Optimised generation of heralded Fock states using parametric down conversion

The generation of heralded pure Fock states via spontaneous parametric down conversion (PDC) relies on perfect photon-number correlations in the output modes. Correlations in any other degree of freedom, however, degrade the purity of the heralded state. In this paper, we investigate spectral entanglement between the two output modes of a periodically poled waveguide. With the intent of generating heralded 1- and 2-photon Fock states, we expand the output state of the PDC to second order in photon number. We explore the effects of spectral filtering and inefficient detection, of the heralding mode, on the count rate, g(2) and purity of the heralded state, as well as the fidelity between the resulting state and an ideal Fock state. We find that filtering can decrease spectral correlations, however, at the expense of the count rate and increased photon-number mixedness in the heralded output state. As a physical example, we model a type II PP-KTP waveguide pumped by lasers at wavelengths of 400 nm, 788 nm and 1930 nm. The latter two allow the fulfillment of extended phase matching conditions in an attempt to eliminate spectral correlations in the PDC output state without the use of filtering, however, we find that even in these cases, some filtering is needed to achieve states of very high purity.