Interference of matter waves is at the heart of quantum physics and has been observed for a wide range of particles from electrons to complex molecules. Here, we demonstrate matter wave interference of single positrons using a period-magnifying Talbot-Lau interferometer based on material diffraction gratings. The system produced high-contrast periodic fringes, which were detected by means of nuclear emulsions capable of determining the impact point of each individual positron with submicrometric resolution. The measured energy dependence of fringe contrast in the range of 8 to 16 keV proves the quantum-mechanical origin of the periodic pattern and excludes classical projective effects, providing the first observation to date of antimatter wave interference. Future applications of this interferometric technique include the measurement of the gravitational acceleration of neutral antimatter systems exploiting the inertial sensing capabilities of Talbot-Lau interference.

In 1923, de Broglie ( 1 , 2 ) introduced the concept of wave-particle duality: the Planck constant h relates the momentum p of a massive particle to its de Broglie wavelength λ dB = h/p. The superposition principle is one of the main postulates of quantum mechanics; diffraction and interference phenomena are, therefore, predicted and have been observed on objects of increasing complexity, from electrons ( 3 , 4 ) to neutrons ( 5 , 6 ) and molecules ( 7 – 9 ). Beyond the early electron diffraction experiments ( 3 , 4 ), the demonstration of single-electron double-slit–like interference was a highly sought-after result. Initially proposed by Feynman as a thought experiment, it was last carried out in 1976 ( 10 ). A few years later, positron diffraction was first observed ( 11 ). However, an analog of the double-slit experiment has not been performed to date on any system containing antimatter. To bridge this gap, we designed and realized a Talbot-Lau interferometer ( 12 ) suited to a low-energy positron beam. This interferometric technique could also lead to studies of fundamental interest on neutral antimatter systems. For instance, several techniques are being considered to undertake the measurement of the gravitational acceleration of positronium, muonium, and antihydrogen ( 13 – 20 ). Talbot-Lau–based inertial sensing ( 21 ) offers the capability to work with low-intensity, weakly coherent beams similar to the one used in this work. This is a highly desirable feature for beam-based experiments with antimatter.

RESULTS

The experiment makes use of the variable energy positron beam facility of the L-NESS (Laboratory for Nanostructure Epitaxy and Spintronics on Silicon) in Como (Italy). Positrons (e+) from the beta decay of a 22Na radioactive source are implanted on a monocrystalline tungsten film and emitted with a kinetic energy of about 3 eV, determined by the work function of the material (22). Slow positrons are then accelerated up to 16 keV by means of a purely electrostatic system. A monochromatic and continuous beam is thus formed with an energy spread less than 0.1%, limited by the stability of the power supplies. The positron rate is (5 ± 1) × 103 e+/s, and beam focusing can be tuned to reach a spot size of the order of several millimeters of full width at half maximum (FWHM), an approximately Gaussian intensity profile (23) and an angular divergence at the level of a few milliradians (24). A suitable model (25) of grating-based interferometry with a partially coherent beam exploits the analogy with Gaussian Schell-model beams of classical optics (Materials and Methods). Within this formalism, incoherence translates physically into a broad transverse momentum distribution and mathematically to a short transverse coherence length l. The L-NESS beam features a coherence length of the order of a few nanometers. An effective configuration for these conditions was obtained with a novel period-magnifying two-grating interferometer (21, 26). It exploits an intermediate working regime between the standard Talbot-Lau setup, where the two gratings and the detector are equally spaced, and the so-called Lau interferometer (27), which has more stringent coherence requirements (25). By means of unequal grating periodicities, the system provides sizable period magnification in a relatively compact setup (21). In particular, we used gold-coated 700-nm-thick silicon nitride (SiN) gratings with periodicity d 1 = (1.210 ± 0.001) μm and d 2 = (1.004 ± 0.001) μm to produce a d 3 = (5.90 ± 0.04) μm periodic interference pattern. Both gratings have a nominal open fraction of 50%.

The periodic spatial distribution generated by the interferometer (Fig. 1) is revealed by a nuclear emulsion detector. Nuclear emulsions (28) offer submicrometer-level position resolution in the detection of ionizing particles (24, 29). They work as photographic films by exploiting the properties of silver bromide crystals embedded in a 50-μm-thick gelatin matrix. For this experiment, we developed a glass-supported emulsion detector (fig. S1) and experimentally demonstrated its capability to resolve periodic patterns at the micrometric scale even with low signal contrast and on large areas (24).

Fig. 1 Schematics of the Talbot-Lau interferometer. Positrons traverse two circular 2-mm-wide collimators 10.2 cm apart. The interferometer is composed of two SiN diffraction gratings with periodicity d 1 and d 2 , respectively, separated by L 1 = (118.1 ± 0.2) mm. Interference fringes with d 3 periodicity are expected at L 2 = (576 ± 5) mm. The emulsion is tilted so that the Y axis in the reference frame of the emulsion surface (X, Y) forms a 45° angle with the y axis of the laboratory. Gamma rays (511 keV) from positron annihilation in the emulsion are monitored with a high-purity germanium (HpGe) detector for rate measurement.

