration of positronium (Ps) beam diffraction on few-layer graphene at kinetic energies\nof 3.3 keV and 2.3 keV. By employing a positron/electron-repelling MCP combined with a delay-line detector for spatially\nresolved detection, the authors present unambiguous evidence of Ps wave-like behaviour via de Broglie matter-wave\ndiffraction. The work identifies the first order diffraction peak for Ps, whereas in control measurements with electrons on the\nsame/similar graphene target at scaled energies at least two diffraction rings are observed. The contrast between Ps and\nelectron diffraction efficiency is attributed to a difference in scattering cross-sections, inviting for more detailed studies in the\nfuture. This nice experimental demonstration constitutes an important step in establishing Ps as a probe in quantum-optics-\nlike experiments and opens avenues for Ps interferometry. No similar experimental demonstration exists in the published\nliterature to my knowledge, making the work original and impactful.\nThe experimental execution seems solid. The authors are well known for their excellent Ps- source, the Ps beam\npreparation through photo-detachment, and the used detection scheme. The identification of the first diffraction maximum is\nwell supported by the quantitative analysis of the data which meets the expected standards in the field. The absence of\nhigher-order peaks in Ps compared with electron data is surprising, but it is not explicitly attributed to the low Ps scattering\ncross-section they found – see my detailed comments. The electron data shows an unexplained first peak that is not labeled\n“first peak” – again see my detailed comments.\nThe manuscript is clearly written, logically structured, and quite accessible to readers in the field, sometimes lacking a few\ndetails in my opinion. Prior work on Ps beams and on electron diffraction is appropriately cited, and the novelty of the\npresent contribution is evident. The references appear generally appropriate.\nDetailed comments:\np1, left, l8 Maybe add for clarity that Ps also forms at the surfaces of solids.\np1, right, l14 Since your matter wavelength depends on the velocity as well, I would appreciate to see a value for your Ps\nwavelength just for information.\np2, left, l43 Since you list the Ps beam properties, why not also state your current number of Ps atoms in the beam – it will\nalso fit later with your presented measurement time.\np2, right, l2 Can you give an example for why it is an important feature? Or do you mean it is important for your experiment?\np2, right, l10 Could you highlight the experimental differences to the measurement you did in [36] Ps transmittance through\ngraphene? I presume the choice of 2-3 layers of graphene in this work is based on the results of that publication?\np3, left, l29 I would remove that last sentence: “As a result […] it took two years”; 600h measurement time and maintenance\nis a tough experimental challenge, in particular to keep systematic drifts under control. It takes the time it takes.\np3, right, l2 Please mention in values the lattice spacings you used for the calculation, and if not before, at latest here report\nthe used matter wavelengths. My over the thumb calculation for Ps at 3.3keV finds the first maximum at 8.1 mm, the second\nat 14.1mm the third at 16.3mm.\np4, left, l4 and l31 The reduced chi-squared is consistently <1, I assume your model is overfitting a little. Is it possible to fix n\nin the fit?\np4, left, l20 Please state the new matter wavelength as well.\np4, right, l21 Can you use your found efficiencies as an explanation why you cannot see the second order diffraction signal?\nIf this is not the issue, what is your explanation that you do not see it?\np9, caption, g4 What is the smoothed background? Is this the outcome of the polynomial fit without the Gaussian? Please\nclarify.\nGeneral: Can you specify the type of your delay-line detector or give a reference in the methods section?\np10, left, l17 At what distance from the graphene, and at what energy (later you write that for Fig. 4b it was 1.6keGV not\n6keV). Please clarify.\np10, left, l29 Again the smooth background is a polynomial fit?\np10, left, l34 Typo: taking _into_ account\nFig. 4b There is an unexplained peak at around 2.5mm, whereas at 3rd order nothing is to be seen. I would like to know why\nthere is this first peak in your electron data and where it comes from. it seems significant enough to not be able to leave it\nwithout explanation.", "reviewer_3": "Dear Editor,\nI have enjoyed reading the article “First observation of positronium diffraction” by Nagata et al. This is a very hard experiment\nand will be of interest due to the subject matter being anti-matter. The carrying out of the experiment and the methods used\nare incontrovertible. However, in the wide field of anti-matter diffraction there is little that can be called new, except that it has\nnow been done with Positronium. This paper is probably out of date since the publication of “Diffraction of atomic matter\nwaves through a 2D crystal” by Kanitz et al., but it should definitely be published in a widely read journal like Physical\nReview.