Variation on Dopfer Experiment - Why Won't This Work?

[peter0302]

FTL Signaling with Dopfer - Why Won't This Work?

I think at the outset we will all agree that the following will not work, but the more interesting question will be why.

Take the Dopfer experiment described at:
http://www.quantum.univie.ac.at/publications/thesis/bddiss.pdf

On page 36, Dopfer shows an apparatus consisting of an "upper half" and a "lower half", each corresponding to one of two entangled photons. The "upper photon" is sent to a Heisenberg lens which focuses parallel photons to a single point on the focal plane. The detector D1 is placed fixed at the focal point. Photons in the "lower half" are sent through a double slit and then to a detector D2 which scans the x-axis. An interference pattern emerges when the photons are correlated with those in the upper half that have been detected at the focal point. The detection at the focal point results in position information being utterly destroyed, thereby allowing the interference pattern in the lower half to emerge.

JesseM and I have discussed extnesively why the coincidence circuitry is necessary and the conclusion we both reached is that with each position of D1, only a subset of photons is detected with position information destroyed; for the rest, it is possible, at least in principle, to detect position information. The only way to correlated that subset in the upper half with image-generating photons in the lower half is with slower-than-light coincidence circuitry.

And so I asked, what would happen if we destroyed position information for ALL the upper half photons corresponding to lower half photons that went through the slits? Would ALL of the lower-half photons going through the slits therefore generate an interference pattern? If so, could this not be used as the coveted switch to send a binary FTL signal?

So the means I thought of to destroy the position information in the upper half photons is as follows. In points in space in the upper half corresponding to the slits in the lower half, place two small biconvex lenses such that the focal point of those lenses is placed exactly where the slits would be in the lower half. All photons passing through those focal points will emerge parallel from the lenses. Then when those photons reach the Heisenberg lens shown on page 36 of Dopfer's thesis, ALL of them will strike the detector D1, thereby destroying position informaiton.

The question is: will an interference pattern then emerge from ALL photons striking detector D2, eliminating the need for coincidence counting? If not, why not?

Jesse suggested that placing the small converging lenses in the upper half might be problematic due to the HUP, that by isolating the photons to two small points in space, there is too much uncertainty in their momentum to ensure that they reach the correct lens and emerge parallel. However, the radius of the lenses could be large enough to account for the uncertainty, since the focal points would be quite small (5 mm or so).

So, why won't this work?

 

[peter0302]

For those that might object that we can't control which directions the entangled photons will travel out of the crystal, and therefore we can't actually ensure that all of the photons that reach the slits have had their twins' position information destroyed, Cramer's experiment actually addresses this:

"An argon-ion laser producing vertically polarized 351 nm UV light pumps a beta barium borate (BBO) crystal, producing two 702 nm infrared photons that are collinear with and momentum-entangled with the pump beam by the process of Type II collinear spontaneous downconversion."

He then uses a polarizing splitter to separate the two entangled photons, and therefore can ensure that each member of the pair is directed to the appropriate apparatus.

 

[peter0302]

Wow. More than a day and no one has any thoughts? :)

Another point I'd ask about is what can we say about the photons once they've been collimated by means of the two small lenses? Do they now have very precise momentum and therefore imprecise position? Can we no longer be sure they'll all strike the Heisenberg lens?

On the flip side, couldn't we make the Heisenberg lens arbitrarily large to compensate for this?

 

[JesseM]

peter, since we had a little PM exchange on this, would you mind if I just quoted our PMs and then offered my response to your last PM here?

 

[peter0302]

Oh, of course. :)

 

[JesseM]

OK, here was our PM discussion (anyone interested in following this discussion should consider taking a look at my discussion with peter starting in post #40 of this thread):

Hey Jesse -
I've been thinking more about our discussion about Dopfer and wanted to run an idea by you

We agreed, I think, that any photons incident orthogonal to the Heisenberg lens would strike a detector located at the focal point of the lens, and that any entangled twins corresponding to those particular photons would create an interference pattern if they passed through a double slit.

