Characterizing entanglement in pulsed parametric down-conversion using chronocyclic Wigner functions

Abstract

Pulsed parametric down-conversion (PDC) processes generate photon pairs with a rich spectral-temporal structure, which offer an attractive potential for quantum information and communication applications. We investigate the four-dimensional chronocyclic Wigner function ${W}_{\mathrm{PDC}}({\ensuremath{\omega}}_{\mathrm{s}},{\ensuremath{\omega}}_{\mathrm{i}},{\ensuremath{\tau}}_{\mathrm{s}},{\ensuremath{\tau}}_{\mathrm{i}})$ of the PDC state, which naturally lends itself to the pulsed characteristics of these states. From this function we derive the conditioned time-bandwidth product of one of the pair photons, a quantity which not only is a valid measure of entanglement between the PDC photons but also allows us to highlight a remarkable link between the discrete- and continuous-variable descriptions of PDC. We numerically analyze PDC processes with different conditions to demonstrate the versatility of our approach, which is applicable to a large number of current PDC sources.