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

Abstract

Pulsed parametric downconversion (PDC) processes generate photon pairs with a rich spectraltemporal structure, which offer an attractive potential for quantum information and communication applications. In this paper, we investigate the four-dimensional chronocyclic Wigner function WPDC(ωs, ωi, τs, τi) 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 is not only a valid measure of entanglement between the PDC photons but also allows 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.

Fast time-domain measurements on telecom single photons

“…by taking a cut through the JSI along the lines indicated in Figure 3, and asuming a flat spectral phase (compare Ref. [3]). Only in the case of perfect decorrelation is this cut equal to the complete marginal spectrum.…”

“…If the TBP is higher than the Fourier limit, it is not possible to reconstruct if there are non-linear spectral phases present or how multimode the state has gotten. To get the exact purity one would need to calculate a "conditioned TBP" as shown in Ref [3]: Is is calculated from conditioned bandwidths, i.e. the temporal duration of the signal when the arrival time of the idler is fixed, and spectral bandwidths with fixed idler frequency.…”

“…For a precise measurement without the need of further numerical modeling, one would also need a conditioned marginal temporal intensity, or a JTI measurement, as pointed out in Ref. [3]. This would mean to duplicate the entire QPG setup for the second PDC photon.…”

“…The established means of measuring this quantity is through interference with a known reference pulse [2]. There is also the possibility to ascertain the photon purity in biphoton states generated through spontaneous parametric down-conversion (PDC) by measuring the joint spectral intensity, but only if no non-linear phases terms on the PDC pump and phasematching function are present [3]. This is due to the fact that one needs the complete spectral or temporal information including phase information, which is hard to measure, even though it is possible in both the spectral and the temporal domain.…”

“…There is, however, also the option of combining temporal and spectral intensity information. As the time-bandwidth-product (TBP) can be extracted from intensity measurements, it can be determined with relative ease, and it was shown that the time-bandwidth-product contains information on purity [3,7]. As spectral intensity measurements can be determined both by time-of-flight dispersive fiber spectrometers [8] and commercially available single photon sensitive spectrometers, one only needs a means of temporal intensity characterisation.…”