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.

Figure 1

(a) Joint spectral intensity for a nearly factorable two-photon-pair source, which fulfills asymmetric group-velocity matching (based on a KDP crystal). (b) Corresponding joint temporal intensity. (c) Joint spectral intensity for a nearly factorable two-photon-pair source, which fulfills symmetric group-velocity matching (based on a BBO crystal). (d) Corresponding joint temporal intensity.

Figure 2

(Color online) Schematic diagrams for the single-source [panel (a)] and double-source [panel (b)] Hong-Ou-Mandel interferometer setups. While these diagrams show, for graphical clarity, PDC sources emitting into noncollinear signal and idler modes, all calculations in our paper refer to type-II collinear sources where the two modes can be distinguished by polarization rather than by optical path.

Figure 3

(Color online) Fourth-fold coincidences (normalized to a unit background) vs temporal delay between two separate heralded single photons interfering through a Hong-Ou-Mandel interferometer. These curves result from (a) a numerical simulation, based on Eq. (22), and (b) the analytic expression in Eq. (29). The different curves in each panel correspond to the following levels of pump chirp: (i)${\beta }_p=0$ (shown in red), (ii) ${\beta }_p=1.19\times {10}^{-26}{\text{s}}^2$ (shown in green), (iii) ${\beta }_p=2.39\times {10}^{-26}{\text{s}}^2$ (shown in blue), (iv) ${\beta }_p=3.58\times {10}^{-26}{\text{s}}^2$ (shown in orange), and (v) ${\beta }_p=4.77\times {10}^{-26}{\text{s}}^2$ (shown in magenta); in this family of curves, lower visibilities correspond to higher levels of pump chirp.

Figure 4

This figure shows the effect of pump chirp on the temporal properties of the two-photon state. In particular we show, plotted as a function of the signal and idler times of emission, (a) the Gaussian function in Eq. (12) for the case of no pump chirp, (b) the rect function in Eq. (12), (c) the square modulus of the product of the functions in the previous two panels, and (d) same as in (c) but for a chirped pump with ${\beta }_p=3.04\times {10}^{-27}{\text{s}}^2$.

Figure 5

(a)This plot shows the behavior of the Schmidt number, which quantifies the degree of entanglement, as a function of the pump chirp parameter for the same source as in Figs. 1a, 1b; this illustrates that pump chirp may be used as a powerful tool to control the degree of entanglement. Panels (b)–(d) show the joint temporal intensity for three different values of pump chirp: ${\beta }_p=0$, ${\beta }_p=2.07\times {10}^{-26}{\text{s}}^2$, and ${\beta }_p=1.01\times {10}^{-25}{\text{s}}^2$ identified along the curve in panel (a).

Figure 6

(Color online) (a) Chronocyclic Wigner function plotted for a source identical to that assumed for Figs. 1a, 1b computed numerically from Eq. (33). Inset: plot resulting from the analytic expression in Eq. (34). (b) Spectral intensity of heralded single photon, in the signal mode, computed as one of the marginal distributions of the CWF. (c) Temporal intensity of heralded single photon, in the signal mode, computed as one of the marginal distributions of the CWF.

Figure 7

Panels (a)–(c): Joint temporal intensity for three different levels of chirp ($\beta =0$, $\beta =-{\beta }_p$, and $\beta =-2{\beta }_p$, corresponding to the following lengths of propagation through fused silica fiber: $L=0$, $L=2.64\text{m}$, and $L=5.28\text{m}$) experienced by the signal and idler photons, and a pump chirp level of ${\beta }_p=-4.77\times {10}^{-26}{\text{s}}^2$. While these plots were computed numerically through Eq. (2), the insets show the corresponding plots computed from our analytical expression [see Eq. (12)]. [(d)–(f)] Chronocyclic Wigner function corresponding to panels (a)–(c). While these plots were computed numerically through Eq. (33), the insets show the corresponding plots computed from our analytical expression [see Eq. (34)].

Figure 8

(Color online) (a) Temporal duration of heralded single photon in the signal mode, as a function of the chirp parameter which characterizes the propagation of signal and idler photon pairs. While the dashed black line was computed through a numerical calculation, the red line was computed through our analytic expression [see Eq. (41)]. In this case the pump is assumed to be chirped with ${\beta }_p=-4.77\times {10}^{-26}{\text{s}}^2$. Note that the temporal duration as a function of chirp parameter exhibits a minimum at the propagation distance corresponding to $\beta =-{\beta }_p$. The dash-dot blue line represents the temporal duration for no external dispersion. (b) Closeup of the previous panel in the region of the minimum.

Figure 9

(Color online) This plot shows, as a function of the propagation distance $z$, the Schmidt number $K$ and the reduced temporal Schmidt number $K_{m,T}$; we present both numerical (dashed black line) and analytic (continuous red line) curves for these functions. The minimum allowed value $K=1$ is indicated with a dash-dot line. While $K$ remains constant during propagation, $K_{m,T}$ exhibits a rich structure. From these plots we may infer that entanglement migrates from the modulus of the joint temporal amplitude (at $z=z_1$) to its phase (at $z=z_2$), back to the modulus (at $z=z_3$), and slowly to the phase once more (for large $z$).