Phonon-phonon interactions due to non-linear effects in a linear ion trap

We examine in detail the theory of the intrinsic non-linearities in the dynamics of trapped ions due to the Coulomb interaction. In particular, the possibility of mode-mode coupling, which can be a source of decoherence in trapped ion quantum computation, or can be exploited for parametric down-conversion of phonons, is discussed and conditions under which such coupling is possible are derived.