Cross-Kerr interaction ($H_{\mathrm{int}}=\hslash \chi n_1{n}_2$)

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.

A one-photon polarization qubit can conditionally rotate another mode's coherent amplitude: the horizontal branch leaves $\alpha$ unchanged and the vertical branch gives $\alpha e^{i\Theta }$. The output is generally entangled. Measuring the polarization in a superposition basis can erase the branch label and prepare a coherent superposition of the two amplitudes.

Normalize the initial polarization qubit and its analysing wave-plate transformation. For a conditional phase rotation of $\pi$, a vertical detection has the displayed probability and prepares $N(|\alpha ⟩+e^{i\varphi }|-\alpha ⟩)$. At $\alpha =0,\varphi =\pi$, that outcome has zero probability, explaining the singular normalization of the absent branch.