My current idea is to replicate the Dopfer experiment, the "top half" being the Heisenberg lens component, and the "bottom half" being the double slit component. In the top half, we'd place two very small converging lenses at points such that the focal points of those lenses correspond to the locations of the slits in the bottom half. Any photons passing through the other end of those lenses would, therefore, be parallel and orthogonal to the plane of those lenses, i.e. collimated. We'd then place a third, larger converging lens in the path of those photons, such that ALL of those photons would then arrive at a detector D! placed at the focal point of that third, larger lens.

In that configuration, in the bottom half, we should expect to see an interference pattern behind the slits because the entangled twin of _every_ photon that passes through the slits would have been detected at D1 with which-path information destroyed, without the need for coincidence circuitry. And if we move D1from the focal point to the imaging plane, thereby making which path detection theoretically possible, the interference pattern should disappear.

Do you see any reason why this couldn't work?

Thanks for your input!

Peter

Hey Jesse -
I've been thinking more about our discussion about Dopfer and wanted to run an idea by you

We agreed, I think, that any photons incident orthogonal to the Heisenberg lens would strike a detector located at the focal point of the lens, and that any entangled twins corresponding to those particular photons would create an interference pattern if they passed through a double slit.

My current idea is to replicate the Dopfer experiment, the "top half" being the Heisenberg lens component, and the "bottom half" being the double slit component. In the top half, we'd place two very small converging lenses at points such that the focal points of those lenses correspond to the locations of the slits in the bottom half. Any photons passing through the other end of those lenses would, therefore, be parallel and orthogonal to the plane of those lenses, i.e. collimated.

We'd then place a third, larger converging lens in the path of those photons, such that ALL of those photons would then arrive at a detector D! placed at the focal point of that third, larger lens.

In that configuration, in the bottom half, we should expect to see an interference pattern behind the slits because the entangled twin of _every_ photon that passes through the slits would have been detected at D1 with which-path information destroyed, without the need for coincidence circuitry.

It can't be true that this is what orthodox QM predicts, because we know in other cases the total pattern of hits at the bottom half doesn't show interference, and for it to change based on what apparatus we put on the top half would imply FTL communication which Eberhard's theorem proves is not possible if the equations of orthodox QM are correct. But it is an interesting puzzle to think about what QM would predict about the scenario you present--would it be a violation of complementarity, with the which-path information erased on the top half but no way of finding an interference pattern in any subset of the data in the bottom half? I doubt it, but I'm not to sure about what the reason would be.

My best guess would be that the problem has to do with your "two small lenses". A photon is free to travel along a path going at any angle from the positions corresponding to the slits, so what's to prevent a photon going from the position of one slit to the "wrong" small lens, and therefore not being properly collimated by it? I suppose you could place a barrier between the lenses and extending up in between the positions corresponding to the two slits, but I wonder if you might not then have a significant number of photons which do come from one of the two slit-positions but are then absorbed by the barrier instead of going through the small lens, so that a significant proportion of photons at the bottom half would not have corresponding detector pings on the top half. It may be that when dealing with a double-slit apparatus, the only way to guarantee that virtually all the photons that make it through the slit also go through the lens is to make the lens quite large compared to the distance between the slits.

My best guess would be that the problem has to do with your "two small lenses". A photon is free to travel along a path going at any angle from the positions corresponding to the slits, so what's to prevent a photon going from the position of one slit to the "wrong" small lens, and therefore not being properly collimated by it?

I thought the same thing at first, but remember that the crystal (generating the entangled photons), the respective focal points, and the respective lenses must form a straight line, so the directions aren't entirely arbitrary. As long as the apparatus is arranged properly, a photon passing through the focal point of a parituclar lens will not pass through the other lens - it might pass through _neither_ lens, but then the entangled twin would pass through _neither_ slit as well.

It's a fun puzzle indeed, although I tend to agree with you that this can't be right...

Thanks again for your input!

I thought the same thing at first, but remember that the crystal (generating the entangled photons), the respective focal points, and the respective lenses must form a straight line, so the directions aren't entirely arbitrary.

