Tracking photon color in Bell experiments

Question

In parametric down-conversion, it is said that a driving photon is converted into two entangled photons whose frequencies add up to the driving frequency. Yet in discussions about entanglement experiments, I have not seen anything about the frequency at the point of detection. What is the story here? Do you have pairs of red and green entangled photons? If Alice detects a red photon does Bob detect a green one and vice versa? Is the color of the photon even known or measured at the point of detection?

EDIT: Thanks to Slavic for the awesome picture posted below. I am struggling now to understand what I see, so let's start with an easy question: obviously, the blue is the driving frequency; what is the mechanism for the cone separation (just ordinary prism effect??) and if so, why are the color rings reversed in the complementary light cones?

Answer 1

Only those photons that travel in the same spatial mode are used, and these are located at the intersection of the cones in which the down-converted photons can be found. As you can see, momentum and energy conservation imply that the colors at the intersection points are equal. Quoting: "Along the intersections of the cones of the same wavelength (in our photograph the green circles) polarization-entangled photon states can be observed."

Answer 2

The cone separation in the picture above has nothing to do with a prizm effect. Spontaneous parametric down-conversion (SPDC) may be considered as a spontaneous version of a nonlinear optical effect of difference frequency generation. It occurs in media with non-zero second order non-linearity, usually non-centrosymmetric crystals. This process is parametric, which means that atoms of the crystal are not excited, leading to conservation of energy for photons: $\omega_p=\omega_s+\omega_i$, where $\omega_p$ is the frequency of the laser pump, and $\omega_{s,i}$ are frequencies of down-converted photons. Moreover, for the process to be efficient the phase-mathcing condition should be satisfied: $\mathbf{k}_p=\mathbf{k}_s+\mathbf{k}_i$, which may be interpreted as momentum conservation for photons.

This condition is hard to satisfy in homogeneous isotropic materials, since due to frequency dispertion $\mathbf{k}(\omega)=\omega n(\omega)/c$ depends on $\omega$ in a nonlinear way. To overcome this difficulty one may use anisotropic crystals, where refractive index for a wave polarized in the plane of optical axis depends on the propagation direction $n=n(\omega,\theta)$ where $\theta$ is the angle between $\mathbf{k}$ and optical axis. For a so-called Type II down-conversion pump and one of the photons are extraordinary, while the other one is ordinary ($e\rightarrow o+e$ phase-matching). So we have the phase matching condition in the following form: $$\mathbf{k}_e(\omega_p,\theta_p)=\mathbf{k}_e(\omega_s,\theta_s)+\mathbf{k}_o(\omega_i,\theta_i),$$ For a fixed $\theta_p$ this equation determines the dependence $\theta_{s,i}(\omega_{s,i})$ of propagation directions for the photons of different frequencies. This gives the cones, shown in the picture we discuss. The pump beam should be exactly in the center of the picture between the two cones for ordinary and extraordinary photons. So the pump is not shown in the picture.

Photons in each cone have a well defined linear polarization - in plane and orthogonal to the plane of optical axis of the crystal, respectively. An exception is the direction where cones for photons with equal frequencies $\omega_i=\omega_s=\omega_p/2$ (green ones in the picture) intersect. That is where the photons entangled in polarization are generated. The original paper describing the first experiment using this scheme may be found here.