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INTRODUCTION
General relativity and quantum mechanics are two of the very foundations of modern physics. Yet, as waspointed out early on by Wigner [1], the two are to some degree incompatible. The most obvious discrepancy is the dynamic spacetime geometry of general relativity, which clashes with the fixed-background approach of quantum field theory. In order to repair some of these perceived contradictions and to properly incorporate gravity into the Standard Model, some scientists actively pursue a unification of gravity with the other fundamental interactions within what has been called a ‘‘Theory of Everything” [2].
A hypothetical quantum theory of gravitation necessarily constitutes a departure from the Einsteinian view of gravity as a geometric phenomenon. As in other quantum field theories, the interaction is mediated by exchange particles. The spins of these exchange bosons as well as the signs of the charges to which they couple determine whether a force is repulsive or attractive. Generally speaking, evenspin particles create an attractive force between all types of charges, whereas the exchange of odd-spin particles leads to a repulsive force between like charges. Hence, the formulation of a quantum theory of gravitation automatically brings about the possibility of different types of exchange particles as well as negative mass charge. Attempts at formulating a quantum gravity have mainly been hampered by the fact that such a theory is non-renormalizable, though a renormalization within the framework of an effective field theory may turn out to be feasible [3].
When constructing such a theory, ordinary ‘‘Newtonian” gravity is associated with a massless tensor (spin-2) exchange boson, as the force has an infinite range and is always attractive. In addition to this tensor part, gravity could have scalar (spin-0) and/or vector (spin-1) components. Unlike the tensor and scalar parts, a vector component would lead to a repulsive force acting between like charges. Such a force would thus produce a dramatic effect on antimatter particles in the Earth’s gravitational field and constitute a violation of the weak equivalence principle (also called the universality of free fall).
There are a number of arguments against ‘‘anti-gravity,” i.e. a tensor-type gravitational interaction with opposite sign for antimatter [4–7]. The most intuitive of these is Morrison’s argument [8], which elegantly demonstrates that such a phenomenon would violate conservation of energy. Quantitative limits on possible anti-tensor gravity effects can be obtained, among others, from estimates of the effects of virtual antiparticles in ordinary matter [9,10] or the absence of changes in the cyclotron frequency of antiprotons () confined in a Penning trap [11]. Most, if not all, of these arguments do not apply to more elaborate models involving vector and scalar gravitons. If the hypothetical vector and scalar charges, as well as their masses (and thus the ranges of the interactions) are carefully chosen, such contributions can be strongly suppressed in ordinary matter. Finally, it should be stated that the problem of the gravitational interaction of antimatter is completely independent from the – equally fascinating – question of matter–antimatter symmetry (CPT), as CPT invariance merely dictates the equality of the inertial masses of particle and antiparticle pairs, but places no restriction on the gravitational masses.
Unlike ordinary matter [12], the behavior of antimatter particles in a gravitational field has never been tested experimentally. Two attempts, at Stanford [13] and CERN’s Low-Energy Antiproton Ring [14] were thwarted by the overwhelming effect of stray electric and magnetic fields upon the electrically charged test particles. The recent production of copious amounts of cold antihydrogen () at CERN’s Antiproton Decelerator (AD) [15,16] has paved the way for high-precision gravity experiments with neutral antimatter. We have proposed the AEGIS experiment (Antimatter Experiment: Gravity, Interferometry, Spectroscopy), to be performed at CERN/AD, in order to address this important question.
The primary scientific goal of AEGIS is the direct measurement of the Earth’s local gravitational acceleration  on . In a first phase of the experiment, a gravity measurement with  relative precision will be carried out by observing the vertical displacement of the shadow image produced by an  beam as it traverses a Moiré deflectometer, the classical counterpart of a matter wave interferometer. In spite of its limited precision, this measurement will represent the first direct determination of the gravitational effect on antimatter.

The essential steps leading to the production of  and the measurement of  with AEGIS are the following:

  • Production of positrons () from a Surko-type source and accumulator;

  • Capture and accumulation of  from the AD in a cylindrical Penning trap;

  • Production of positronium () by bombardment of a nanoporous material with an intense  pulse;

  • Excitation of the  to a Rydberg state with principal quantum number ;

  • Recombination of  by resonant charge exchange between Rydberg  and cold ;

  • Formation of an  beam by Stark acceleration with inhomogeneous electric fields;

  • Determination of  in a two-grating Moiré deflectometer coupled with a position-sensitive detector.

The feasibility of the first two points has been conclusively demonstrated by the ATHENA and ATRAP collaborations (see, in particular, [17,18]). In the following, we will discuss the remaining aspects of the proposed technique in more detail.

