Generating path entangled states in waveguide systems with second-order nonlinearity

Spontaneous parametric down-conversion in coupled nonlinear waveguides is a flexible approach for generating tunable path entangled states. We describe a formalism based on the Cayley-Hamilton theorem to compute the quantum states generated by waveguide arrays for arbitrary system parameters. We find that all four Bell states can be generated in directional couplers with non-degenerate photons. Our method enables one to efficiently explore the phase space of waveguide systems and readily assess the robustness of any given state to variations in the system's parameters. We believe it represents a valuable tool for quantum state engineering in coupled waveguide systems.