Spatio-spectral engineering of entangled and single photons in parametric down-conversion

Photon pairs generated through SPDC inherently exhibit spatio-spectral coupling, which implies that photons with different spatial DOFs possess varying spectra. While quantum optics applications often focus on either spatial or spectral DOFs independently, the correlation between them poses a fundamental challenge in protocols involving entangled photon sources or single-mode photon states. Theoretical studies on SPDC, that address both space and spectrum together, are mostly limited to approximate wave functions of photon pairs or involve numerical computations. Such theoretical studies usually consider either monochromatic signal and idler photons (the narrowband approximation), loosely focused pump and collection beams (the plane wave approximation), or infinitesimally thin crystals (the thin crystal approximation). This dissertation aims to bridge the gap between the fundamental theory of SPDC and its practical applications. In particular, we have developed a comprehensive theory that does not rely on a specific pump beam or nonlinear crystal and goes beyond the common narrowband, plane wave, and thin crystal approximations. The developed approach accurately describes the inseparability of spatial and spectral DOF and applies to a wide range of experimental setups. Furthermore, we show that the origin of the spatio-spectral coupling is closely related to the Gouy phase of the interacting beams. We utilize the developed theory, taking into account the spatio-spectral coupling insights, to control the entanglement of photon pairs from SPDC. As an application, we shape the spatial distribution of the pump beam to design an efficient source of high-dimensional entangled states in the spatial DOF. In our second application, we tailor simultaneously the effective nonlinearity of the crystal and spatial distribution of the pump, to engineer single-mode photons.