Understanding Coherence vs. Entanglement in Quantum Eraser

[Erik Ayer]

TL;DR Summary

There is a tradeoff between coherence and entanglement which makes it necessary to use coincidence detection to observe interference of lack of interference. How does this work in...

As I understand it, either coherence or entanglement can be in an experiment or some of each, varying continuously between the two. Because of this, coincidence detection is needed to pick out interference patterns among all the data. Coherence would refer to quantum waves of photons taking different path not having a consistent phase relationship, as I understand it, so sending one of a pair of photons through a double-slit would mean they aren't in phase at the slits, and therefore the interference pattern would shift for each photon.

If that's the case, how is one specific pattern filtered out via coincidence? It would seem like the photons that are not going through the double slit would need to be separated somehow according to phase shift. How is this done for the quantum eraser? Here's the wikipedia diagram:

The upper beam is going directly to a detector. There is the 0.3mm mask in front of it, as well as a 1nm filter. Does that thin slit somehow correlate to a specific interference pattern on the lower part? I can't see how that would happen.

Looking at Birgit Dopfer's experiment:

Again the lower beam goes through a double slit, but in this case the entanglement in through momentum. Depending on the position of D_2, the momentum, and hence which slit an entangled partner went through, can be preserved or lost. Without coincidence there is no interference, and I think this is because the photons are not coherent and there are infinite overlapping interference patterns that add up to a Gaussian distribution. Assuming that D2 is at f such that which-way information is lost.

 

[Cthugha]

The double slit is essentially a momentumsensitive measurement. A beam that enters the double slit at normal incidence will create a different interference pattern compared to a beam that hits the slit at an angle due to the path length difference to the two slits that arises, when the beam arrives at an angle. If the range of possible momenta is large, then also the range of possible phase differences is large, which means that the light is spatially incoherent. For any momentum, you will get a well defined interference pattern. The sum over all momenta will not result in a well-defined interference pattern.

If you now use a spatially narrow detector on the other side of the setup (or place a spatially narrow filter in front of it) in the correct manner, it acts as a momentum filter. In that case, you postselect only on this subset of photons with well-defined momentum which has better spatial coherence compared to the set of photons of all possible momenta.

 

[Erik Ayer]

I've been puzzling over this for a while and think I understand how photons with specific momenta are being selected. In the upper arm, the lens focuses all photons with a certain momentum to a certain spot in a plane at the focal length of the lens. So photons arriving at the center spot directly behind the lens all were going straight down the axis between the center of the BBO and lens. Coincidence with the lower-arm photons gives the interference of photons traveling straight down that axis between the center of the BBO and double-slit.

If the detector in the upper arm is at 2f, this would be like a blurry image of the downconverted light from the BBO, wouldn't it? So the detector would pick up photons from some range of momenta and even if each set of photons from a given momentum were creating interference patterns, there would be coincidences with phtons over a range of momenta which would add up to a mess. Is that correct?

If a screen with a pinhole were put at f behind the lens to only allow photons with the direct-along-the-axis momentum to get through, that would eliminate the problem where the detector could get photons of a range of momenta when it was at 2f. The pinhole would be the momentum filter. It seems like there would still be no interference pattern because where a photon was detected in the x-direction would carry information as to which slit its entangled partner went through. Both photons would still be moving parallel to the axes - their momenta would still be straight along those axes. But the entanglement would then seem to be one of position, like was a photon to the left and going through the left slit vs. being on the right and going through the right slit. Putting the upper detector at f, right behind the pinhole, erases that position-within-the-beams entanglement.

I think this is wrong. Momentum is entangled, not position. What is the correct way to think about this?

 

[Erik Ayer]

Ha, I think I figured it out. For objects that are far from a lens, the light from a point on the object is close to parallel when it gets to the lens. If the object is close, then the light from a point on the object is not parallel. Where the image focuses will be different - I think, further away from the lens. That also means that the momentum of the (non-parallel) light can carry information of its "position," or which slit it goes through.

That would explain why interference could be controlled by where the detector was positioned. I read in an older physics forums post that when the slits were far from the BBO, there was interference without coincidence but it couldn't be controlled because entanglement was gone. Hence, the tradeoff between coherence and entanglement.