Entanglement swapping for generation of heralded time-frequency-entangled photon pairs

Photonic time-frequency entanglement is a promising resource for quantum information processing technologies. We investigate swapping of continuous-variable entanglement in the time-frequency degree of freedom using three-wave mixing in the low-gain regime with the aim of producing heralded biphoton states with high purity and low multipair probability. Heralding is achieved by combining one photon from each of two biphoton sources via sum-frequency generation to create a herald photon. We present a realistic model with pulsed pumps, investigate the effects of resolving the frequency of the herald photon, and find that frequency-resolving measurement of the herald photon is necessary to produce high-purity biphotons. We also find a trade-off between the rate of successful entanglement swapping and both the purity and quantified entanglement resource (negativity) of the heralded biphoton state.

Figure 1

Entanglement-swapping setup. The infinity symbols denote entangled photon pairs. Pulsed pump beams, denoted in blue, are directed into PPKTP-waveguide-based SPDC sources. The active fields comprise the signal from source 1 ($s_1$) and idler from source 2 ($i_2$), while the remaining $i_1$ and $s_2$ are the bystander fields. Dichroic filters (DC) separate the signal and idler fields within the sources, and combine the active fields. ${\text{PPKTP}}_{\text{SFG}}$ is the SFG PPKTP waveguide and ${\text{DC}}_{\text{SFG}}$ is the dichroic filter separating the sum-frequency converted photons from unconverted active photons. Entanglement swapping is performed when exactly one SFG photon is detected with an ideal spectrometer and there is no detection at the fail detector $D_{\mathrm{fail}}$ (a single-photon counting module).

Figure 2

$|\mathrm{\Phi }({\omega }_i,{\omega }_s){|}^2$, the joint spectral intensity (JSI) from a single source with the phase matching function and parameters described in Sec. 4. The two input sources are modeled to be identical. The dashed line is the difference-frequency axis and the dot-dashed line is the sum-frequency axis.

Figure 3

3D contour surface showing $|\psi ({\omega }_{\mathrm{b}1},{\omega }_{\mathrm{a}2},{\omega }_{\mathrm{SFG}}){|}^2$ with contour plots of the projections on the back planes. The 3D contour surface connects values of one-tenth of the maximum value of ${\left|\psi \right|}^2$. The tilt angle resulting from the correlation between ${\omega }_{\mathrm{SFG}}$ and the sum frequency ${\omega }_{\mathrm{b}1}+{\omega }_{\mathrm{b}2}$ is more apparent in the output JSIs of Fig. 6.

Figure 4

Scaling of the (a) ${\omega }_{\mathrm{SFG}}$ spectrum $p\left({\omega }_{\text{SFG}}^n\right)$, and the corresponding (b) rate of entanglement swapping events as $L_{\text{SFG}}$ varies. In (a) straight lines connect calculated points to serve as eye guides. Frequency-resolving measurement of ${\omega }_{\mathrm{SFG}}$ gives discrete outcomes described after Eq. (36) and labeled with the central frequency of the range of frequencies that constitute the corresponding frequency bin.

Figure 5

Figures of merit with frequency-resolving detection. The (a) negativity $\mathcal{N}\left({\omega }_{\text{SFG}}^n\right)$ and (b) purity $\mathcal{P}\left({\omega }_{\text{SFG}}^n\right)$ for the output biphoton state prepared by each measurement outcome are shown for five $L_{\text{SFG}}$ values. A single SPDC source used in this experiment has a purity of 0.82 (indicated with the dashed reference line) and a negativity of 2.89 (not shown). The points indicate calculated values and are connected by lines to serve as eye guides, which should not be taken to represent the actual shape of the curve.

Figure 6

Output JSIs with $L_{\text{SFG}}=0.5$ mm. The dashed line is the difference-frequency axis and the dot-dashed line is the sum-frequency axis. Energy conservation requires that the sum frequency of the output biphoton decreases as the measured ${\omega }_{\mathrm{SFG}}$ increases. The ${\omega }_{\mathrm{SFG}}$ spectrum is relatively narrowband, so these differences are visible but small.

Figure 7

Comparison of the (a) negativity and (b) purity of output states prepared via either frequency resolving or frequency nonresolving heralding over five $L_{\text{SFG}}$ values. The effect of frequency resolution using the same measurement configuration as used in Fig. 5 and the weighted average of Eq. (43) is shown with the blue points, while the orange points show the values when the SFG photon is detected but ${\omega }_{\mathrm{SFG}}$ is unresolved.

Figure 8

Negativity as a function of pair-production efficiency $\eta$. Coherences between the vacuum and the biphoton subsystem are included. Straight lines connect calculated points to serve as an eye guide.

Figure 9

Negativity $\mathcal{N}\left(\rho \right)$ vs $\eta$ for multiple numbers of frequency modes, $N$. The input density matrices for this calculation have no coherence between the vacuum and biphoton subsystems.