\nThe manuscript presents an interesting topic, namely wave particle duality, which is one of the key cruxes of quantum\nmechanics. The wave nature of electrons was empirically verified by the Davisson-Germer experiment in 1927. Wide\ninterest in the topic has led to similar\nconfirmations from matter constituents (neutrons, helium, larger molecules) scattering off of structures, potentials, crystals,\netc. as noted in the manuscript. It has even been shown that anti-matter (positron) has similar particle-wave nature in 2019\n(doi/10.1126/sciadv.aav7610).\nThe production of Ps from photo-ionizing the Positronium-minus ion is well understood, as are the detection methods. A\ngreat deal of time is needed in collecting statistically significant data, and as such, the results are not ambiguous and the\nconclusion incontrovertible. Details or a reference for the delay line detector should be given. I enjoyed reading this\nmanuscript, but I am left with the following thought: is there a newfound technique or analysis developed through the\nexperimental methods, or new knowledge resulting from this measurement, namely that -much like the hydrogen atom, a\nbare electron or a positron- Ps also has a wavelike nature as a bound state.\nDetailed comments\n• Line 5 through 8 on page one (left), the authors describe Ps as an atom and go on to say that Ps behaves as a neutral\natom; however, Ps acts like a neutral atom, however when Ps atoms move through a gas they scatter like a lone electron\n(Brawley et al. 2010 doi/10.1126/science.1192322]\n• Throughout the paper, the authors claim that the observation of Ps interference opens new avenues for fundamental\nmeasurements (see page 2 left line 25 or page 4 (right) line12). The authors should continue by highlighting some potential\nmeasurements. The authors could consider spin sensitive measurements with polarized positron production. Even so, how\nexactly does this Ps diffraction measurement open a new avenue, and what new avenue is there that the reader would have\nnot known before?\n• On page 4, line 45-46 (left) the authors note that their \"findings have prompted ongoing theoretical investigations\" but I\nwonder if there is any disagreement. It seems that the authors did not measure anything surprising.\n• In figure 3, is the increased signal between the 2nd and 3rd peaks explained by the detection of gamma rays?\n• In figure 4, there is a prominent peak before the first electron diffraction peak. This should be addressed.\n• Lastly, I am curious as to why the diffraction efficiency was orders of magnitude lower than the electron diffraction (see\npage 4 line 29-31 right). Maybe detailing the variance of graphene samples (e.g. lattice spacing) might be helpful.\nI thoroughly enjoyed reading this manuscript and I am happy that the authors were able to complete such a difficult\nmeasurement. Even though the results are neither ground breaking nor surprising, I believe that following slight revisions,\nin particular highlighting potential new experiments, this manuscript should be published in a widely read journal such as\nNature.", "reviewer_4": "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."}, "author_response": "Reply to referee's comments\nWe are pleased to note the favourable comments provided by the four referees. We\nthank them for their careful reading of our manuscript and for their valuable\nconstructive feedback. We have addressed all points in a detailed, point-by-point\nresponse and revised the manuscript accordingly.\nSincerely,\nYugo Nagata, Riki Mikami, Nazrene Zafar, Yasuyuki Nagashima\n------------------------------------------------------------------------------------------------------\nReviewer #1\nThis manuscript presents an experimental demonstration concerning an old and well\nstudied topic, the diffraction of matter waves. The actual specific interest of this current\nresearch is that the system under analysis is the positronium (Ps), the atom composed\nby a positron and an electron. It is very important that the demonstration\nof a quantum interference is performed on Ps atoms, which are in fact\nunstable composite particles. On my knowledge, this is the first time\nthat quantum interference experiments concern unstable combinations\nof leptons and antileptons elementary particles, affected by an\nannihilation behavior. That this is a groundbreaking experimental advance is correctly\nclaimed in the title and in the abstract of the manuscript.