Studying Antimatter Gravity with Muonium

The gravitational acceleration of antimatter, gbar, has yet to be directly measured; an unexpected outcome of its measurement could change our understanding of gravity, the universe, and the possibility of a fifth force. Three avenues are apparent for such a measurement: antihydrogen, positronium, and muonium, the last requiring a precision atom interferometer and novel muonium beam under development. The interferometer and its few-picometer alignment and calibration systems appear feasible. With 100 nm grating pitch, measurements of gbar to 10%, 1%, or better can be envisioned. These could constitute the first gravitational measurements of leptonic matter, of 2nd-generation matter, and possibly, of antimatter.


Introduction
The question of antimatter gravity, first raised in the 1950s [1], is of continuing interest [2,3]. To date, decades of experimental effort have yet to yield a statistically significant direct measurement. Antimatter gravity studies using antihydrogen are ongoing [4][5][6], and experiments with positronium have been discussed [7]. We report here on progress towards a measurement with muonium (Mu), an exotic atom consisting of an electron bound to an antimuon.
The most sensitive limits on antimatter gravity currently come from indirect tests (for example, equivalence principle tests using torsion pendula [8] or masses in Earth orbit [9]), relying on the expected amounts of virtual antimatter in the nuclei of various elements. Stringent, albeit model-dependent, limits have thereby been set on the gravitational acceleration,ḡ, of antimatter on Earth (e.g., ∼ 10 −7 [10]). The virtual muon-antimuon component of nuclear wave functions being negligible, the extent to which these indirect limits apply to muonium is far from obvious. Another limit, |α g − 1| < 8.7 × 10 −7 [11], has been derived from the measured cyclotron frequency of magnetically confined antiprotons, compared with that of H − ions, based on the gravitational redshift due to Earth's gravitational potential in the field of the local galactic supercluster [12,13]; it too need not apply to antimuons. 1 A direct test of the gravitational interaction of antimatter with matter is desirable

Method
The measurement requires a new muonium beam that is under development, a precision atom interferometer, and a low-background muonium detection system. The experimental challenge stems from the short muon (and muonium) lifetime: τ Mu . = τ µ = 2.2 µs at rest [38]. In one lifetime, if it obeys the equivalence principle, a muonium atom initially at rest thus falls by ∆y = 1 2 gτ 2 Mu = 24 pm -at the edge of measurability using modern atom-interferometry techniques. Fortunately, the exponential 2 The only published direct test so far [4] has yielded the limit −65 <ḡ/g < 110. 3 For example, some authors [39][40][41][42] have suggested that antimatter be identified with the r < 0 solutions of GR's Kerr-Newman equation; in this caseḡ = −g would be the expected GR result. 4 This also suggests a solution to what has been called "the worst prediction in physics": that the gravitational zero-point energy of the universe seems to exceed the size of the cosmological constant by a factor ∼ 10 120 [43]. Figure 1: Principle of three-grating muonium-gravity interferometer, shown schematically in elevation view, with gravitational deflection and phase shift ∆φ exaggerated for clarity. Muonium beam enters from left, slow-electron detector is at right. Not shown: ring electrodes, which accelerate slow electrons onto their detector, starting downstream of grating 3 and continuing within (helically wound) scintillating-fiber-barrel positron-tracking detector; and scintillating-bar hodoscope surrounding positron-tracking detector. decay of the muon affords a statistics-precision tradeoff: the gravitational deflection increases as t 2 , while the number of events decreases as exp(−t/τ µ ). The statistical optimum is thus to measure after two lifetimes; with systematic uncertainties taken into account, the optimal time will be somewhat greater than this. This leads to a deflection to be measured of ≈ 0.2-0.5 nm, which we argue is within the state of the art.
