Theory of cross phase modulation for the vibrational modes of trapped ions

We analyze nonlinear coupling between individual vibrational quanta for trapped ions. The nonlinear Coulomb interaction causes a Kerr-type Hamiltonian, for which we derive an analytical expression for the coupling constant χ. In contrast to a previously published formula [C.F. Roos, T. Monz, K. Kim, M. Riebe, H. Häffner, D.F.V. James, R. Blatt, Phys. Rev. A 77 (2008) 040302(R)], our result is in close agreement with experimental data.