\nComing now to discuss about the core of the manuscript, it is to be noted that the writing\nis well conducted and extremely clear, starting with a wide discussion on the usefulness\nof the diffraction and interference phenomena in various research fields, with hint also\non recent and future advances, as exposed in literature. The description of the\nexperimental apparatus is vivid and exhaustive, showing how this difficult experiment\nis based on a suitable construction of a coherent Ps beam, a technology developed by this\nresearch group in recent years.\nThe analysis of the diffraction signal is performed by separating the background\ncontribution of gamma ray events from the radial density distribution of Ps events, with\na suitable fitting model. Hence the first order diffraction peak results well evident.\nMoreover, the radial position of the diffraction peak also demonstrates that the quantum\ninterference involves the whole atom and not the single components, as usefully\ndiscussed in the manuscript. Just a small comment: the diffraction experiment uses\ngraphene samples with 2 or 3 layers as gratings. For the purpose of helping the reader,\nit can be useful to write explicitly the lattice spacing of the graphene sample, and add\nsome information over the particular samples employed and if it is possible to conduct\nsimilar experiments with single layer graphene.\nConcluding, in my opinion this manuscript reports about fundamental aspects of\nquantum wavelike properties on a particular instable system, never analyzed before.\nPublication is warmly recommended.\nReply: We thank the referee for the favourable assessment, and for recognising the\nsignificance of this work for fundamental physics.\nComment 1) For the purpose of helping the reader, it can be useful to write explicitly the\nlattice spacing of the graphene sample, and add some information over the particular\nsamples employed and\nReply: Lattice constant of graphene is 246 pm.\nIn page 2, right, line 37, we inserted the lattice spacing of 246 pm.\nComment 2) if it is possible to conduct similar experiments with single layer graphene.\nReply: The diffraction intensity of graphene depends on the number of layers of graphene\nin the sample used. Consequently, employing monolayer graphene results in a\ncorresponding decrease in the intensities of the diffraction peaks. As evident from the\npresent data, if the diffraction intensity halves, the peak would become obscured by\nstatistical scatter. Indeed, measurements using monolayer graphene yielded data where\nthe diffraction peak was not clearly discernible.\nWe added the following sentence in page 5, left, line 16 as,\n“In this study, we used a graphene thin film consisting of 2–3 layers rather than a\nmonolayer. The diffraction intensity from graphene depends on the number of layers in\nthe sample; consequently, using a monolayer leads to a corresponding reduction in peak\nintensities. As evident from the present data, halving the diffraction intensity would\nrender the peak comparable to the statistical scatter. To perform the experiment with\nmonolayer graphene, it will therefore be necessary to increase the Ps beam intensity\nfurther.”\nReviewer #2\nThis manuscript reports the first demonstration of positronium (Ps) beam diffraction on\nfew-layer graphene at kinetic energies of 3.3 keV and 2.3 keV. By employing a\npositron/electron-repelling MCP combined with a delay-line detector for spatially\nresolved detection, the authors present unambiguous evidence of Ps wave-like behaviour\nvia de Broglie matter-wave diffraction. The work identifies the first order diffraction\npeak for Ps, whereas in control measurements with electrons on the same/similar\ngraphene target at scaled energies at least two diffraction rings are observed. The\ncontrast between Ps and electron diffraction efficiency is attributed to a difference in\nscattering cross-sections, inviting for more detailed studies in the future. This nice\nexperimental demonstration constitutes an important step in establishing Ps as a probe\nin quantum-optics-like experiments and opens avenues for Ps interferometry. No similar\nexperimental demonstration exists in the published literature to my knowledge, making\nthe work original and impactful.