Question about Other Tests of EPR Paradox

[Caroline Thompson]

... So the beam coming out on the left looks like classical light with a limited coherence length (such as all light!) and you're trying to use a Young's experiment outside of the coherence length, so you shouldn't be surprised to find no interference ...

Just a suggestion: Could it be that in many cases it's a matter of appearing to get no interference because the two beams each consist of a mixture of signals, some with identical phase, some with phase differing by 180 deg? The two interference patterns wash each other out.

And why should this be so? Because I think there is good reason to think that there is an almost determinstic effect going on here. The pump laser is causing two output signals to come into existence. Because the pump frequency is twice that of the outputs, the phase of pump and output cannot exactly match. They can be *related*, though. Half the outputs can be effectively in phase with the even laser wave peaks, half with the odd ones. Both members of any given output pair have the same phase relationship to the pump and so are always in phase with each other.

This is the mechanism behind "induced coherence". It also lies behind many other experiments, including the recent proposal for a loophole-free Bell test. See my website for my draft paper on the subject.

Caroline
http://freespace.virgin.net/ch.thompson1/

 

[DrChinese]

...

If you bring the slits closer together, you WILL see an interference pattern (you're now within the coherence length) but I suspect that the angular correlation between the two photons in the entangled pairs is now such, that the other photon will not allow you anymore to decide through which slit the first one went (they are now within the angular uncertainty on the entanglement).

...

cheers,
Patrick.

I would *think* we could align the left and the right sides to sufficiently narrow slits that we could - if there was no detector at all - get an interference pattern on both sides. But you have me doubting this now... :)

 

[vanesch]

I would *think* we could align the left and the right sides to sufficiently narrow slits that we could - if there was no detector at all - get an interference pattern on both sides. But you have me doubting this now... :)

Well, more I think about it, more I believe that you can't.
Indeed, you know that there must be 2 relations satisfied in PDC:

One is the energy relationship:

E_pump = E_idler + E_signal,

the other is the momentum relationship:

k_pump = k_idler + k_signal

this means that you get an angular "rainbow" in the idler and signal beams, no ? So this is polychromatic light, of which each angular sector will not interfere with another angular sector. Or am I mistaking ?

cheers,
Patrick.

 

[seratend]

Yes, but photons that come from the upper hole do NOT give an interference pattern, and so do those from the lower hole. It is only when we have superposed states that we see that, and as you calculate yourself, their combination is absent.
cheers,
Patrick.

Ok, Now I understand where we have the misunderstanding. What I have called |interference+> is the interference pattern produced by a plane wave photon of + polarization interacting through 2 slits and not only one slit (the interference pattern produced by a single source of linear polarization light through two slits of adequate geometry).

As you have written in later posts, you may assume that the photons states do not diffract through the 2 slits, but it is another assumption and you get your result: no interference at all.

However, if we accept, as an hypothesis for this thought experiment (as I understand Dr Chineese figure), we have an ideal EPR correlated source producing near plane waves of entangled photons of polarisation |+> and |->, we get on the left screen the superposition of the interference pattern of + and - polarized light:

|psi_end_interf(x)>=P_interf_left(x).|psi_end> =<x|interference+>|x>|L:+>|R:->|detector->+<x|interference->|x>|L:->|R:+> |detector->
And we have the norm of this vector:
<|psi_end_interf(x)|psi_end_interf(x)>=|<x|interference+>|²+|<x|interference->|²~2. |<x|interference>|²

where |<x|interference>|² is the well known interference pattern (and not a blob) of photons passing through the 2 slits (the 2 slit experiment).
(if the light intensity is low enough, we will get single clicks in the experiment that will construct this interference pattern, but we can’t say that the photon + has passed through the upper or lower slit as in a classical 2 slit experiment)

Anyway, what is the most important (at least for me), is that this state does not depend on what is done on the right side of the experiment (we can remove the right detector or add it: we still have the same result).
The problem of getting an interference pattern on the left screen depends only on local conditions (and not on what occurs on the right side of the experiment): i.e. the “size” of the beam (e.g. light coherence or the size of the local plane wave, related to the capacity of the PDC source) and the geometry of the slits (your following posts). It is the same thing in a classical 2 slit experiment with a single light source where obtaining an interference pattern depends on the light source characteristics, the slits geometry and the detector sensitivity.