Interferometer alignment is particularly challenging, and it is intimately connected to beam coherence. The Talbot-Lau interferometer can produce high-contrast fringes if the resonance condition (21) L 1 L 2 = d 1 d 2 − 1 (1)is met, even for a fully incoherent (l → 0) beam. However, as this regime is approached, the required accuracy at which the above condition should be ensured increases. To improve the tolerance to possible misalignments, we collimated the beam by means of two circular openings (Fig. 1). Regardless of beam coherence, the experimental uncertainty on the optimal (resonance) value of the ratio L 1 L 2 that stems from the errors on the measured grating periods amounts to σ L 1 / L 2 = 0.002 . In the adopted geometry, this corresponds to an uncertainty of 5 mm on the ideal detector location L 2 . To circumvent this issue, we operated emulsion films tilted by 45° in such a way that the Y coordinate of the emulsion plane is correlated to L 2 , which varies along the z axis in the laboratory reference frame (Fig. 1). We thus performed a position scan in a single exposure by analyzing different horizontal slices of the emulsion detector.

The relative rotational alignment of the two gratings is a critical parameter, as contrast follows a Gaussian modulation as a function of the ϕ angle formed by the slits, with standard deviation (25) σ ϕ = 1 2 π d 2 l det L 2 λ dB (2)where ldet is the coherence length computed at the detector plane by an analytical model (Materials and Methods) (25). Throughout the measurements, the ratio l det λ dB ∼ 800 remained approximately constant, which yielded a tolerance σ ϕ ~ 550 μrad for the collimated beam. The second grating was mounted on a piezoelectric rotation stage with nominal resolution of 0.8 μrad; the adopted alignment protocol (Materials and Methods) ensures that ϕ < 70 μrad for the full duration of the exposures. All the components of the interferometer, including the rotation stage, were nonmagnetic, and the interferometer was surrounded by a mu-metal shield designed to reduce residual Earth magnetic field to less than 0.5 μT. Parallelism between the interferometer optical axis and the beam propagation axis was controlled at the level of 1 mrad by means of alignment lasers. The apparatus operated at a vacuum pressure between 10−7 and 10−6 mbar during each measurement. We configured the interferometer for maximum contrast at E = 14 keV (λ dB = 10.3 pm) by setting L 1 = d 1 d 2 λ dB (21). Since this geometry satisfies the Talbot-Lau resonance conditions, single-slit diffraction from the first to second grating plane is not negligible (as it would instead occur for L 1 λ dB d 1 ≪ d 2 ). Therefore, a quantum-mechanical description of the system is required, which predicts a peculiar contrast dependence on the positron energy. To ensure uniform working conditions throughout the explored energy range, we used nuclear emulsions prepared without the standard surface protective gelatin layer. The presence of even a micrometric layer would have introduced detection efficiency (23) and positron trajectory smearing (24) that are both energy dependent. Thermally induced grains are the dominant source of background noise in emulsion detectors. Nonetheless, no measurable increase in the average density of noise grains was observed for the unprotected emulsions compared to standard detectors used under the same conditions.

We now summarize the results of five beam exposures performed for the positron energies of 16, 14, 11, 9, and 8 keV. Since the transport efficiency through the collimated L-NESS electrostatic beam decreases rapidly below 8 keV, lower energies would have required unpractically long exposure times. On the other hand, energies higher than 16 keV would have run into power supply stability and vacuum discharge limitations. The residual beam rate at the detector after collimation (~80% loss) and the passage through the gratings (~90% loss) was about 100 e+/s. Exposure times between 120 and 200 hours were required to accumulate sufficient statistics (about 2 × 107 grains in the analysis region). These time intervals were selected to match the total counts under the 511 keV annihilation peak on the calibrated HpGe detector (23) measured at 14 keV. Beam focusing was tuned to ensure that beam spot size, and hence the geometrical features of the beam such as angular divergence, did not appreciably vary with the energy. Spot sizes deviated by less than 10% from the average value of 6.5 mm FWHM on the detector plane. Silver bromide crystals activated by the passage of the positrons through the emulsion become visible with optical microscopes after chemical development; we then processed the emulsions at the microscope scanning facility of the University of Bern. For each given X-Y position, the microscope grabbed a series of images by shifting the focal plane in the direction normal to the emulsion surface (Z). Grain clusters were reconstructed (30) to assign X, Y, and Z coordinates to the positron impact point (see Fig. 2). The data were subdivided in 370 × 294 μm2 wide regions, which we refer to as views.

Fig. 2 Representative example of view analysis. All panels refer to the highest contrast view for E = 14 keV. (A) Three-dimensional distribution of the reconstructed clusters limited to a 100-μm-wide region along X. A hint of the periodic fringes is even appreciable by visual inspection. (B) Raw microscope image cropped to a region of 250 × 160 μm2. (C) Histogram of the Z position of the clusters and Gaussian fit used to determine the analysis region.

