A moiré deflectometer for antimatter

The precise measurement of forces is one way to obtain deep insight into the fundamental interactions present in nature. In the context of neutral antimatter, the gravitational interaction is of high interest, potentially revealing new forces that violate the weak equivalence principle. Here we report on a successful extension of a tool from atom optics—the moiré deflectometer—for a measurement of the acceleration of slow antiprotons. The setup consists of two identical transmission gratings and a spatially resolving emulsion detector for antiproton annihilations. Absolute referencing of the observed antimatter pattern with a photon pattern experiencing no deflection allows the direct inference of forces present. The concept is also straightforwardly applicable to antihydrogen measurements as pursued by the AEgIS collaboration. The combination of these very different techniques from high energy and atomic physics opens a very promising route to the direct detection of the gravitational acceleration of neutral antimatter.

T he precise measurement of forces between objects gives deep insight into the fundamental interactions and symmetries of nature. A paradigm example is the comparison of the motion of matter in the gravitational field, testing with high precision that the acceleration is materialindependent, that is, the weak equivalence principle [1][2][3][4] . Although indirect experimental evidence suggests that the weak equivalence principle also holds for antimatter [5][6][7] , a direct observation for antimatter is still missing. First attempts in this direction have recently been reported by the ALPHA collaboration 8 , who used the release of antihydrogen from a magnetic trap to exclude the absolute value of the gravitational acceleration of antihydrogen to be 100 times larger than for matter. An alternative approach is followed by the GBAR collaboration 9 , which is based on sympathetic cooling of positive antihydrogen ions and their subsequent photodetachment. One of the specified goals of the AEgIS collaboration (antihydrogen experiment: gravity, interferometry, spectroscopy) is the direct detection of the gravitational acceleration using an antihydrogen beam 10,11 combined with a moiré deflectometer 12 , a device with high sensitivity for acceleration measurements.
Here, we present the successful realization of such a device for antiprotons. This has been achieved using slow antiprotons from the Antiproton Decelerator (AD) at CERN, the technology of emulsion detectors developed for recent high-energy neutrino experiments 13 and a novel referencing method employing Talbot-Lau interferometry 14,15 with light. The observation is consistent with a force at the 500 aN level acting on the antiprotons. This demonstration is an important prerequisite for future studies of the gravitational acceleration of antimatter building on an antihydrogen beam.

Results
Moiré deflectometer. The principle used in the experiment reported here is visualized in Fig. 1a. A divergent beam of antiprotons enters the moiré setup consisting of three equally spaced elements: two gratings and a spatially resolving emulsion detector. The two gratings with periodicity d define the classical trajectories leading to a fringe pattern with the same periodicity at the position of the detector. If the transit time of the particles through the device is known, absolute force measurements are possible by employing Newton's second law of mechanics 16 .
As indicated in Fig. 1b, the position of the moiré pattern is shifted in the presence of a force with respect to the geometric shadow by where F || represents the force component along the grating period, m is the inertial mass of the test particle, a is the acceleration and t is the time of flight between the two gratings. It is important to note that the shift has two contributions. The velocity of the particle after the second grating in the direction of the acceleration is non-zero and the particle is also accelerated in the second half of the moiré deflectometer. The relevant parameter for precision measurements is the sensitivity, that is, the minimal detectable acceleration a min . This can be estimated by considering the maximal signal S to noise ratio possible in this scenario. Since the influence of a pattern shift is most sensitively detected at the steepest gradient of the pattern the visibility u ¼ (S max À S min )/(S max þ S min ) should be maximized and the periodicity minimized. The noise of the signal is intrinsically limited for classical particle sources to the shot noise which scales as 1/ ffiffiffiffi N p , where N is the number of detected particles. Consequently, the minimal detectable acceleration 12 is given by It is important to note that this device works even for a very divergent source of particles as shown in Fig. 1a, and thus is an ideal device for the highly divergent beam of antihydrogen atoms that is expected in the AEgIS apparatus.

Talbot-Lau interferometry with light as absolute reference.