Fig. 1. Laser system for the Rydberg excitation of  prior to  recombination.
METHOD
Positronium production and excitation
In recent years, the potential for nanoporous insulator materials to be used as highly efficient Ps converters has been recognized, and the relevant formation mechanisms have been studied extensively [19]. When e+ are implanted into such a material at kinetic energies ranging from several  to a few , they scatter off atoms and electrons () in the bulk and are slowed to eV energies within a few ps. With efficiencies ranging from  to , the slow  capture either bound  or those liberated in prior collisions and form . These tend to accumulate in defects of the material due to the reduced dielectric strength and hence increased  binding energy. In the pores,  repeatedly bounces off the cavity walls and eventually approaches complete thermalization with the target material.
While some  are lost due to so-called pick-off annihilations of  with the molecular  of the cavity walls, a sizable fraction diffuses out of the film at thermal energies. The overall  yield as well as the final velocity distribution depend upon the characteristics of the target material (in particular, its pore structure), the implantation depth, and the target temperature. Measurements using positron annihilation lifetime spectroscopy have shown that the  fraction (outside the sample) can reach  in silicon-based polymer materials cooled to  [20]. In other experimental work, it was shown that the energy profile of  emitted from the surface of a silica film at room temperature followed a Maxwell–Boltzmann distribution and was compatible with the Ps being fully thermalized [21]. We are currently conducting experiments in order to determine the optimal converter material and  energy in terms of  yield. Furthermore, we are investigating how well the emitted Ps is thermalized at very low target temperatures.
The photo-excitation of  to Rydberg states requires photon energies close to the binding energy of . Laser systems at the corresponding wavelengths () are not commercially available. We are therefore planning to perform a two-step excitation, from the ground state to the  state (), and then to the  Rydberg band (). Two pulsed-laser systems, both of which are pumped by a Q-switched Nd:YAG laser (), are currently under development. Both systems must provide sufficient power to excite the emitted  within a few  and must be geometrically matched to the expanding cloud. Furthermore, the bandwidths of the lasers must be tailored to the transition linewidth broadened due to the Doppler effect as well as level splitting due to the motional Stark effect and the linear and quadratic Zeeman effect (the latter three creating the Rydberg band).
An overview sketch of the planned laser setup is shown in Fig. 1. The first system is a dye laser whose optical cavity uses prisms as selective elements in order to produce the large bandwidth required. After up-conversion with a succession of second and third harmonic crystals, the laser will produce  radiation with pulse energies of . The second system combines an optical parametric generator with an amplification stage, both of which make use of commercially available periodically poled crystals (i.e. materials with birefringent layers in alternating orientation). In these non-linear media, the pump photons are down-converted to an idler and a signal photon (signal wavelength , pulse energy ). The wavelengths can be precisely controlled by adjusting the crystal temperature. Due to the broad bandwidth, the coherence time of the second system will be several orders of magnitude shorter than the pulse length. Nevertheless, excitation fractions in the Rydberg band of  or higher can be expected.
Antihydrogen recombination and beam formation

An  recombination scheme based on resonant charge exchange with  was first proposed almost twenty years ago [22]. The reaction proceeds according to the equation

             (1)

where the star denotes a highly excited Rydberg state. This reaction owes its appeal to the fact that the cross-section scales approximately with the fourth power of the principal quantum number. In addition, it creates  in a narrow and well-defined band of final states. Most importantly,  formed with  at rest is created with a velocity distribution dominated by the , hence the surrounding (cryogenic) environment [23]. This is in stark contrast to the rather high  temperature observed when using the nested-well technique pioneered by ATRAP and ATHENA [24,25]. Our proposed technique is conceptually similar to a charge exchange technique based on Rydberg cesium [26] which has been successfully demonstrated by ATRAP [27].
The principle is illustrated in Fig. 2. The  emitted from the porous insulator material are excited to Rydberg states. They then traverse a Penning trap region in which several  have been accumulated and stored. The charge exchange cross-section is very large ( for ) and reaches a maximum when the  and  relative velocities are matched. Taking into account the corresponding kinetic energy, as well as a smaller contribution due to converted internal energy,  is created at velocities of .

Fig. 2. Proposed method for  recombination and subsequent acceleration.