\nThe experimental execution seems solid. The authors are well known for their excellent\nPs- source, the Ps beam preparation through photo-detachment, and the used detection\nscheme. The identification of the first diffraction maximum is well supported by the\nquantitative analysis of the data which meets the expected standards in the field. The\nabsence of higher-order peaks in Ps compared with electron data is surprising, but it is\nnot explicitly attributed to the low Ps scattering cross-section they found –see my\ndetailed comments. The electron data shows an unexplained first peak that is not labeled\n“first peak” – again see my detailed comments. The manuscript is clearly written,\nlogically structured, and quite accessible to readers in the field, sometimes lacking a few\ndetails in my opinion. Prior work on Ps beams and on electron diffraction is appropriately\ncited, and the novelty of the present contribution is evident. The references appear\ngenerally appropriate.\nReply: We thank the referee for the favourable assessment, and for recognising the\nsignificance of this work for fundamental physics.\nComment 1) p1, left, l8 Maybe add for clarity that Ps also forms at the surfaces of solids.\nReply: Thank you for the helpful suggestion. Our intent here was to give a flavour of the\nstates in which Ps exists, rather than the details of its production.\nIn page 1, left, line 7, we have rewritten this sentence as “it behaves as a neutral atom\nin vacuum. Ps can also exist within insulators and may exhibit unique behaviour\ngoverned by quantum mechanics.”\nComment 2) p1, right, l14 Since your matter wavelength depends on the velocity as well,\nI would appreciate to see a value for your Ps wavelength just for information.\nReply: In page 2, right, line 36, we added “With a lattice constant of 246 pm, graphene\nis a suitable material with which to perform diffraction experiments in this study, where\nthe Ps de Broglie wavelength is approximately 0.87 nm/√E/eV. “\nComment 3) p2, left, l43 Since you list the Ps beam properties, why not also state your\ncurrent number of Ps atoms in the beam – it will also fit later with your presented\nmeasurement time.\nReply: The Ps beam count rates were 0.02 cps at 3.3 keV and 0.01 cps at 2.3 keV. These\nlow values are due to: (i) the Ps beam being collimated through a 2 mm diameter aperture\nimmediately after the generation, (ii) being further collimated by the holder of the\ngraphene sample having a diameter of 2.6 mm, (iii) the combined geometrical\ntransmittance of the lacey carbon and Cu grid being approximately 40% and (iv) the\nnumber of Ps being reduced by about half in 2–3-layer graphene (Mikami et al., Eur.\nPhys. J. D 77, 205 (2024)). Please note that as the MCP detection efficiency has not been\nmeasured, we confine the metric to count rate rather than the Ps beam intensity.\nIn page 3, left, line 9, we added “The Ps count rates were ~0.02 s⁻¹ at 3.3 keV and ~0.01\ns⁻¹ at 2.3 keV. The low rates arise from cumulative losses due to : (i) initial collimation\nby a 2.0 mm-diameter aperture immediately downstream of the source; (ii) further\nlimitation by the 2.6 mm aperture in the graphene sample holder; (iii) a combined\ngeometric transmittance of ~40% for the lacey carbon support and 300-mesh Cu grid;\nand (iv) an additional ~50% reduction in flux through the 2–3-layer graphene sample\n[30].”\nComment 4) p2, right, l2 Can you give an example for why it is an important feature? Or\ndo you mean it is important for your experiment?\nReply:\nThe vacuum is important for this experiment and, more generally, for Ps diffraction\nfrom 2D crystals. Ultra-high vacuum is required to suppress the adsorption of\ncontaminants on the graphene surface; otherwise adsorbates attenuate or blur the\ndiffraction peaks and alter the surface potential, compromising the measurement.”\nIn page 2, right, line 15, we added as “to keep the graphene surface clean so that\ndiffraction can be clearly observed.”\nComment 5) p2, right, l10 Could you highlight the experimental differences to the\nmeasurement you did in [36] Ps transmittance through graphene? I presume the choice\nof 2-3 layers of graphene in this work is based on the results of that publication?\nReply: The Ps beam intensity gradually decreases over time during the day. For this\nreason, in the transmittance experiment [36], it is necessary to measure data under\nvarious conditions of 1-layer, 2-layer, 3-5 layer, 6-8 layer graphene and no sample\nalternately in a short period of time in order to normalise by no sample data correctly.\nOn the other hand, we measured only under the one condition of the 2-3 layer graphene\nin the diffraction experiment as it did not require us to normalize the data in the same\nway. We chose 2–3-layer graphene for the diffraction experiment informed by [36]: it\nprovides significantly higher diffracted intensity than a monolayer (while keeping\nmultiple scattering and attenuation modest), and it is mechanically more robust as a\nfreestanding membrane. Using a monolayer would reduce the peak heights sufficiently\nthat, at the current beam intensity, the first-order ring would be comparable to statistical\nnoise.”