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

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 $W_{\mathrm{PDC}}({\omega }_{\mathrm{s}},{\omega }_{\mathrm{i}},{\tau }_{\mathrm{s}},{\tau }_{\mathrm{i}})$ 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.

Figure 1

(a) JSI of the decorrelated PDC process. It is not possible to infer the signal frequency from measuring the idler frequency. (b) Corresponding JTI, exhibiting no correlations between signal and idler photon arrival times. (c) Conditioned SPWF of the signal photon. For further information see the text.

Figure 2

(a) JSI of the correlated PDC process. (b) Corresponding JTI, exhibiting inverted correlations. (c) Conditioned SPWF of the signal. For further information, see the text.

Figure 3

The ICTBP and the cooperativity $K$, calculated from the JSA and the JSI function for different degrees of spectral-temporal correlation between the signal and idler. The insets show the respective JSI (left) and JTI (right) for three distinct spectral widths of the pump pulses. For further information, see the text.

Figure 4

The ICTBP and the cooperativity $K$, calculated from the JSA and the JSI function for different degrees of spectral-temporal correlation between the signal and idler under the assumption of a chirped pump pulse. The insets show the respective JSI (left) and JTI (right) for three distinct spectral widths of the pump pulses. For further information, see the text.