Seratend.

P.S. In my previous post I have written (2 errors in a line! :):
P_interf_left(x)=|in_x><in_x| Where |in_x>= |R:+>|x>+|R:->|x>
However, everyone (I hope ; ) has corrected it and use the correct value:
P_interf_left(x)= |L:+>|x>< L:+|<x|+|L:->|x><L:-|<x| (we are measuring the left photons and not the right ones! :)
P.P.S. In my previous posts I was concentrating on the left interference pattern result.

 

[DrChinese]

However, if we accept, as an hypothesis for this thought experiment (as I understand Dr Chineese figure), we have an ideal EPR correlated source producing near plane waves of entangled photons of polarisation |+> and |->, we get on the left screen the superposition of the interference pattern of + and - polarized light:

|psi_end_interf(x)>=P_interf_left(x).|psi_end> =<x|interference+>|x>|L:+>|R:->|detector->+<x|interference->|x>|L:->|R:+> |detector->
And we have the norm of this vector:
<|psi_end_interf(x)|psi_end_interf(x)>=|<x|interference+>|²+|<x|interference->|²~2. |<x|interference>|²

where |<x|interference>|² is the well known interference pattern (and not a blob) of photons passing through the 2 slits (the 2 slit experiment).
(if the light intensity is low enough, we will get single clicks in the experiment that will construct this interference pattern, but we can’t say that the photon + has passed through the upper or lower slit as in a classical 2 slit experiment)

Anyway, what is the most important (at least for me), is that this state does not depend on what is done on the right side of the experiment (we can remove the right detector or add it: we still have the same result).
The problem of getting an interference pattern on the left screen depends only on local conditions (and not on what occurs on the right side of the experiment): i.e. the “size” of the beam (e.g. light coherence or the size of the local plane wave, related to the capacity of the PDC source) and the geometry of the slits (your following posts). It is the same thing in a classical 2 slit experiment with a single light source where obtaining an interference pattern depends on the light source characteristics, the slits geometry and the detector sensitivity.

Seratend.

OK, we have could potentially have interference on the left side under appropriate conditions... But isn't the interference on the left (ignoring the right for the moment) due to superposition of states? And if there is entanglement, doesn't the right side get involved so that it is no longer totally localized to the left? It seems like there would be a superposition of states which are themselves identical (mirrored) on the two sides.

 

[vanesch]

Ok, Now I understand where we have the misunderstanding. What I have called |interference+> is the interference pattern produced by a plane wave photon of + polarization interacting through 2 slits and not only one slit (the interference pattern produced by a single source of linear polarization light through two slits of adequate geometry).

Eh, we're apparently talking about 2 different experiments !
The way I understood DrChinese proposal had nothing to do with polarisation of the photons, and the |L+> state corresponded to "the photon goes to the upper left hole". It was because they are emitted "back-to-back" (this is not realistic, but there can be a certain angular condition) that L+ is entangled with R- (pure geometry if they are strictly back to back).

cheers,
Patrick.

 

[seratend]

Eh, we're apparently talking about 2 different experiments !
The way I understood DrChinese proposal had nothing to do with polarisation of the photons, and the |L+> state corresponded to "the photon goes to the upper left hole". It was because they are emitted "back-to-back" (this is not realistic, but there can be a certain angular condition) that L+ is entangled with R- (pure geometry if they are strictly back to back).

cheers,
Patrick.

I understand now better and beter your previous posts and the possible interpretations of this "simple" experiment!
Now, Dr Chinees, you have to describe better what you want to say. And the best way is to give the ideal state outputted by the source of your experiment.

OK, we have could potentially have interference on the left side under appropriate conditions... But isn't the interference on the left (ignoring the right for the moment) due to superposition of states? And if there is entanglement, doesn't the right side get involved so that it is no longer totally localized to the left? It seems like there would be a superposition of states which are themselves identical (mirrored) on the two sides.

