Maximizing Purity and Heralding Efficiency of Type-I Down-Converted Photons Using Beam Focal Parameters

We demonstrate theoretically that correlated bi-photons can be generated with high heralding efficiency and high spectral purity for non-collinear Type-I spontaneous parametric down-conversion. In Type-I down-conversion, the generated photons have the same linear polarization that is perpendicular to the linear pump polarization. Previously, it was thought that high efficiency and purity could not be obtained for this configuration. We show that the non-collinear geometry gives an additional degree-of-freedom that allows for simultaneous optimization of these source metrics. We predict near-unity ($\approx0.97$) heralding efficiency and single-photon spectral purity by adjusting the beam focal parameters, which can be obtained over a wide range of pump, signal, and idler wavelengths without requiring special crystal dispersion characteristics. As an example, we predict a heralding efficiency of 0.97, a single-photon purity of 0.97, and a pair production rate of 0.50 pairs/(s$~$mW$~$THz) using a 400-$\mu$m-long $\beta$-barium borate crystal pumped by a 355-nm-wavelength pulsed laser with a bandwidth of 8-THz. Our work offers a simple and universal approach for producing high-quality quantum photonic states for a wide variety of quantum information science applications.