 Cross-Kerr interaction

An ideal two-mode intensity-intensity coupling makes the phase of each mode depend on the other's photon number. With this sign convention, evolution for time $t$ sends a mode-1 coherent amplitude to $\alpha e^{-i\chi t{n}_2}$ when mode 2 has definite occupation. This single-mode unitary model does not by itself establish a strong, noiseless interaction between real travelling single photons.