The interference on the left side is the superposition of 2 identical interference patterns (the interference pattern of incident +/- left polarisation photons, i.e supposing that the the plane waves and the slits geometry are correct).
Each left photon state ("before" the diffraction) is entangled in polarisation with a right photon: the global state is:
|psi_end>=|interference+>|L:+>|R:->|detector->+|interference->|L:->|R:+> |detector+>

and if we agree that the spatial distribution of photons of + or - polarization are indentical, we have (the double slit interaction is insensitive to the polarization):
|psi_end> = |interference>[|L:+>|R:->|detector->+|L:->|R:+> |detector+>]

The difference with this experiment and a 2 slit experiment with a detector measuring the which slit the photon passes through is the following one (for example the diffraction of |+> polarization photons):

Before the slits, the photon states may be described by:
|psi>=|space>|+> where |space> is the spatial distribution of the photon plane wave <x|space> ~ 1 on the vicinity of the slits (plane wave condition).
If you like you can change and select a statistical mixture of |+> and |-> photons:
|psi><psi|=|space><space|(|+><+|+|-><-|)
That mixture state is formally analog to the state |psi>=|1>|space>|+>+|2>|space>|-> (where |1> and |2> are for example the states of DrChinese)

We call |interference> the spatial distribution of the interference pattern of a diffracted photon (|<x|interference>|².

We call s1,s2 the slit number 1,2.

We have |space>=|space_nots1s2>+|space_s1>+|space_s2> with <space_s1|space_s2>=0
|space_s1> is the part of the input wave plane covering the slit1
|space_s2> is the part of the input wave plane covering the slit2
|space_nots1s2> is the part of the input wave plane not covering the slit1 and slit2.

Thus, only |space_s1> and |space_s2> propagates through the slit s1 and s2 (this causal analysis, may be formally derived from the SE as well as from Maxwell equations).
i.e just after the slits, we have the state:

|psi_as>=(|space_s1>+|space_s2>)|+>
Now because <x|space_s1>=/=0 (idem for s2) only in the small area of the size of the slits, the propagation of the plane wave (attention y axis) through the SE or Maxwell equations, will enlarge the distribution of state |space_s1> and |space_s2>. At a distance of y of the slits (the screen), we have the new states
|space_s1_y>, |space_s2_y> and

|psi_y>=(|space_s1_y> + |space_s2_y>)|+>

Where this time <space_s1_y|space_s2_y>=/=0 i.e.

|space_s1_y> + |space_s2_y>= |interference> is the well known interference pattern i.e.
|psi_y>=|interference>|+>

(and for the mixture: |psi_y><psi_y|=|interference><interference|(|+><+|+|-><-|)

=> interference pattern at position x on the screen is |<x|interference>|²

Now, if we introduce a local detector just after the slit s1, we will have an interaction between the spatial part of the photon and the detector that we can model with the following projector:
P_s1=|spatial_s1><spatial_s1|(|photon-s1><IS|+(1-|spatial_s1><spatial_s1|)|nophoton-s1><IS|

(state |photon_s1> if photon is located on slit s1, |nophoton_1> if no photon of s1 slit)
<photon_s1|nophoton_s1>=0 (orthogonal state by construction of measurement).
Where |IS> is the initial state of the projector.

Before the slits we have the global state (we include now the state detector+its interaction with the photons:
|psi>=|space>|+>|IS>
Just after the slits and before the detector (wave propagation view):
|psi_as>=(|space_s1>+|space_s2>)|+>|IS>
Just after the detector:
|psi_as_detect>=(|space_s1>|photon-s1>+|space_s2>|nophoton-s1>)|+>

At the screen, we have:
|psi_as_detect>=(|space_s1_y>|photon-s1>+|space_s2_y>|nophoton-s1>)|+>

We now see the difference: |space_s1_y> and |space_s2_y> are not interference patterns, just the spatial extension (blobs) of 2 almost plane waves that have a common spatial extension (the domain of the interference pattern, when there is no slit detector). However, now because the states |photon-s1> and |nophoton-s1> are orthogonal, we have the spatial distribution seen by the screen:

|<x|spatial_distribution>|²=|<x|space_s1_y>|²+|<x|space_s2_y>|²

As we can see, the interferences of photons are created locally at the 2 slits locations together with the propagation of the waves. If you had a local interaction that avoids the superposition of the spatial extension of local plan waves, later (on the screen measurement), you will have not interference. It does not depend on anything else outside this local area (where the photon is located). QM has not modified this physical property.

