The GBAR experiment (Gravitational Behaviour of Anti hydrogen at Rest) at CERN, aims to measure the free fall acceleration of ultracold neutral anti hydrogen atoms in the gravity field of the Earth. Anti hydrogen ions (one anti proton and two positrons) are produced in the interaction of antiprotons and positronium in two subsequent charge exchange reactions. These ions will be sympathetically cooled with Be + ions to less than 10 µK. The ultracold ions will then be photo-ionized just above threshold, and the free fall time over a known distance measured.
The general theory of multiphoton ionization of hydrogen atoms by simultaneous absorption of several photons is discussed. In the nonrelativistic dipole approximation, the calculations lead to a recursion relationship concerning angular functions and involve radial matrix elements containing many summations over intermediate states. A general method for exactly evaluating several sums is described in detail and applied to the photo-ionization of atomic hydrogen in the ground state. Numerical results are presented as dispersion curves of the transition rate $\frac{{\ensuremath{\sigma}}_{n}}{{I}^{N\ensuremath{-}1}}$ for a range of wavelengths from 1000 up to 7200 \AA{}.