Nonlinear coupling of continuous variables at the single quantum level

We experimentally investigate nonlinear couplings between individual quanta of the 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=χ ̂ \^nᵣ ̂ \^nₛ, where ̂ \^nᵣ, ̂ \^nₛ 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 over a range of trap frequencies. With 2.970.3em0exMHz stretch mode and 3.610.3em0exMHz rocking mode frequencies, we observed a coupling of 20.50.3em0exHz∕phonon. Implications for ion trap experiments aiming at high-fidelity quantum gate operations are discussed.