The precision we require can be achieved using a three-grating interferometer in a Mach-Zehnder-like arrangement (Fig. 1), 5 the two inter-grating separations being equal [44,45]. The sensitivity of such an experiment is estimated as [46]: where C = 0.1 is the (conservatively estimated) fringe contrast (see below), N the number of events detected, d the grating pitch, and t the Mu transit time between gratings. While Eq. 1 indicates that finer grating pitch would be helpful, we have chosen d = 100 nm as a compromise between sensitivity and systematic error due to grating variations over the ∼ cm 2 grating area. At a nominal rate of 10 5 incident Mu/s, and taking into account decays and estimated inefficiencies, the statistical measurement precision is ≈ 0.3g per √ N d , N d being the exposure time in days. Although conceptually similar to previous atom-beam interferometers [47][48][49], the proposed precision and source size call for improvements beyond those previously built. We note that, assuming (for the sake of illustration)ḡ = g and 2-lifetime grating separation, the interferometric gravitational phase difference ∆φ = πḡt 2 /2d = 3 mrad corresponds to gt 2 ≈ 0.2 nm of gravitational deflection. A simple way to see this is to consider that from grating 1 to 2, the deflection is y 2 = − 1 2 gt 2 , and at grating 3, it is four times as large, y 3 = 4y 2 . The phase difference at grating 3 with respect to the diffracted beams from grating 2, which start at y 2 , and at grating 3 have fallen an additional distance 5 While we will not have the separated beams typical of Mach-Zehnder interferometers, as shown below, we expect quite visible interference fringes nonetheless [48,50]. This geometry is often referred to as a Talbot interferometer by the X-ray optics community. y 2 , thus corresponds to y 3 − 2y 2 = 190 pm of gravitational deflection, where t = 2τ µ = 4.4 µs. 6 The gratings' relative vertical alignment should therefore to be controlled with an accuracy better than 10 pm (1 pm) for a 10% (1%) measurement.
We are engaged in R&D to develop the needed instrumentation and demonstrate the feasibility of the technique, with three focus areas: 1. Development of improved low-velocity muon and muonium beams; 2. Development of a sufficiently precise interferometer; and 3. Development of a sufficiently precise interferometer alignment and calibration technique.
The success of this work will culminate in MAGE: the Muonium Antimatter Gravity Experiment.

Interferometer
In a Mach-Zehnder atom interferometer, the de Broglie wave corresponding to each incident atom diffracts at the first grating, and again at the second, interfering with itself at the third grating. This produces a sinusoidally varying image at the location of the third grating, with period equal to the grating pitch. As illustrated in Fig. 1, the image is slightly displaced vertically due to the gravitational deflection of the muonium atoms. With 100 nm grating pitch, the image is too fine to be captured by a detector, but by scanning the third grating vertically using piezoelectric actuators, the image is translated into the time domain as a sinusoidally varying beam intensity. With ≈ 50%-open gratings, 7 even orders (except zero) are suppressed, and the three diffraction orders shown contain most of the transmitted intensity. Since each atom's de Broglie wave interferes with itself, and the interference patterns from all atoms are in phase with each other, this configuration accommodates an extended, incoherent source, easing alignment and beam requirements [50][51][52][53]. We have modeled the performance of such an interferometer using the procedure of Refs. [48,54] and find a maximum expected contrast of 20% for the case of overlapping beams. This will of course be degraded in practice due to such factors as grating imperfections, thus the 10% contrast assumed above is plausible.
To accommodate the anticipated beam size and divergence in various muonium source configurations, the grating area should be ∼ cm 2 and the open fractions as close to the optimal values as practicable. The structures would ideally be grids of ≈ 1 cm-wide horizontal slits. For mechanical stability, however, buttresses between lines of the grating are required, producing an array of mini-slits that form a grid approximating an ideal grating. The grid structures will be made in a thin film of Si 3 N 4 or ultrananocrystalline diamond (UNCD), coated with higher-Z metal, as membrane windows on silicon wafers. The fabrication techniques required are e-beam lithography and reactive-ion and wet etching, all available to us via our collaboration with the Center for Nanoscale Materials (CNM) at Argonne National Laboratory, as well as within the Laboratory for Micro-and Nanotechnology at the Paul Scherrer Institute (PSI). The CNM has developed specific process technology that has been demonstrated in UNCD and Si 3 N 4 for membrane aperture structures and in UNCD for zone-plate structures very similar to that needed for the grid structures we require [55,56].
Given the small deflections to be measured, mechanical and thermal stability of the interferometer are paramount. To that end, the gratings will be mounted on an extremely stiff, channel-cut single-crystal silicon optical bench (Fig. 2) like those developed for use in cryogenic monochromators at synchrotron radiation facilities [57], and operated at a temperature at which the thermal expansion coefficient of silicon is near zero [58]: T < ∼ 4 K (or, for initial testing, the more convenient T ≈ 125 K). Sub-10 pm grating alignment is also required, which is challenging using established techniques. We have therefore proposed to use the semiconductor-laser tracking frequency gauges (TFGs) recently   developed for other equivalence-principle tests [59][60][61]. These employ Pound-Drever-Hall locking [62] of a laser to the length of an interferometer (which can be nonresonant or resonant, i.e., a cavity), such that small changes in length are translated into shifts of a laser frequency. The beat frequency between the measurement laser and a reference laser locked to a stable length can be measured to high precision. A precision of 2 pm has been demonstrated using a non-resonant Michelson interferometer, and 46 fm using a Fabry-Perot cavity with a finesse of 130 [59]. More recently [63], a precision of 1 pm was demonstrated by some of us (D.M.K., J.D.P., T.J.R., and R.D.R.) at Illinois Institute of Technology (IIT) using the same non-resonant Michelson interferometer (Fig. 3). Figure 4 shows a possible arrangement of the optics that form the laser interferometers used to monitor the grating alignment.
