Analysis of a quantum nondemolition measurement scheme based on Kerr nonlinearity in photonic crystal waveguides

We discuss the feasibility of a quantum nondemolition measurement (QND) of photon number based on cross phase modulation due to the Kerr effect in photonic crystal waveguides (PCW's). In particular, we derive the equations for two modes propagating in PCW's and their coupling by a third order nonlinearity. The reduced group velocity and small cross-sectional area of the PCW lead to an enhancement of the interaction relative to bulk materials. We show that in principle, such experiments may be feasible with current photonic technologies, although they are limited by material properties. Our analysis of the propagation equations is sufficiently general to be applicable to the study of soliton formation, all-optical switching and can be extended to processes involving other orders of the nonlinearity.