Right, but because of the uncertainty principle, any precise measurement of the position introduces a lot of uncertainty into the momentum, so you can't assume it's like classical particles where the direction prior to traveling through the slit is the same as the direction coming out of the slit (if you have the Feynman Lectures on Physics, you can look at his introductory analysis of the double-slit experiment, where he sums up two paths which go in straight lines from the source to the slits, but then each go at totally different angles from the slits to a point on the screen, in order to calculate the probability the particle will end up at that point on the screen). Of course there is no actual doubl-slit on the top apparatus, but if we're looking at hits on the top apparatus that correspond to hits on the bottom one, the fact that we know the bottom photons went through one of the slits would also mean that in retrospect the corresponding photons at the top apparatus behave like their positions were narrowed down to those of the two slits.

Right, but because of the uncertainty principle, any precise measurement of the position introduces a lot of uncertainty into the momentum, so you can't assume it's like classical particles where the direction prior to traveling through the slit is the same as the direction coming out of the slit

Sure, but it's only a problem if the uncertainty is great enough to put the possible path of the photon outside the appropriate lens. But this wouldn't be the case.

If the wavelength is 702 nm, like Dopfer's experiment, and the slit width is .75 um, also like Dopfer's experiment, then the uncertainty in momentum of the photon after it passes through the focal point of the lens will be much less than that necessary to put it outside of the radius of the lens.

Since
delta_X * delta_P <= h/4pi,
and delta_X = slit width "a", then
delta_P <= h / (4 pi * a)

delta_P is also equal to (h / lambda) * sin(theta), where theta is the angle of deviation from the focal point that the photon might undergo due to Heisenberg uncertainty. So:

h / lambda * sin(theta) <= h / (4 pi * a)
sin(theta) <= lambda / (4 pi * a)

lambda = 702x10-9 m
a = .75x10^-6 m

theta <= 4.3 degrees

So the Heisenberg uncertainty in the angle is only 4.3 degrees - very small considering how close the lens would be to the "virtual" slit in the upper half of the experiment. Therefore HUP shouldn't prevent those photons from striking the correct lens.

In response to your last comments, I had two thoughts:

1. Normally when you see a calculation of how much uncertainty is added to the photon's momentum when it goes through the slits, it's assumed that the photon's wavefunction impinging on the slits was a plane wave with a perfectly well-defined momentum, and total uncertainty about position. In contrast, in this experiment the photons came from a known initial position, so there must already have been a certain amount of uncertainty in their momentum before going through the slits. Perhaps this point isn't too important though--ideally it might be possible to make the initial uncertainty in momentum arbitrarily small by making the source sufficiently far away.

2. Your argument is based on running the Heisenberg microscope backwards, so that photons emerging from a single point in the focal plane of each of the two small lenses are perfectly collimated by the lenses. But this only works perfectly if all the photons going to one of these lens came from precisely one point in the focal plane--if the slits have some finite width, then the photons coming out of the lenses won't be perfectly collimated. You can make the slits smaller, but this makes the uncertainty in momentum of the photons coming out the slits greater, which takes us back to the problem I mentioned that some photons might go to the "wrong" lens or miss the lenses altogether. Perhaps if we actually did the detailed math here, there would be some tradeoff between photons that aren't detected because they aren't perfectly collimated by the two small lens and thus aren't focused to the position of the detector behind the third larger lens, and photons that aren't detected because they didn't actually go to the small lens in front of the slit they came from (of course all this talk about which slit a photon came from in a quantum experiment is suspect, but I think you can understand this as something like shorthand for which paths contribute significantly to the final outcome in a path integral).

 

[peter0302]

Thanks Jesse!

On the first comment, there is still some uncertainty in position of the photons coming out of the entangled photon source. I found this paper on Beamlike Type II SPDC, which explains how the photons make a "blob" of coherent light rather than a cone of random photons like in normal SPDC:

http://www.ornl.gov/~webworks/cppr/y2001/pres/119242.pdf

And as you say, as long as the source is far enough away from the two apparatuses, we should be able to assume a fairly straight direction - but not so perfect that we lose position certainty. :) Of course the formal experiment will have to take this into account.

