SPDC pair state ($|\Psi ⟩\simeq |0,0⟩+\eta |1,1⟩$)

The weak-gain state generated by an undepleted pump contains a vacuum term and a photon-pair amplitude. A continuum pair amplitude combines the pump spectrum with a phase-matching factor $L\mathrm{sinc}(\Delta kL/2)e^{i\Delta kL/2}$. Its joint spectrum determines spectral correlations and the purity attainable by heralding one partner.