\nIn page 5, left, line 10, we inserted the following sentence as, “The present measurement\nwas conducted using apparatus similar to [30]. Whereas [30] required auxiliary\nmeasurements to correct for a slight decrease in count rate in order to determine the\nabsolute Ps transmittance, the objective here was a long-duration run that required no\nsuch correction. Thus, the data were acquired independently. In this study, we used a\ngraphene thin film consisting of 2–3 layers rather than a monolayer. The diffraction\nintensity from graphene depends on the number of layers in the sample; consequently,\nusing a monolayer leads to a corresponding reduction in peak intensities. As evident\nfrom the present data, halving the diffraction intensity would render the peak\ncomparable to the statistical scatter. To perform the experiment with monolayer\ngraphene, it will therefore be necessary to increase the Ps beam intensity further.”\nComment 6) p3, left, l29 I would remove that last sentence: “As a result […] it took two\nyears”; 600h measurement time and maintenance is a tough experimental challenge, in\nparticular to keep systematic drifts under control. It takes the time it takes.\nReply: We removed “As a result […] it took two years”.\nComment 7) p3, right, l2 Please mention in values the lattice spacings you used for the\ncalculation, and if not before, at latest here report the used matter wavelengths. My over\nthe thumb calculation for Ps at 3.3keV finds the first maximum at 8.1 mm, the second\nat 14.1mm the third at 16.3mm.\nReply: In page 2, right, line 36, we added “With a lattice constant of 246 pm, graphene\nis a suitable material with which to perform diffraction experiments in this study, where\nthe Ps de Broglie wavelength is approximately 0.87 nm /√E /eV.”\nWe also added “which corresponds to a de Broglie wavelength of 15pm.” in page 3, right,\nline 20.\nComment 8) p4, left, l4 and l31 The reduced chi-squared is consistently <1, I assume\nyour model is overfitting a little. Is it possible to fix n in the fit?\nReply:\nWe thank the referee for raising this point. To test whether the reduced chi-squared\nvalues ( ) below unity indicate over-fitting, we repeated the fits while fixing the\n2\nbackground-shape parameter to a series of values around the best-fit, and examined\n𝜒𝜒 /𝜈𝜈\nboth and the fitted first-order peak position.\n𝑛𝑛\n2\n𝜒𝜒 /𝜈𝜈\nFigures (a) and (c) show results for the 3.3 keV data. In (a), the circles give (with\n2\n= 15) as a function of fixed the square marks the value obtained when is allowed\n𝜒𝜒 /𝜈𝜈\nto float (best-fit =7.9; in the manuscript this appears as = 11.5/14—see\n𝜈𝜈 𝑛𝑛; 𝑛𝑛\n2\ncorrection below). As is moved away from 7.9, increases and exceeds 1 at both\n𝑛𝑛 𝜒𝜒 /𝜈𝜈\n2\nends of the scan, as expected if the floating- solution is preferred. In (c), the\n𝑛𝑛 𝜒𝜒 /𝜈𝜈\ncorresponding first-order peak position shows only a weak dependence on , varying by\n𝑛𝑛\n< 4% across the scanned range.\n𝑛𝑛\nFigures (b) and (d) present the same test for 2.3 keV. The behaviour is similar:\n2\nrises toward and above unity as is displaced from the best-fit value (8.3), while the\n𝜒𝜒 /𝜈𝜈\npeak position changes by < 3%. These checks indicate that, although is somewhat\n𝑛𝑛\n2\nbelow 1 when floats, the physical observables are insensitive to the precise\n𝜒𝜒 /𝜈𝜈\nbackground parameterisation, and there is no evidence that over-fitting biases the\n𝑛𝑛\npeak location.\nWe also correct a typographical error in the degrees of freedom: = 14 (not 13).\nAccordingly, the values quoted in the manuscript should read = 11.5/14 (p. 4, left,\n𝜈𝜈\n2\nl. 17) and = 9.0/14 (p. 4, right, l. 13).\n𝜒𝜒 /𝜈𝜈\n2\n𝜒𝜒 /𝜈𝜈\nIn page 4, left, line 33, we have added the following clarifying sentences: “As the reduced\ngave a value slightly less than 1, the possibility of overfitting was considered. To test\n2\nrobustness against the background parameterisation, we repeated the fit with the background\n𝜒𝜒 /𝜈𝜈\nparameter n fixed to values about ±25% from that at the best-fit of n=7.9 such that ~ 1. The\n2\ninferred peak position changed by <4%, and the remaining parameters stayed within their\n𝜒𝜒 /𝜈𝜈\nuncertainties, indicating that the result is insensitive to the precise background model.”\nIn page 4, right, line 25, we added the following sentence as,\n“Again was less that 1, therefore we repeated the test for robustness by refitting with\n2\nbackground parameter ±35% from the best-fit of ≈\n𝜒𝜒 /𝜈𝜈\n2\n1. The peak position shifted by <3%, indicating insensitivity to the precise background\n𝑛𝑛 fixed to values 𝑛𝑛=8.3, chosen to bring 𝜒𝜒 /𝜈𝜈\nparameterisation.”