Heralding pure single photons: a comparision between counter-propagating and co-propagating twin photons

We investigate different strategies suitable to generate pure heralded single photons through spontaneous parametric down-conversion, comparing the counter-propagating geometry studied in [1] with more conventional co-propagating configurations which enhance the purity of the heralded photon state through the technique of group-velocity matching. Our analysis is based on the correlation of twin photons in the temporal domain,a non-standard approach that provides a physical view of the mechanisms that permit to eliminate the temporal entanglement of the state and to generate high-purity heralded photons. The Schmidt number associated to the temporal modes, which provides a more quantitative estimate of the purity, is then calculated. The efficiency of the various strategies and the individual properties of the heralded photons thereby generated are also compared.