Control, measurement, and propagation of entanglement in photon pairs generated through type-II parametric down-conversion

We present a study of the effects of dispersion on the properties of photon pairs generated by type-II collinear spontaneous parametric down-conversion. Specifically, we take into consideration the effects of a chirped pump, as well as of dispersive propagation of the photon pairs. We present expressions for the joint amplitude both in the spectral and temporal domains, as well as for the chronocyclic Wigner function of heralded single photons, which fully characterizes the single-photon spectral (temporal) properties. On the one hand, we show that unwanted effects of pump chirp in terms of the heralded single-photon duration can be suppressed for states designed to be factorable and spectrally elongated. On the other hand, we show that pump chirp constitutes an effective tool for the control of the degree of photon-pair entanglement. We show that when frequency-entangled photon pairs propagate through a dispersive medium, entanglement can "migrate" between the modulus and phase of the joint temporal amplitude.