  We study the counterpart to the multi-photon down conversion in the quantised
motion of a trapped atom. The Lamb-Dicke approximation leads to a divergence of
the mean motional excitation in a finite interaction time for k-quantum down
conversions with k>=3, analogous to the situation in the parametric
approximation of nonlinear optics. We show that, in contrast to the Lamb-Dicke
approximation, the correct treatment of the overlap of the atomic
center-of-mass wave function and the driving laser waves leads to a proper
dynamics without any divergence problem. That is, the wavy nature of both
matter and light is an important physical property which cannot be neglected
for describing the motional dynamics of a trapped atom, even for small
Lamb-Dicke parameters.
