We investigate the spatiotemporal structure of the biphoton entangled state produced by parametric down-conversion (PDC) at the output face of the nonlinear crystal. We analyze the geometry of biphoton correlation for different gain regimes (from ultralow to high), different crystal lengths, and different tuning angles of the crystal. While for collinear or quasicollinear phase matching a X-shaped geometry, nonfactorizable in space and time, dominates, in the highly noncollinear conditions we observe a remarkable transition to a factorizable geometry. We show that the geometry of spatiotemporal correlation is a consequence of the angle-frequency relationship imposed by phase matching and that the fully spatiotemporal analysis provides a key to control the spatiotemporal properties of the PDC entangled state and in particular to access a biphoton localization in time and space in the femtosecond and micrometer range, respectively.
Caspani, L., Brambilla, E., & Gatti, A. (2010). Tailoring the spatiotemporal structure of biphoton entanglement in type-I parametric down-conversion. Physical Review A, 81(3), . https://doi.org/10.1103/PhysRevA.81.033808