\nComment 9) p4, left, l20 Please state the new matter wavelength as well.\nReply:\nIn page 4, right, line 11, we added “At this energy, the de Broglie wavelength was 18\npm.”.\nComment 10) p4, right, l21 Can you use your found efficiencies as an explanation why\nyou cannot see the second order diffraction signal? If this is not the issue, what is your\nexplanation that you do not see it?\nReply:\nThank you for your insightful comment. On its own, the measured first-order Ps\ndiffraction efficiency does not determine the second-order efficiency, because the relative\norder intensities depend on the (unknown) Ps elastic amplitudes. However, if we\nassume—as in electron diffraction—that the second-to-first order intensity ratio is ~0.7\n(see Shevitski et al., Phys. Rev. B 87, 045417 (2013)), then the event density at the ring\nposition scales as the diffraction efficiency divided by the area of 2pi r dr. Therefore, the\nevent density for the 2nd order diffraction decreases by 0.7 * 8.1/14.1=0.4, so the expected\nsecond-order peak height would be only ~40% of the first-order peak in the ρ(r) plot.\nBecause the γ-ray background is approximately uniform for r > 4 mm, this reduced signal\nis comparable to the background fluctuations at r ≈ 14 mm, making the second-order\npeak statistically difficult to resolve with the present data set. Any additional\nsuppression of Ps elastic scattering relative to electrons would further decrease the\nvisibility of the second order.\nComment 11) p9, caption, g4 What is the smoothed background? Is this the outcome of\nthe polynomial fit without the Gaussian? Please clarify.\nReply: Yes, the smoothed background is a polynomial fit without the Gaussian.\nIn page11, caption, g4, we added as ”which is a polynomial fit without the Gaussian”.\nComment 12) General: Can you specify the type of your delay-line detector or give a\nreference in the methods section?\nReply:\nWe used a DLD80 (Roentdek). In page 3, left, line7, we added “(Roentdek, DLD-80)”.\nComment 13) p10, left, l17 At what distance from the graphene, and at what energy\n(later you write that for Fig. 4b it was 1.6keV not 6keV). Please clarify.\nReply: Thank you for pointing this out. Fig. 4a and 4b both correspond to the 6 keV\nelectron data taken with the same sample and geometry. We chose to present the figures\nbecause the electron-gun brightness at lower energies was insufficient to produce as\nclean a pattern and peak integration.\nThe 1.6 keV value mentioned in the text refers to a separate measurement used solely\nfor the equal-velocity comparison with 3.3 keV Ps; it is not the dataset shown in Fig. 4.\nComment 14) p10, left, l29 Again the smooth background is a polynomial fi t?\nReply: Yes, we used a quadratic polynomial for the smoothed background.\nIn page 6, left, 5, we added as “using a Gaussian function with a smooth background of\na quadratic polynomial.”\nComment 15) p10, left, l34 Typo: taking _into_ account\nReply: Thank you very much. We fixed it.\nComment 16) Fig. 4b There is an unexplained peak at around 2.5mm, whereas at 3rd\norder nothing is to be seen. I would like to know why there is this first peak in your\nelectron data and where it comes from. it seems significant enough to not be able to leave\nit without explanation.\nReply: Thank you for drawing attention to these features.\nA third-order ring is expected but its intensity is weak for our thickness and electron\nenergy, and its radius lies close to that of the second order. In the present geometry\nthese two orders partially overlap, so the third order is not resolved as a separate\nmaximum.\nThe small shoulder at ∼2.5 mm is not a lattice reflection. It is an instrumental artefact\nof the electron gun, as the same feature appears in calibration runs without any sample\ninserted.\nWe added the following in the figure 4 caption:“The small shoulder preceding the first\ndiffraction peak is a characteristic of the electron gun, as it appears even in the absence\nof a sample.”\nReviewer #3 (Remarks to the Author):\nDear Editor,\nI have enjoyed reading the article “First observation of positronium diffraction” by\nNagata et al. This is a very hard experiment and will be of interest due to the subject\nmatter being anti-matter. The carrying out of the experiment and the methods used\nare incontrovertible. However, in the wide field of anti-matter diffraction there is little\nthat can be called new, except that it has now been done with Positronium. This paper is\nprobably out of date since the publication of “Diffraction of atomic matter waves through\na 2D crystal” by Kanitz et al., but it should definitely be published in a widely read\njournal like Physical Review.