To suppress background in the muonium detector, a coincidence technique will be used. Decaying muons emit a fast positron, leaving behind the (now unbound) electron, which can be accelerated electrostatically onto a microchannel plate (MCP) or pixel detector and counted [64]. Due to the Q value of the µ → eνν decay, the fast positrons typically emerge at a substantial angle to the beam direction, so can be efficiently detected in a barrel scintillating-fiber tracking detector downstream of the third grating (Fig. 5). The detection in coincidence of an accelerated electron and a fast positron with track pointing back to the "decay region" unambiguously signals a muonium decay in that region, which extends from the third grating to the slow-electron detector. To minimize the electronics channel count, the scintillating fibers are laid out as helices and read out on both ends, so that the end-to-end amplitude and time differences can be used to localize the hit to one turn of the helix. The surrounding scintillating-bar hodoscope (with wavelength-shifting fibers for readout) is used to determine the azimuthal angle φ of the hit, allowing determination of z to about a fiber width (1 mm). G4beamline [65] simulations of the muonium detector indicate a ≈ 70% acceptance for the coincidence requirement, which can be increased if desired by extending the barrel detector further upstream or downstream. (Due to the electrostatic acceleration of the electron, its detection efficiency is near 100%.)

Muonium Beam
We have identified two approaches for producing the needed muonium beam, both of which employ a superfluid He (SFHe) µ + → Mu converter. SFHe has been shown to be a highly efficient converter of stopping positive muons to muonium (Fig. 6) [66]; indeed, based on their experimental results, the Ref. [66] authors conclude that below 0.7 K essentially all µ + stopping in helium form muonium. The interferometry requirements dictate the properties needed for the beam: it should be Figure 3. Semiconductor-laser tracking frequency gauge (TFG) precision in picometers vs. averaging time in seconds, determined by measuring the half-meter optical path-length difference of a Michelson interferometer using two TFGs simultaneously, operated at 1,560 nm with a common optical path (from [63]). The demonstrated stability (Allan deviation less than 3 pm for averaging times ranging from about 1 second to about 1,000 seconds) implies that X-ray calibrations need not be repeated more often than every 10 to 20 minutes; given operation in a cavity of modest finesse (as in Fig. 4), the calibration interval extrapolates to 10 days.   . Muonium production vs. temperature in liquid helium (from [66]); open circles are results in pure 4 He, filled circles, 4 He + 0.2% 3 He, and triangles, pure 3 He. (Quantity plotted is observed muon decay asymmetry; an observed asymmetry of 0.105 corresponds to 100% muonium formation.) traveling in vacuum in a region free of electric and magnetic fields. Since it is neutral, and has an electric polarizability similar to that of hydrogen (among the smallest of any atom), Mu is intrinsically insensitive to such fields; nevertheless, for the effects of gravity to stand out, even small, higher-order electromagnetic effects should be suppressed. To optimize the stability of the interferometer structure, the Mu atoms should be moving slowly, such that the distance traveled in a few Mu lifetimes is < ∼ 10 cm, and with a small enough range in velocity that the interference phase is not significantly smeared out. Because the loss of contrast is given by C/C 0 = − exp (σ 2 φ /2) [52], and the RMS width σ φ of the gravitational phase is quite small, in practice we do not expect the Mu velocity range to contribute significantly to the uncertainty of the measurement.
