Exploring Energy-Time Entanglement Using Geometric Phase

Using the signal and idler photons produced by parametric down-conversion, we report an experimental observation of a violation of the Bell inequality for energy and time based purely on the geometric phases of the signal and idler photons. We thus show that energy-time entanglement between the signal and idler photons can be explored by means of their geometric phases. These results may have important practical implications for quantum information science by providing an additional means by which entanglement can be manipulated.

Figure 1

(a) Schematic of the experimental setup. $Q_{s1}$ and $Q_{i1}$ are quarter-wave plates with their optic axes oriented at 45° from the horizontal polarization direction. $Q_{s2}$ and $Q_{i2}$ are rotatable quarter-wave plates with their optic axes oriented at angles $135°+{\beta }_s$ and $135°+{\beta }_i$ from the horizontal polarization direction, respectively. $F$ is an interference filter with 10-nm bandwidth, centered at 727.6 nm; ID is an iris diaphragm. The signal and idler photons are collected into multimode fibers and detected using two avalanche photodiodes. A weak lens ($f=1\mathrm{m}$) focuses the pump beam onto the pump mirror $M_p$. (b) Two-photon path diagrams illustrating how the geometric phases $2{\beta }_s$ and $2{\beta }_i$ influence the two-photon interference.

Figure 2

(a) The signal photon path $ABCDA$ through the two quarter-wave plates $Q_{s1}$ and $Q_{s2}$, in alternative 1. (b) Poincaré-sphere representation of polarization states of the signal photon corresponding to path $ABCDA$. Here, RCP represents right circular polarization while LCP represents left circular polarization. A signal photon acquires a phase $2{\beta }_s$ in going through path $ABCDA$, which is equal to half of the solid angle subtended by the closed circuit $ABCDA$ on the Poincaré sphere.

Figure 3

Measured number of coincidence counts as a function of $2{\beta }_s$, the geometric phase of signal photon in alternative 1, for four different fixed values of $2{\beta }_i$, the geometric phase of idler photon in alternative 1. (a) $2{\beta }_i=0$ and $\pi$, (b) $2{\beta }_i=\pi /2$ and $3\pi /2$. The solid lines are sinusoidal fits. The distance of the dashed line from the origin is a measure of the fixed value of $k_0(x_s+x_i)$ to within the period $2\pi$.