\nReply:\nThank you for your encouraging comments. We appreciate your recognition of the\nexperimental difficulty and your positive assessment of our methodology.\nRegarding novelty, our central contribution is not merely that “it has now been done\nwith positronium,” but that we provide the first direct demonstration of quantum\ninterference for a lepton–antilepton bound atom. Positronium is qualitatively different\nfrom ordinary atoms used in prior diffraction studies (including Kanitz et al.): it is\nneutral antimatter, unstable (142 ns triplet lifetime), and its elastic amplitude in first-\nBorn approximation cancels due to the equal masses and opposite charges of its\nconstituents. This leads to a strongly suppressed coherent signal and makes diffraction\nof Ps uniquely challenging. Observing a clear first-order peak therefore establishes\nthat Ps interferes as a single quantum entity, rather than as independently diffracting\nelectron and positron—an issue with no analogue in standard atomic beams. It also\nprovides the first quantitative constraint on Ps–graphene elastic scattering in the keV\nregime, relevant for Ps interferometry and gravity tests.\nConcerning Kanitz et al., we agree it is an important advance in ordinary-matter atom\ndiffraction through 2D crystals. The subject of our paper is the interference of matter\nwaves in a bound system of particles and antiparticles, rather than of ordinary matter\natoms. This has truly fundamental importance and requires experimental verification\nOur experimental concept is easily understood and the relationship between particles\nand antiparticles is the subject that may interest many readers. We are confident that\nthis paper is suitable to Nature Communications.\nReply: We thank the referee for their favourable assessment, and for recognising the\nsignificance of this work for fundamental physics.\nComment 1) Line 5 through 8 on page one (left), the authors describe Ps as an atom and\ngo on to say that Ps behaves as a neutral atom; however, Ps acts like a neutral atom,\nhowever when Ps atoms move through a gas they scatter like a lone electron\n(Brawley et al. 2010 doi/10.1126/science.1192322]\nReply: Thank you for this helpful clarification. We agree that, at low collision energies,\nPs–molecule scattering cross sections closely follow those for electron impact at equal\nprojectile velocity (Brawley et al., Science 330, 789–792, 2010). This important\nobservation does not imply that Ps ceases to be an atom; it reflects similarities in the\ninteraction at low energies and does not contradict the fact that Ps is a neutral bound\ne e atom whose centre-of-mass wave can diffract as a single quantum entity. The\nsentence on p.1 was intended to emphasise Ps’s neutrality and field insensitivity in\n⁺ ⁻\nvacuum, and not to make a statement about gas-phase scattering. To avoid ambiguity\nwe have revised it to: “it behaves as a neutral atom in vacuum.” in page 1, left, line 7.\nComment 2) Throughout the paper, the authors claim that the observation of Ps\ninterference opens new avenues for fundamental measurements (see page 2 left line 25\nor page 4 (right) line12). The authors should continue by highlighting some potential\nmeasurements. The authors could consider spin sensitive measurements with polarized\npositron production.\nReply: Thank you—we agree and have expanded revised the text to note the feasibility\nof spin-resolved surface studies using polarised Ps derived from beta decay positrons.\nIn page 2, left, line 24, we added as “Moreover, positrons produced in β decay are spin-\npolarised; Ps beams generated using these positrons should inherit this polarisation [13],\nopening opportunities for spin-resolved studies of magnetic surfaces.”\nComment 3)Even so, how exactly does this Ps diffraction measurement open a new\navenue, and what new avenue is there that the reader would have not known before?\nReply: Thank you for your comment. First, the feasibility of coherent Ps interference is\nan experimental question—non-trivial for a short-lived, annihilating, lepton–antilepton\natom. Our measurement demonstrates that Ps does diffract as a single quantum entity,\nthereby enabling interferometric schemes that were previously only proposed.\nSecond, the data show that the elastic (coherent) signal is strongly suppressed relative\nto electrons at equal velocity, implying different scattering amplitudes for Ps–crystal\ninteractions. This quantitative insight is actionable: it sets requirements for beam\nintensity, sample thickness, and background control in planned Ps interferometers.