Azimuthal spectrum after parametric down-convertion with radial degrees of freedom

Abstract

Considering the quantum state produced in type I spontaneous parametric down-conversion with collinear, degenerate signal and idler beams, and a Gaussian pump, we show that the azimuthal Schmidt number in the Laguerre-Gaussian (LG) basis increases when the radial indices of the LG modes detected in the signal and idler beams are different. These observations are confirmed by the good agreement between theoretical and experimental results. The theoretical results are obtained by deriving expressions for the probability amplitude to detect LG modes with any combination of azimuthal and radial indices in a down-converted photonic quantum state.