Each view was independently analyzed to search for periodicity in the impact point distribution by maximizing the so-called Rayleigh test function (31) R ( α , d 3 ) = | 1 n ∑ j = 1 n exp ( 2 π i X j ( α ) d 3 ) | (3)with respect to the period d 3 and the angle α between the fringes and the microscope reference frame. We introduced X j (α) as the X coordinate of the jth grain in the view rotated by α. The effectiveness of this period-finding approach was experimentally demonstrated for similar applications (24, 32). Values of the parameters maximizing R, namely, ( α * , d 3 * ) , were searched in the range −0.05 rad < α < 0.05 rad and 5.7 μm < d 3 < 6.1 μm. If a consistent periodic signal spanned a large area, then a distinctive peak would be expected in the ( α * , d 3 * ) distribution over the analyzed views (24). A representative example is shown in Fig. 3 for the 14-keV exposure. A least-squares Gaussian fit to the data (Fig. 3) yields the detected period for each exposure, which is reported in Table 1. The statistical error (68% confidence interval) comes from the fit procedure, whereas the estimated 0.8% systematic error descends from the conversion between camera pixels and physical size (24). The measured period is compatible with the expected value d 3 = (5.90 ± 0.04) μm. Once the optimal period and angle were found, a histogram of X j (α*) mod d 3 * was constructed, where mod is the modulo operation on floating point numbers. The signal contrast was estimated by fitting the histogram with a d 3 * -periodic sinusoidal function appropriate for the phase structure of the fringes (Fig. 4A). We performed the subtraction of a constant background from the histogram by measuring the grain density in the bulk, as positrons can only penetrate a few micrometers of emulsion. This yielded an estimate of the number of grains due to the intrinsic emulsion noise entering the analysis region. This is limited to 340 × 270 μm2 on the X-Y plane to remove areas affected by sizeable optical aberration. The depth in Z was determined with a Gaussian fit on the positron implantation profile (24). The average width of the selected region along the Z direction was 2.9 μm. We performed this procedure only for views contained in the 3σ elliptical region defined in Fig. 3. As the signal-to-noise ratio decreased moving away from the center of the beam spot, maximization of the Rayleigh test essentially converged to random values of ( α * , d 3 * ) for views outside the elliptical region; contrast estimation for these views would give unphysical results. The contrast measured after noise subtraction is, in principle, independent from the signal-to-noise ratio. An average noise density ranging from 9 to 6 grains/1000 μm3 was measured in the different exposures. The average number of grains per view entering the analysis fluctuated within 10% from the mean value of 11,000. A two-dimensional heatmap of the measured contrast covering the scanned surface area is shown in Fig. 4B. Since Y is correlated to L 2 , a contrast modulation was observed in the Y direction. The contrast dependence on X displayed a marked asymmetry, likely as a result of the limited beam coherence and alignment accuracy. We emphasize that this feature would have been hidden in a moving mask–based detection scheme. On the other hand, emulsions allowed direct fringe detection on a substantially larger area than the surface of the gratings (3 × 3 mm2): The scanned area contained a sizable number (~104) of consecutive high-visibility d 3 -periodic fringes. Visual inspection of the two-dimensional maps already suggests that contrast is decreasing with the energy. To provide a quantitative estimate of the peak contrast, we selected views in a 1-mm-wide region (indicated by the dashed lines in Fig. 4B). The corresponding contrast for energies in the range of 8 to 14 keV encompassing the largest contrast modulation is shown in Fig. 4A as a function of the position of the geometrical center of the view. This is given in the reference frame of the center of the beam along the Y axis, which was estimated by means of a Gaussian fit to the intensity profile. The position of the beam is stable in the laboratory reference frame at the level of 0.5 mm. However, this reference frame is more robust with respect to manual positioning errors of the emulsion in the interferometer and in the microscope reference frame, both of which can be larger than 0.5 mm. The data were fit with a Gaussian function plus a constant background (Fig. 4A), which was used to estimate the maximum contrast for each energy. This is shown in Table 1, alongside the position of the contrast peak (Y 0 ) extracted from the same fit. Uncertainties on the contrast were computed from the fit parameters, whereas the error on Y 0 is dominated by the estimated stability of the beam and by the uncertainty in determining its center.

Fig. 3 Optimal angle and period found via the Rayleigh test. Scatter plot and profile histograms of the optimal period and angle ( α * , d 3 * ) for 14 keV. A total of 1620 views covering a surface of about 10 × 14 mm2 were analyzed. Histograms are fit with a Gaussian function plus a constant background. The black ellipse indicates the corresponding 99.7% confidence level region. The corresponding plots for the other energy points are shown in fig. S2.

Table 1 Summary of the results at different energies. Measured fringe period d 3 , contrast C, and peak contrast position Y 0 , as discussed in the text. View this table: