  We experimentally investigate nonlinear couplings between vibrational modes
of strings of cold ions stored in linear ion traps. The nonlinearity is caused
by the ions' Coulomb interaction and gives rise to a Kerr-type interaction
Hamiltonian H = n_r*n_s, where n_r,n_s are phonon number operators of two
interacting vibrational modes. We precisely measure the resulting oscillation
frequency shift and observe a collapse and revival of the contrast in a Ramsey
experiment. Implications for ion trap experiments aiming at high-fidelity
quantum gate operations are discussed.

Quantum Physics