To determine the magnitude of the fringe pattern shift, knowledge of the undeflected fringe position (indicated as grey trajectories in Fig. 1b) is required. Due to the neutrality and high speed of photons, it is favourable to measure this position independently with light so that the action of forces is negligible. Unlike the case of classical particles described above, geometric paths are not applicable for visible light as diffraction at the gratings has to be taken into account. Figure 1c depicts the corresponding light field pattern where the distance between the gratings is given by the Talbot length L Talbot ¼ 2d 2 /l. This configuration is known as Talbot-Lau interferometer 14 , which is based on the near-field Talbot effect 15 -the rephasing of the pattern in discrete distances behind a grating illuminated with light. The final pattern is not a classical distribution, but an interference pattern and coincides with the pattern of the moiré  Finally, the antiprotons are detected with a spatially resolving emulsion detector. To infer the force, the shifted position of the moiré pattern has to be compared with the expected pattern without force. (c) This is achieved using light and near-field interference, the shift of which is negligible. A grating in direct contact with the emulsion is used to reference the antimatter and the light measurements.
deflectometer experiencing no acceleration. Thus, light provides the required absolute zero-force reference. The only prerequisite is that the Talbot length (or a multiple integer of it) is matched to the distance between the gratings and the detector. With that, the absolute shift of the antimatter pattern can be directly accessed and systematic errors can be significantly reduced as the moiré deflectometer and Talbot-Lau interferometer use the same gratings. We would like to stress that Talbot-Lau interferometry is also possible for matter waves such as atoms and molecules 17,18 if their de Broglie wavelength is long enough.
Experimental implementation. The experiment was performed within the AEgIS apparatus designed to produce antihydrogen for a future measurement of the gravitational acceleration 10,19 . A beam of antiprotons with a broad energy distribution, delivered by the AD at CERN, is realized after the 5.3 MeV antiprotons are transmitted through degrader foils with a total thickness of 225 mm (170 mm of aluminium and 55 mm of silicon). The simulated distribution has a mean energy of 106 keV and a root mean squared value of about 150 keV (see Methods). After traversing a 3.6-m long tube within two homogeneous magnetic fields of 5 T and 1 T, the antiprotons enter the deflectometer. We estimate the mean de Broglie wavelength to be 8.8 Â 10 À 14 m, which implies that the concept of classical paths for the trajectories of the antiprotons is applicable for our gratings with a periodicity of 40 mm.
The grating holder is compact (25 mm distance between the gratings) so that the passive stability of the relative positions between the gratings for the long measurement time of 6.5 h is ensured. The slit arrays are manufactured in silicon by reactive ion etching, leading to a 100-mm thick silicon membrane with a slit width of 12 mm and a periodicity of d ¼ 40 mm. Low-energy antiprotons hitting the slit array annihilate on the surface of the array and do not reach the detector. For this measurement, the final pattern, that is, the annihilation positions of antiprotons after passing two gratings, is detected by an emulsion detector. The moiré deflectometer and the annihilation detector are mounted in a vacuum chamber (10 À 5 mbar) on the extraction line of the AEgIS apparatus. After the exposure to antiprotons, the emulsion detector is removed, developed and analysed with an automatic microscope available at one of the participating institutions to determine the location of single annihilations. This facility was initially developed for the detection of neutrinoinduced t-leptons by the OPERA experiment 13 . The development of emulsion detectors for the application presented here, which involves operation in vacuum, is described in refs 20,21. After removal of the emulsion detector the pattern of the Talbot-Lau interferometry with light was recorded in a subsequent measurement. For this purpose, the grating holder was homogenously illuminated by an incoherent light source (red light-emitting diode with spatial diffuser). For a wavelength of l ¼ 640 nm, the Talbot distance is L Talbot ¼ 2d 2 /lE5 mm. Thus, for our setup (L ¼ 25 mm), we analyse the fifth rephasing of the light waves. The light pattern was directly recorded at the plane of the emulsion with a high-resolution flatbed charge-coupled device scanner (2.7 mm resolution). To align the antiproton and light measurement in the experiment reported here, an independent spatial reference is implemented. For that purpose we installed an additional transmission grating in direct contact with the detector plane. Contact grating and moiré deflectometer (see Fig. 1a) were simultaneously illuminated: first with antiprotons and subsequently with light. In each case, the pattern behind the contact grating is a simple shadow without any force dependence, and thus can be used as a reference for alignment.