On the second, my response is to look at the Dopfer experiment. The detector at the focal point has some finite size - it's not a point. Yet the interference pattern generated is close to perfect. Which indicates to me that there is a small amount of wiggle-room here in terms of being exactly on the focal point versus being a millimeter or fraction thereof away from it. In other words, yes, passing through the focal point ensures a collimated beam, but passing very close to the focal point results in a photon that is very close to being parallel, if not perfectly so, which, when it strikes the third lens, may still strike the detector.

Further, when the detector D1 in Dopfer is placed between the focal plane and the imaging plane we start to see a slow transition from a Gaussian pattern to an interference pattern, indicating further to me that we don't need perfect accuracy to get the interference pattern - only enough to make the maxima and minima significant enough to be detected outside the margin of error.

What the experimenter must do (i.e., what I'd better do if I want this to result in anything useful is to do the math and figure out how much wiggle room there is. If 90% or so of the photons whose twins pass through the slits in the bottom half wind up striking the detector in the upper half - heck, if 50% of them do - we should still see something resembling an interference pattern in the lower half.

 

[JesseM]

On the second, my response is to look at the Dopfer experiment. The detector at the focal point has some finite size - it's not a point. Yet the interference pattern generated is close to perfect.

What does the interference pattern have to do with it, though? You'd predict an interference pattern in the double-slit experiment even if you explicitly assume the slits have a finite width. I thought we were talking about the issue of whether 100% of the photons that were detected on the bottom apparatus would also have corresponding hits in the focal plane of the large lens.

Which indicates to me that there is a small amount of wiggle-room here in terms of being exactly on the focal point versus being a millimeter or fraction thereof away from it.

Well, by making the detector behind the large lens in the upper apparatus wider than a point, you open up the possibility that even though there'd be interference in the subset of photons in the lower apparatus whose twins went to a particular position in the upper detector's range, the total pattern of photons in the lower apparatus wouldn't show interference because the sum of all these subsets would be a non-interference pattern. In this way complementarity could be preserved without opening up the possibility that the total pattern at the lower apparatus would vary depending on what happened at the upper apparatus. I thought your argument critically depended on the idea that all the photons in the upper apparatus would be detected at a single position, so there'd be no way to mark out subsets of hits at the lower detector corresponding to distinguishable subsets of hits at the upper detector, and thus no way to have the total pattern of photons at the lower detector show non-interference without violating complementarity.

 

[peter0302]

Well, by making the detector behind the large lens in the upper apparatus wider than a point, you open up the possibility that even though there'd be interference in the subset of photons in the lower apparatus whose twins went to a particular position in the upper detector's range, the total pattern of photons in the lower apparatus wouldn't show interference because the sum of all these subsets would be a non-interference pattern.

Yes, the interference pattern in the bottom half reflects the superposition of many interference patterns of subsets of photons. But as we see in Dopfer, if those photons are detected only a short distance from the focal plane, those interference patterns will only be slightly out of sync with one another, resulting in the near-perfect, but not totally-perfect, pattern we see when D1 is moved slightly away from the focal plane, and then we see the pattern continue to morph into a gaussian pattern as D1 is placed further away.

That is the only explanation of how Dopfer gets an interference pattern by placing the detector D1 - which is wider than a point - and keeping it fixed there. Yes, if D1 were _very_ wide, the pattern of photons in the lower half would be totally Gaussian, but like I said, there must be a margin of error for Dopfer's experiment to have worked at all, and I think the key to understanding it is to look at how the pattern slowly morphs as D1 is moved - otherwise, if what you're saying is right, we'd expect either a perfect interference pattern when D1 is at the focal point, or none at all as soon as D1 is moved a wavelength away.

My hypothesis is that the photons that are slighly off from the focal point will make an interference pattern that is slightly out of phase with the "central" one. Indeed, the wider the detector D1 is, the more of a problem this will be.

 

[peter0302]

Here's a picture of what I'm talking about. At the focal plane, the perfectly collimated beams pass perfectly through the focal plane. Photons that are slightly astray still come very close to the focal point. As long as all photons are detected there the interference pattern should resemble that of Dopfer's.