A way to produce antihydrogen atoms which allows to measure their fall in the gravity field of the Earth has been studied in DAPNIA/SPP . The path proposed will produce positively charged antihydrogen ions (H+), much easier to manipulate than the neutral atoms (H). This allows to slow them down enough to perform the gravitational measurement.
This path consists in the chain of reactions opposite. The antiprotons (p) interact with positronium (Ps) in order to produce atoms of antihydrogen in their fundamental state. The neutral atoms thus formed interact again with Ps to produce the positively charged ion. The target for the antiprotons is then a dense gas of positronium atoms, instead of the cloud of positrons currently used in the antihydrogenexperiments at CERN. The neutral antihydrogen atoms produced in the first reaction, with a rate 1000 times higher, may also be sued for other types of experiments.
Note that positronium may be excited with a laser system in order to control the distribution of excitation levels ( quantum number n). The excited antihydrogen atoms are then produced with a much higher cross section, proportional to n4.
An example of an experiment was proposed where these H+ ions may be slowed down to a velocity of order 1 m/s [*]. Their extra positron is expelled via a laser excitation which produces the energy necessary to detach them. Their low velocity allows to measure the gravity acceleration of these atoms made of antimatter over a distance of a few centimetres compatiblewith an apparatus of reasonable size.
* J.Walz & T. Hänsch, General Relativity and Gravitation, 36 (2004) 561.