\nWith these two points, the work opens concrete paths that a reader would not have been\nable to assume a priori: (i) spin-resolved Ps interferometry (using polarised β-decay\npositrons) for magnetic-surface studies; (ii) Mach–Zehnder Ps interferometers for\nprecision tests of gravity and short-range forces; and (iii) surface metrology with neutral\nantimatter, exploiting Ps’s intrinsic surface sensitivity in UHV. Future designs must\naccount for the observed Ps specific scattering properties revealed by the present\ndiffraction measurement.\nComment 4) On page 4, line 45-46 (left) the authors note that their \"findings have\nprompted ongoing theoretical investigations\" but I wonder if there is any disagreement.\nIt seems that the authors did not measure anything surprising.\nReply: Thank you—we did not intend to imply any disagreement with established theory.\nRather, the relevant theory is incomplete: most Ps–scattering work treats molecules\nbelow a few hundred eV, whereas keV-energy Ps interacting with crystalline solids (and\nPs diffraction from 2D lattices) lacks a predictive framework. In this regime the first-\nBorn elastic amplitude cancels, so the diffracted intensity depends on higher-order terms\nand on the competition with inelastic channels—issues that have not yet been quantified.\nComment 5) In figure 3, is the increased signal between the 2nd and 3rd peaks explained\nby the detection of gamma rays?\nReply: Thank you for the question. In Fig. 3, the apparent excess between the nominal\nsecond- and third-order radii lies within the 1–2 σ scatter of neighbouring bins. Our TOF\nsidebands indicate a nearly flat -ray background for >4 mm; together with a small\ncontribution from the undiffracted/forward-scattered Ps tail, this accounts for the\n𝛾𝛾 𝑟𝑟\nobserved level. An unresolved second-order component cannot be excluded, but with the\npresent statistics it cannot be distinguished from the background.\nComment 6) In figure 4, there is a prominent peak before the first electron diffraction\npeak. This should be addressed.\nReply: Thank you for pointing this out. This feature is not a lattice reflection: it appears\nwith the sample removed, so it originates from the incident electron beam (gun/optics)\nrather than graphene. We have noted this explicitly in the text and caption (Fig. 4).\nWe added the following in the figure 4 caption:“The small shoulder preceding the first\ndiffraction peak is a characteristic of the electron gun, as it appears even in the absence\nof a sample.”\nComment 7) Lastly, I am curious as to why the diffraction efficiency was orders of\nmagnitude lower than the electron diffraction (see page 4 line 29-31 right). Maybe detailing\nthe variance of graphene samples (e.g. lattice spacing) might be helpful.\nReply: The key difference from electrons is the elastic scattering amplitude. In the keV\nregime one normally applies first-Born (kinematic) diffraction theory. For positronium\nin a static potential, however, the first-Born elastic amplitude cancels because the\nelectron and positron have equal masses and opposite charges; the leading non-zero\nelastic term therefore arises only at second order (or higher) [H. Ray, Phys. Lett. A 252,\n316 (1999)]. At the same time, inelastic channels (break-up, internal excitation, pickoff\nannihilation) enter already at first order, depleting the coherent flux. These two effects\nnaturally suppress the diffracted intensity for Ps by orders of magnitude relative to\nelectrons at equal velocity.\nRegarding sample variance: the graphene lattice constant is essentially fixed (246 pm)\nand does not vary appreciably between our membranes; it sets the ring positions but\ndoes not explain the efficiency difference. What can affect the intensity are thickness\n(number of layers), cleanliness, and support-grid open fraction. We therefore used 2–3-\nlayer graphene (as informed by Ref. [36]) to improve the signal while keeping multiple\nscattering modest.\nIn page 4, right, line 36, we added the following sentences as,\n“In single-scattering theories, the diffracted signal scales with the elastic scattering\namplitude. For positronium, however, the 1st Born elastic amplitude in a static potential cancels\nbecause the electron and positron have equal masses and opposite charges, so the leading non-zero\nelastic contribution arises only at higher order [31]. By contrast, inelastic channels such as internal\nexcitation, breakup, or pickoff annihilation appear already at first order. These features naturally\nsuppress the coherent elastic signal and provide a consistent explanation for the small Ps diffraction\nefficiency observed here.”\nIn page 2, right, line 37, we inserted the lattice spacing of 246 pm."}]

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