Antimatter fringe patterns. With the emulsion detector, the positions of the annihilation vertices can be detected with a typical resolution of 2 mm (see Fig. 2a). The fragments produced by the annihilation of antiprotons lead to a characteristic starshaped pattern, which can be observed with the microscope (an example is depicted in Fig. 2a). The first observation of such an annihilation star succeeded shortly after the discovery of the antiproton using emulsions 22 . This allows for very robust and high-quality particle identification, which makes this detector practically background-free. In addition, this detector can detect the arrival of antiprotons over a large area and thus is compatible with an upscaling of the grating area necessary for experiments with a divergent antihydrogen beam.  Fig. 2c. The high visibility implies that the periodicity is well-defined in an area as large as 15 Â 6 mm 2 since the data collapses onto one fringe by taking the detected position modulo the extracted periodicity d of the pattern. To extract the periodicity, we employ the Rayleigh test 23 that is also widely used in astronomy 24 . The periodicity d and the relative rotation a of the pattern is found by maximizing where n is the total number of antiprotons and y i ¼ y 0 Á cos a þ x 0 Á sin a depicts the antiproton's projected coordinate. This leads to an inferred periodicity of 40.22±0.02 mm, which is consistent with the expected emulsion expansion of B1% and the nominal periodicity of 40 mm. It is interesting to note that the analysed area corresponds to 368 slits and, on average, only in every second slit an antiproton is detected. In Fig. 2b, the observed moiré pattern for antiprotons is shown. The 241 events associated with antiproton annihilations were accumulated during the 6.5-h run of the experiment. The Rayleigh tests on sub-segments of the detected patterns reveal local distortion due to the expansion/shear of the emulsion and allow the identification of regions with negligible distortion. We have restricted the areas to two-thirds of their initial size, which ensures a position uncertainty due to shear to be smaller than ± 1.2 mm.
Absolute deflection measurement. To determine the absolute position of the antiproton fringe pattern (parameter a in Fig. 2b), we conduct a comparison with the measurement with light. The results are represented in Fig. 3a,b where the detected intensity is indicated by the red shading. The alignment is achieved by overlaying the contact patterns as depicted on the right of Fig. 3b. The moiré pattern can now be directly compared with the Talbot-Lau pattern (left of Fig. 3b) to extract a possible deflection.
For the quantitative analysis, we extract the orientation of the antimatter (Rayleigh test) and light patterns (Fourier transformation as the data is discrete in space). We find that the relative angle of the two antiproton patterns, which are 15 mm apart, deviates from the angle measured between the two corresponding light patterns by Dy ¼ 0.92 ± 0.27 mrad.
This observation is consistent with independent systematic studies of the distortion of emulsions on this large scale 25 . It is important to realize that this angle implies an intrinsic systematic uncertainty in the determination of the relative shift between the light and antimatter patterns since one cannot know which part has undergone the deformation. Assuming that both areas of the emulsion corresponding to contact and moiré have changed the same way on the centimetre scale, that is, half of the angular deviation for each pattern, we can compare the relative positions of the antiprotons with that of the light pattern as shown in Fig. 3b. The contact patterns on the right overlay as these are direct shadows of the grating (no force dependence), while an upward shift of the antiprotons in the force sensitive moiré pattern is noticeable. For quantitative analysis, we collapse the data onto one fringe (see Fig. 3c) and deduce the relative shift of Dy mean ¼ 9.8 ± 0.9 mm (stat.) where the error is due to the uncertainties (one sigma) of the involved fits. Estimating a bound on the systematic uncertainties, we repeat our analysis assuming that either the contact or the moiré pattern has been changed due to the distortion. With that we find a minimal shift of Dy min ¼ 3.7 ± 0.9 mm (stat.) and maximal shift of Dy max ¼ 16.4 ± 0.9 mm (stat.) leading to a shift of Dy mean ¼ 9.8 ± 0.9 mm (stat.) ± 6.4 mm (syst.).

Discussion
The observed shift of the moiré pattern is consistent with a force acting on the antiprotons. With the assumption of a mean velocity of v ¼ 4.5 Â 10 6 ms À 1 implying a transit time of t ¼ 5.6 ns, we find a mean force of F ¼ 530 ± 50 aN (stat.) ± 350 aN (syst.). It is important to note that the mere observation of a pattern sets an upper bound for the force being present. The impinging antiproton beam has a very broad velocity distribution due to the degrading process in the foils. Thus, in the case that a force is present, the experimentally observed moiré pattern is an ensemble of differently shifted patterns corresponding to the transit times t for different velocities. The results of a simulation of the performance of the moiré deflectometer are depicted in Fig. 4 and clearly reveal how the visibility vanishes for increasingly large forces (a force of 10 fN reduces the visibility below u ¼ 10%). The observed visibility of 71% is consistent with a mean force of B500 aN. The visibility of the antiproton moiré pattern on its own (not relying on additional referencing) is an independent consistency check that the observed pattern is indeed shifted due to a force. Additionally with the observed high visibility of the moiré pattern, we exclude the possibility that the force has shifted the pattern by more than one period (see Fig. 4).