The first muonium-beam approach is to use the "muCool" µ + beam under development by some of us (A.A., A.K., K.K., T.J.P., and A.S.) [67][68][69][70]. In this scheme (due to D. Taqqu [71]), the surface-muon beam is cooled by a factor ∼ 10 10 in phase-space density prior to stopping, while losing only 3 orders of magnitude in intensity, thus resulting in an improvement of 10 7 in brightness. This reduces the spot size to < 1 mm and dramatically enhances the stopping fraction. The muCool beam is stopped in a micron-thin film of superfluid helium (SFHe) coating the bottom of the cryostat. Mu is formed in the SFHe, and half of it diffuses to the upper surface, where (since it behaves chemically like a light hydrogen isotope), due to the negative chemical affinity of hydrogen in SFHe [72,73], it is expelled into vacuum. The resulting Mu beam will be very nearly parallel (divergence ≈ 27 mrad) and monoenergetic (∆E/E ≈ 0.1%), with a velocity (6.3 mm/µs) determined by the chemical potential of Mu in SFHe [71], whose value (270 K) [74] is the largest for any hydrogen isotope. (Because the Mu atom is in thermal equilibrium with the superfluid helium prior to being ejected, both the energy spread and the angular divergence of the Mu beam are determined by the ratio of the SFHe temperature, here taken to be 0.2 K, to the chemical potential.) The resulting interferometer acceptance is of order unity, leading toḡ sign determination with about one month's worth of beam.
The second option exploits another idea of Taqqu's [74]: use a thicker (∼ 100 µm) SFHe converter with an uncooled subsurface-muon beam, giving aḡ sign determination with two days' worth of beam. Since it avoids the beam loss inherent in the muCool approach, the thick-film approach could enable a < ∼ 10% measurement ofḡ in a month of beam time, again assuming adequate control of systematic uncertainties, and possibly a < ∼ 1% measurement in a future, upgraded surface-muon beam. Since only Mu atoms formed close to the upper surface will emerge upwards from the helium to form the desired beam, an electric field is maintained in the helium (via a pool of negative charge on the upper surface of the SFHe layer) to cause the stopping µ + to separate from their ionization trails and drift to the upper surface, where they then form muonium. Some deadtime may be incurred if the pool of charge needs to be periodically replenished (e.g., via a submerged emitter tip). The ∼ cm-wide beam results in some acceptance loss if cm-wide gratings are employed, thus larger gratings (if feasible) could be beneficial; alternatively, the SFHe deflector (described next) could have a curved surface so as to produce some focusing of the beam into the interferometer [75]. In addition to providing higher beam intensity, the thick-film approach eliminates the technical risk associated with muCool, although it has other uncertainties, such as those associated with creating and maintaining the charge pool. Clarifying these issues will require further muonium-beam R&D at PSI.
Theḡ sensitivity estimates given above (and in Fig. 7) were calculated assuming the use of PSI's πE5 beamline, with 1.2 × 10 8 surface muons/s, and 10% interferometer contrast. 8 In both the thinand thick-film approaches, the converter must be a flat, horizontal pool of SFHe at the bottom of a cryostat see Fig. 8). The resulting vertical Mu beam can be deflected into the horizontal (as required by the interferometer) by reflecting it off of a 45 • -inclined SFHe-coated surface [75]. This has the further advantage that antimuons failing to stop in the SFHe layer pass in front of, rather than through, the 8 Note that the sensitivities of Fig. 7 are somewhat more pessimistic than that of Eq. 1, due to inclusion of estimated decay losses from the Mu source to the first grating. The size of this effect will depend on the final source-interferometer distance, which will depend on cryostat engineering details yet to be determined.
(a) "Thin-film" SFHe beam (b) "Thick-film" SFHe beam Figure 7. Representative MAGE sensitivity estimates vs. grating separation for beam options described in text, with 0.5 µm-thick gratings of 100 nm pitch, assuming 10% contrast and that statistical uncertainties dominate over systematics; shown is beam time required for 5σ determination of the sign ofḡ (i.e., δḡ/g = 0.4). 8 Figure 8. MAGE concept drawing detectors, where they could otherwise be counted, potentially contributing to accidental background coincidences. It also frees up space for a calibration X-ray beam (discussed below).

