Gaussian functions are optimal for waveguided nonlinear-quantum-optical processes

Many nonlinear optical technologies require the two-mode spectral amplitude function that describes them---the \emph{joint spectral amplitude} (JSA)---to be separable. We prove that the JSA factorizes \emph{only} when the incident pump field and phase-matching function are Gaussian functions. We show this by mapping our problem to a known result, in continuous variable quantum information, that only squeezed states remain unentangled when combined on a beam splitter. We then conjecture that only a squeezed state minimizes entanglement when sent through a beam splitter with another pre-specified ket. This implies that to maximize JSA separability when one of the (pump or nonlinear medium) functions is non-Gaussian, the other function \emph{must} be Gaussian. This answers an outstanding question about optimal design of certain nonlinear processes, and is of practical interest to researchers using waveguide nonlinear optics to generate and manipulate quantum light.