The measured force could arise from a Lorentz force either caused by an electric field of B33 V cm À 1 in direction of the grating period or a magnetic field component of B7.4 G perpendicular to the grating period and antiproton direction. The latter is compatible with the measured magnetic field of B10 G at the position of the deflectometer due to the fringe field of the trapping region and stray fields of neighbouring experiments in the AD zone.
The results presented are a crucial step towards the direct detection of gravitational acceleration of antihydrogen with the AEgIS experiment. Its concept is based on the formation of excited antihydrogen through the charge exchange reaction of electromagnetically trapped antiprotons with bunched Rydberg positronium. The resulting dipole moments of the antihydrogen atoms in a weak electric field allow their subsequent acceleration with electric field gradients, thus forming a beam towards the moiré deflectometer. The measurement of the antihydrogen's arrival position is realized by detection of the annihilation of its antiproton-thus using techniques presented here.
It is important to note that the expected absolute shift of the antihydrogen pattern due to gravity is comparable to the one observed in the current experiment. Although the gravitational force acting on antihydrogen is 10 orders of magnitude smaller than the sensitivity level reached with the presented small prototype deflectometer, the resolution of the setup can be simply improved by scaling up the deflectometer and the detector. A detailed discussion of the expected performance can be found in refs 10,19. The main improvement is achieved by increasing the transit time t (see equation (1)). Using a beam of antihydrogen atoms with a significantly lower velocity of B500 ms À 1 and a distance of 1 m between the gratings (this experiment v ¼ 4.5 Â 10 6 ms À 1 and L ¼ 25 mm) will improve the sensitivity by 11 orders of magnitude (eight orders of magnitude due to slower velocity and three orders of magnitude due to increased length of the device), thus allowing the application of this technique to direct measurements of the gravitational force with antihydrogen.
High resolution is a prerequisite for the successful direct detection of the gravitational acceleration of antimatter. For absolute measurements, the sensitivity is the relevant parameter, which is ultimately limited by the intrinsic shot noise due to the detection of single atoms. Since the sensitivity scales with 1/ ffiffiffiffi N p det (N det representing the number of detected particles) increasing the length of the moiré setup implies a similar expansion of the transverse dimensions to keep the throughput, and thus the flux, high. We have already successfully produced high-quality gratings with a transverse extent as large as 100 mm. The currently limiting systematic errors due to the distortion of the emulsion can be overcome by referencing the antihydrogen pattern directly to an in situ-realized light pattern, by employing emulsions on a glass substrate instead of the plastic used for this measurement or by the use of photomasks 25 .

Methods
Orientation of the antiproton pattern. Collapsing the detected annihilation events to a single histogram as shown in Fig. 2 relies on the accurate determination of the periodicity as well as the angular orientation of the two dimensional fringe pattern. For this reason, we conduct the Rayleigh test given by equation (2) for different angles and periodicities and the results are depicted in Fig. 5 for the moiré and the contact pattern. The maximum is well-defined so that periodicity and angular orientation can be extracted for further analysis. The performance of the Rayleigh test has been tested with simulated data sets.
Velocity distribution. The kinetic energy of the antiprotons reaching the deflectometer is estimated with a simulation based on Geant4 (ref. 26). All the materials interposed along the beamline, between the AD antiproton beam (initial kinetic energy of 5.3 MeV) and the deflectometer, as well as the magnetic field and geometry of the AEgIS apparatus, are taken into account. The input parameters of the Monte Carlo simulation are matched to meet the experimentally observed best antiproton trapping efficiency. This measurement was performed by counting the number of trapped antiprotons for various thicknesses of an aluminium degrader placed upstream of the antiproton trap. The thickness of the degrader foils used to slow down the antiprotons was selected in order to maximize the number of antiprotons with a kinetic energy lower than 10 keV that enter the antiproton trap and was set to 55 mm of silicon and 170 mm of aluminium. An additional foil of titanium (2 mm) is placed at the exit of the 1 T magnet. The energy distribution at the position of the deflectometer is depicted in Fig. 6. It is very broad and thus can be employed for setting limits on the maximum force present by analysing the visibility of the moiré pattern.