Interferometer Alignment and Calibration
A key problem that must be solved is how to determine the zero-deflection interference phase. 9 The solution we have adopted is to calibrate the interferometer using soft X-rays. This is most simply accomplished if the MCP used to detect the slow electrons can also serve as an X-ray detector; 10 suitable MCPs are commercially available. An X-ray energy of 2 keV gives the same wavelength as that of 6.3 mm/µs muonium; this is an energy within the range of commercially available benchtop X-ray sources. At this wavelength Si 3 N 4 is partially transparent, reducing undesired heating of the gratings by the calibration source; an interference pattern is nevertheless formed due to the combination of phase shift and partial absorption of the X-rays that pass through the grating material. If need be, the contrast can be enhanced by using a more absorbing grating in the third position. The undesirable heating is further reduced if the laser gauge is long-term stable such that it is unnecessary to repeat the X-ray calibration at short intervals. Figure 3 shows that the interval between X-ray calibrations can be about 15 minutes assuming we require the calibration to be good to 3 pm. Reducing that requirement to 10 pm extends the recalibration interval to 4 hours based on an extrapolation of the TFG performance 9 The "obvious" solution of comparing muonium and antimuonium beams is unfortunately not feasible, since it is impractical to produce a sufficiently positron-rich µ − stopping medium. 10 If necessary, to eliminate the possible ambiguity between Mu and X-ray events, the e − accelerating potential can be turned off during calibration runs. assuming it displays "1/f noise." Figure 4 shows the laser interferometer operating in a cavity, which increases its performance. By similarly extrapolating the Allan deviation for a TFG operating in a cavity (finesse of 130) [59] and requiring the 3 pm calibration, we get a recalibration interval of over ten days. This essentially eliminates the heating problem.

Systematic Uncertainties
Controlling and understanding systematic uncertainties will be critical to the success of the experiment. In particular, it will be crucial to verify that the measured deflection is in fact due to gravity. The interferometer will therefore be designed to be rotatable about the beam axis; with grating lines vertical, there will be no gravitational phase shift. The interferometer will be constructed to rotate by 90 • and 180 • , allowing upright, inverted, and sideways operation. Rotation will result in distortions of the support structure. A 1-cm-tall silicon post supported at the bottom shortens under gravity by 6-9 pm, depending on crystal orientation. A cantilever deflects by a larger distance. We address these distortions with several measures: (1) The optical bench (Fig. 2a) is a short and thick structure suspended at two points along the beam axis chosen to null the gravity effect on the critical grating-position combination y 1 + y 3 − 2y 2 (where y i refers to the vertical position of grating i). (2) We take advantage of a high degree of symmetry in its construction and support. (3) The grating bars themselves are supported at ∼ µm intervals with buttresses (fabricated out of the same membrane), reducing their deflection to a negligible level. (4) The atom diffraction patterns are relatively narrow (first diffraction maximum at 5.6 mrad), and we localize both Mu and X-ray events to ∼ 1 mm, which allows use of the X-ray interference pattern to monitor and correct for gravitationally induced changes due to inversion, as well as other possible effects that may depend on transverse position.
Electric-and magnetic-field gradients shift the observed Mu atom interferometric phase. The electric fields are caused by "patch effect" surface potential variations on the electrostatic shielding. Since the electrostatic shield inverts with the interferometer, the electric effect cancels well with interferometer inversion. For magnetic effects (which do not entirely cancel with inversion), the Kirch group at ETH/PSI can monitor and shield at the < nT/mm level [76], which is sufficient. Table 1 shows that with reasonable shielding measures these effects are a small fraction of 1 g and below the level of concern. The van der Waals effect (vdWE) can also cause an interferometric phase shift. A shift of ≈ 300 mrad was observed by placing a Si 3 N 4 grating in one arm of an atom interferometer [77] (cf. the 12 mrad shift due to gravity in MAGE for a grating spacing of 2 lifetimes). Two factors make the vdWE larger in MAGE: the gratings may be thicker (∼ 3-fold), and we anticipate using metallization on the upstream face (∼< 2-fold enhancement; see Table I of [78]). Several, more significant, factors reduce the vdWE in MAGE. (1) Both interferometer beams see the same gratings, so vdWE-induced phase shift results only from inhomogeneous variation in grating structure. (2) The Mu atom's static polarizability is similar to that of H, 34 times smaller than that of Na, implying that the effect on visibility in MAGE is much smaller than in [77]. (3) The vdWE phase shift cancels with interferometer inversion. Thus, the vdWE will be negligible in MAGE.
A useful systematics test will be to measure, using various atom beams as well as muonium, the Sagnac effect due to the Earth's rotation, which depends on both the compass direction of the beam and the angle of the gratings relative to the horizontal (or vertical). For our geometry, beam velocity, and latitude, the effect is about 10% as big as the gravitational deflection. This will demonstrate the capability of the interferometer to measure phase shifts smaller than that due to gravity. Operating the interferometer with atoms of normal matter also tests the fidelity of calibration using X-rays and inversion.

Prospects
An R&D program on SFHe-produced Mu beams is proceeding at PSI in parallel with the muCool project. It is anticipated that such beams may be available at PSI within about 3 years. We hope to produce the interferometer and detectors in parallel, so as to be ready for the first muonium gravity measurements in the early 2020s.
Author Contributions: All authors contributed equally to this work.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: