Can light be entangled in polarization and still exhibit interference?

[Erik Ayer]

If one beam from SPDC is sent to a double-slit or similar experiment, it can form interference if the which-way information is erased in the other beam. To detect it, photons have to be counted in coincidence because there will be both interference and "anti-interference" that add up to a single bump with none of the structure. This is a fundamental effect of creating entangled beams.

What are the properties of the anti-interference photons? It seems to me that they would have to be 180 degrees out of phase between the two slits in order to reverse the probabilities of where they will land on the screen. When normal light (as in what Thomas Young used, or a non-downconverted laser beam) hits the slits, the wave splits and goes through both slits where each sub-wave is in phase. How can light end up 180 degrees out of phase, or is the mechanism different than this?

 

[Cthugha]

This is not at all a feature specific to SPDC. Even for "normal" light, as you call it, the double slit interference pattern is a function of the phase diefference of the two beams arriving at the slits. You can change this phase difference for example by changing the angle between the beam and the slits. A beam arriving at normal incidence will show a different pattern compared to a beam arriving at 45 degrees because a path length difference from the source to the two slits arises. Accordingly, a spatially extended light source will not show an interference pattern in a double slit experiment, but a superposition of all of them, which adds up to no pattern at all. This is also the reason why there is a pinhole in front of the double slit in the original Young double slit experiment. It creates an effective point source. You can try this experiment at home quite easily.

SPDC emission necessarily comes from a spatially extended source and covers lots of emission angles. Therefore it does not show interference. However, due to entanglement it allows - loosely speaking - to put the narrow pinhole Young used in the path of the twin beam. This is what the coincidence counting does. You pick a narrow range of angles (which is a kind of momentum measurement) and in coincidence counting you get the same narrow range on the other side, which yields the interference pattern in coincidence counting. Usually the other coincidence counting detector in the twin beam path is placed behind a beam splitter, which yields the 180 degree phase difference, but essentially you can get any kind of interference pattern by moving the detector in the twin beam path around. There is not just 0 and 180 degrees, but everything in between.

 

[Erik Ayer]

This is not at all a feature specific to SPDC. Even for "normal" light, as you call it, the double slit interference pattern is a function of the phase diefference of the two beams arriving at the slits. You can change this phase difference for example by changing the angle between the beam and the slits. A beam arriving at normal incidence will show a different pattern compared to a beam arriving at 45 degrees because a path length difference from the source to the two slits arises. Accordingly, a spatially extended light source will not show an interference pattern in a double slit experiment, but a superposition of all of them, which adds up to no pattern at all. This is also the reason why there is a pinhole in front of the double slit in the original Young double slit experiment. It creates an effective point source. You can try this experiment at home quite easily.

SPDC emission necessarily comes from a spatially extended source and covers lots of emission angles. Therefore it does not show interference. However, due to entanglement it allows - loosely speaking - to put the narrow pinhole Young used in the path of the twin beam. This is what the coincidence counting does. You pick a narrow range of angles (which is a kind of momentum measurement) and in coincidence counting you get the same narrow range on the other side, which yields the interference pattern in coincidence counting. Usually the other coincidence counting detector in the twin beam path is placed behind a beam splitter, which yields the 180 degree phase difference, but essentially you can get any kind of interference pattern by moving the detector in the twin beam path around. There is not just 0 and 180 degrees, but everything in between.

Thank you, that explanation makes a lot more sense than what I was thinking. Would adding a pinhole to an experiment using SPDC and double slits - Dopfer's experiment - reduce the overlap of interference patterns such that one pattern is discernible without coincidence? That would, of course, drop intensity to a really low level.

 

[DrChinese]

Would adding a pinhole to an experiment using SPDC and double slits - Dopfer's experiment - reduce the overlap of interference patterns such that one pattern is discernible without coincidence?

If you make the stream coherent using a pinhole, the related entanglement ceases.

 

[Erik Ayer]

If you make the stream coherent using a pinhole, the related entanglement ceases.

That would be because the diffraction from the pinhole alters the photons' paths and destroys the path entanglement, correct? If a lens focused the signal beam to a point, then let is spread out (slightly) before coming to a double-slit, would that give the required spatial coherence without destroying the path entanglement?

 

[DrChinese]

That would be because the diffraction from the pinhole alters the photons' paths and destroys the path entanglement, correct? If a lens focused the signal beam to a point, then let is spread out (slightly) before coming to a double-slit, would that give the required spatial coherence without destroying the path entanglement?

The coherence is complementary to the entanglement. More of one means less of the other.

Nature is funny that way, eh?

 

[Erik Ayer]

I am thinking through these ideas :)

So, when light from a laser is downconverted, the result is two cones of light where one has a short wavelength in the center, long wavelength on the outside, and the other cone has the opposite - long wavelength in the center. In certain configurations, where the sub-cones of light that have 2x the wavelength of the pump beam cross, there are polarization-entangled photons.

Since the pump beam has finite diameter, these two points where the 2x wavelength light crosses also have finite diameter. I would think that putting those beams through a somewhat longish set of tubes that are the width of the beams would filter out a lot of the light that isn't that wavelength. Path entanglement would still be present, and polarization entanglement would not be affected.

If a standard laser is sent through a double slit, it produces interference. In contrast, if sunlight (as an example) were sent to a double-slit without the benefit of an initial single-pinhole to make it spatially coherent, there would be many interference patterns, shifted and overlapped depending on the angle which the light came from (since the sun is not a point source, there is some variation on the angles which its light comes in at - pardon my grammar). In the case of the downconverted beams going through tubes, would those beams be similar to the laser beam in that all the light is going in one direction and thus, form interference?

 

[DrChinese]

So, when light from a laser is downconverted, the result is two cones of light where one has a short wavelength in the center, long wavelength on the outside, and the other cone has the opposite - long wavelength in the center. In certain configurations, where the sub-cones of light that have 2x the wavelength of the pump beam cross, there are polarization-entangled photons.

The short vs long doesn't really work like that. Due to conservation rules, the sum of the output frequencies is equal to the input frequency (of the laser). So generally the output wavelength is doubled, but that is not a strict requirement. There is also a relationship involving the cone angle, although it is more complicated. The point is that the crystal has sufficient size that the splitting does not really occur at a single, definite, well-defined position. You can read more about the process at the below links, which provide more detail than you will probably ever need to know:

https://arxiv.org/abs/quant-ph/0205171
http://xqp.physik.uni-muenchen.de/publications/files/articles_2001/journmodopt_48_1997.pdf
https://arxiv.org/abs/1001.4182

 

[Erik Ayer]

The second link shows type II SPDC, which is what I was thinking of. I will read through the links you posted carefully, thanks!

 

[Erik Ayer]

The short vs long doesn't really work like that. Due to conservation rules, the sum of the output frequencies is equal to the input frequency (of the laser). So generally the output wavelength is doubled, but that is not a strict requirement. There is also a relationship involving the cone angle, although it is more complicated. The point is that the crystal has sufficient size that the splitting does not really occur at a single, definite, well-defined position. You can read more about the process at the below links, which provide more detail than you will probably ever need to know:

https://arxiv.org/abs/quant-ph/0205171
http://xqp.physik.uni-muenchen.de/publications/files/articles_2001/journmodopt_48_1997.pdf
https://arxiv.org/abs/1001.4182

I am still working through these articles. Unfortunately my understanding of the QM formalism is lacking, which I'm trying to remedy and entanglement is the inspiration.

My comment about the wavelengths of photons in the two cones coming from type II SPDC comes from this and similar pictures:

I think the change in wavelength is on the order of 10nm. From this picture, the green cone looks to be where the downconverted photons are twice the wavelength of the pump beam, and where they intersect, there is polarization entanglement. If I understand correctly, other pairs of photons are momentum, path, or spatially entangled (not sure what term is correct). So if one photon from a pair is found to be on the left side of the beam (say, where the wavelength is twice the input, just to make this easier), then it's entangled partner would be on the left of its beam. Is this an accurate model of what's going on?

Say someone were to take an input beam and shoot it at an opaque disk with two pinholes in it, then through a nonlinear crystal to generate two sub-beams/sub-cones of downconverted light. Possibly a better way to do this would be to take a laser and compress it with a reverse beam expander, pass it through a 50-50 beam splitter and reflect the two sub-beams so they were parallel and very close together. If the crystal is cut for colinear SPDC, both photons of the downconverted pairs will be in the same beam line, although both could be in the left sub-downconverted beam or the right sub-downconverted beam. If those two sub-downconverted beams were sent to a double-slit so they diffracted and overlapped, would they create interference?

This would be similar to having a thick beam in that photons would be in a superposition with respect to their position within the beam, so the double-slit would interfere them. However, most of the other light from the downconversion cones would be blocked out. It seems like they overlapping, shifted interference patterns would be reduced.

How much crack am I on?

 

[Cthugha]

Yes, experiments along the line of this idea have been done, but I am on vacation right now and do not know the reference by heart.

Anyway, the basic physics is as follows: The double slit is basically a measurement of the spread in momenta (which translate into phase differences). The better defined the relative momentum is, the better the visibility of your pattern will be. Of course position correlated photons will also be correlated in momentum, so: yes, you can reduce the spread in momenta this way and by filtering downconverted light correctly, you will at some point be able to see an interference pattern. However, this subset of photons will not be entangled in momentum anymore. In order to have entanglement you need - loosely speaking - correlations which are stronger than possible classically. In order to see an interference pattern, you need to filter the emitted photons drastically. Basically you only have two pretty well defined momenta left. You will find that these photons are still classically correlated, but you cannot break Bell's inequality anymore withthis subset, which means there is no entanglement anymore.

What you can get, though, is interference in one variable and entanglement in a completely different variable. There was a paper by Schleich some time in 2012 in PNAS, which was a nice example for that, but I forgot the title.

 

[vanhees71]

Is it maybe this

http://www.pnas.org/content/109/24/9314.abstract

I abhorr already the abstract. It's already full of misleading statements about the socalled "wave-particle duality", which is overcome with modern QT allready more than 90 years ago! The experiment itself may be interesting nevertheless, and it's a good exercise to understand its outcome in terms of modern QED.

 

[Erik Ayer]

Yes, experiments along the line of this idea have been done, but I am on vacation right now and do not know the reference by heart.

You're on vacation and still answered me? Thank you! This is answering questions I've had for a while.

What you can get, though, is interference in one variable and entanglement in a completely different variable. There was a paper by Schleich some time in 2012 in PNAS, which was a nice example for that, but I forgot the title.

This begs the obvious question as to whether light can be entangled in polarization, while the beams are trimmed to give interference and if so, can one turn this into a quantum eraser? For the question of entanglement vs. coherence, this is spatial coherence, correct? I don't understand what, specifically, is being changed to give coherence (instead of entanglement) unless it's that the pump beam is condensed into a thinner beam - I would imagine that with a thin beam, there isn't as much chance for positions in the downconverted light to be in superposition with respect to position.