 

Fig. 3. Electrode geometry and resulting equipotential lines (magenta) as employed for Stark acceleration of Rydberg atoms.
While neutral atoms are not sensitive (to first order) to constant electric fields, they do experience a force when their electric dipole moment is exposed to an electric-field gradient. Since the dipole moment scales approximately with the square of the principal quantum number, Rydberg atoms are especially amenable to being manipulated in this way. This technique is related to the splitting of spectral lines due to the presence of external electric fields (the Stark effect) and is therefore called Stark acceleration.
Recently, Stark acceleration has been successfully demonstrated by one of the AEGIS groups with (ordinary) hydrogen after excitation to the  states [28,29]. In these experiments, accelerations of  were achieved using the electrode geometry shown in Fig. 3. A hydrogen beam traveling at  was stopped within  over a distance of only . We intend to use a similar electrode configuration, adapted to the cylindrical geometry of a Penning trap, to accelerate the created  atoms to about  in the direction of the deflectometer apparatus. Prior to the beam formation, remaining  can be transferred back to the accumulation trap in order to be reused in the next cycle.
 
 

Fig. 4. Principle sketch of the Moiré deflectometry technique with two identical gratings and a position-sensitive detector.
Unlike atomic fountain interferometers, such a device does not necessit a te trapped atoms. Furthermore, neither spatial nor temporal coherence of the incoming particle beam are required. Due to these unique features, an antimatter gravity experiment based on a Mach–Zehnder interferometer was proposed some ten years ago [32]. However, interactions between (anti)-matter waves and material gratings can lead to a number of decoherence effects in matter wave diffraction: quenching of metastable states; deflection of Rydberg states in field gradients; transitions between sub-levels of Rydberg states; and annihilation of antiatoms on the grating. Furthermore, the technique places a very stringent limit on the acceptable beam divergence, which must be smaller than the diffraction angle  , where  is the de Broglie wavelength of the matter wave. All of these limitations can be alleviated by increasing the grating period relative to the de Broglie wavelength. At the point where  , diffraction no longer occurs.
The resulting device is the so-called Moiré deflectometer, in which diffraction on the gratings is replaced by a (classical) shadow pattern of those particles that converge onto the third grating. Interestingly, the gross characteristics of the interferometer are retained [33], in particular, the vertical displacement of the interference pattern according to Eq. (2). A three-grating Moiré deflectometer has been used to measure the local gravitational acceleration to a relative precision of  with a beam of argon atoms traveling at an average velocity of  [33]. In departing from the three-grating deflectometer, we intend to replace the third grating by a position-sensitive silicon strip detector (see Fig. 4). Thereby the overall transmission of the apparatus is increased by the inverse of the grid’s open fraction (roughly a factor of three).
The value of  is extracted from the primary observables (time of flight  and vertical displacement of the fringe pattern ) in the following way, as illustrated by Monte-Carlo simulations performed by us: First, the ensemble of all annihilation events on the detector is plotted as a function of , as shown in Fig. 5(a). These events are binned in symmetric classes of , one of which is shown shaded in dark blue in the figure. Secondly, the vertical displacement  of the fringe pattern is extracted for each of the count classes, as illustrated in Fig. 5(b). Thirdly, the vertical displacement for all count classes is plotted against the mean time of flight in the class. A quadratic fit to that graph, as shown in Fig. 5(c), will then yield . In these simulations, a grating period of  was used, and a finite detector resolution of  was taken into account.
Our simulations have shown that in order to perform a measurement of  to  relative precision, about  atoms at a temperature of  will be required. This could be achieved within about 2–3 weeks of data taking, assuming the AD beam is shared among four experiments. The grids and detector must be kept precisely aligned for the entire data taking period, which can be achieved by an auxiliary laser beam. The zero position of the vertical displacement, (i.e. in the absence of gravity) is most conveniently obtained by performing a calibration measurement with the gratings and detector rotated by  about the beam axis.

Gravity measurements

In matter wave interferometers of the Mach–Zehnder type [30,31], three identical gratings are placed at equal distances  from each other. The first two gratings produce an interference pattern at the location of the third. That pattern has the same period  as the gratings, and its position perpendicular to the diffracted particle beam can be determined precisely by displacing the third grating and recording the overall transmission with a particle detector. Under the influence of gravity, the interference pattern is vertically displaced (it ‘‘falls”) by a distance

            (2)

where  is the local gravitational acceleration and  is the time of flight  between each pair of gratings of a particle beam traveling at velocity .

Fig. 5. Analysis of deflectometer data (Monte Carlo simulation): (a) detector events are binned according to time-of-flight count classes, (b) the vertical displacement of the fringe pattern is determined for each count class, and (c) a quadratic fit to the plot of vertical displacement versus mean time of flight yields the local gravitational acceleration

 
This text was taken from the article Kellerbauer et al. - AEGIS Collaboration, Nuclear Instruments and Methods in Physics Research B 266 (2008) 351-356 – PDF , courtesy AEGIS Collaboration.
Update: A. Kellerbauer (AEGIS collaboration), The AEGIS experiment at CERN Measuring the free fall of Antihydrogen, Hyperfine Interact, 209 (2012) 43 (2012) - PDF
 
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