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, alternatively, can be exploited for parametric down-conversion of phonons, is discussed and conditions under which such coupling is possible are derived. 
Cyrille Marquet, Ferdinand Schmidt-Kaler and Daniel F. V. James