In your experiment proposal (in my understanding), you have a state that is, locally, a superposition of different polarisations photons falling on a double slit, thus you get the linear superposition of the interference pattern (and not the blob pattern) of each type of photon:
|psi>=|interference>|case1 left photon>+|interference>|case2 left photon>

where case1 and case2 are used to condensate the states of the right side apparatus and the left photons polarisation.
In other words, |interference> is the result of a local interaction that takes care only on left photons and not on what is done on the right ones.

I hope this can help.

It seems like there would be a superposition of states which are themselves identical (mirrored) on the two sides.

Please, do not mix a global state and local views of this global state: this leads to confusions, like EPR experiments. The global state always change instantaneously (like the total energy of 2 particles) through local interactions, while the local views (e.g. through a local measurement) does not change, if a local interaction occurs at the end of the universe where the global state may also be present.
QM has not changed this “classical” behaviour. Only the confusion of global state updates versus local views may lead to confusions (like FTL signalling).

In some EPR experiments, like Aspect, non local measurements are done through the collection of signals coming from different space locations. We thus have non local “artificial” observables (e.g. the product of the spins at 2 space like locations). And it is normal that we get instantaneous change of these observables due to some local interactions. We can either use selective filtering on observables to see or not interference. However, this selective filtering always uses local well known interactions (its physics) and we always rediscover the clustering principle: what is done on the moon does not change what is done in the lab (if yes, therefore, we should have an interaction that explains that: this the current physics approach)

Seratend.

 

[DrChinese]

Now, Dr Chinees, you have to describe better what you want to say. And the best way is to give the ideal state outputted by the source of your experiment.

My sincerest apologies for the lack of clarity in my example and diagram.

I did not intend to imply that the photon beams were being measured as to spin polarization, nor that the source beams had much other properties other than they were in an entangled state.

I am still analyzing your and vanesch's comments, and I greatly appreciate your time spent on this. Any continued assistance in helping me to see the solution is welcomed.

-DrC

 

[Caroline Thompson]

... In some EPR experiments, like Aspect, non local measurements are done through the collection of signals coming from different space locations. We thus have non local “artificial” observables (e.g. the product of the spins at 2 space like locations). And it is normal that we get instantaneous change of these observables due to some local interactions. We can either use selective filtering on observables to see or not interference. However, this selective filtering always uses local well known interactions (its physics) and we always rediscover the clustering principle: what is done on the moon does not change what is done in the lab (if yes, therefore, we should have an interaction that explains that: this the current physics approach).

Yes, you're absolutely right, but this means that no Bell test is ever really violated! If it were, then the above would not be true, since in order for it to be true we need hidden variables and this is precisely what Bell showed to be impossible if quantum mechancs is correct.

Anyway, since no "loophole-free" Bell tests has ever yet been conducted, we at present have the choice:

1. Put all present apparent violations of the Bell test down to the "loopholes" [see http://en.wikipedia.org/wiki/Bell_test_loopholes] and declare Einstein et al to have been right in that all correlations are due to ordinary shared ("hidden") variables, quantum mechanics wrong

or

2. Accept the currently-orthodox view that QM is right and a loophole-free experiment would be able to prove this. But then, as I said, your description would not be enough to explain the obervations. Some nonlocal effects would be needed.

[Sorry to butt in like this! I realize this was not your main point, but I'm afraid you (in common with others such as Anton Zeilinger) are deceiving yourselves if you think the solution to entanglement is that easy.]

Caroline
http://freespace.virgin.net/ch.thompson1/

 

[DrChinese]

[Sorry to butt in like this! I realize this was not your main point, but I'm afraid you (in common with others such as Anton Zeilinger) are deceiving yourselves if you think the solution to entanglement is that easy.]

Come on, Caroline... we have plenty of other threads to cover your philosophical point(s). In this thread, Aspect is accepted and variants are being discussed.

 

[seratend]

While DrChinese is working on our posts to make its constructive comments (and may be, to detect also errors :), I will try to give my point of view.

First, in [my] physics, I always try to separate the logic from the interpretation (it is a personal choice : ). For me interpretation is the domain of philosophy or religion or politics (we are free to choose one or several and change if we want).
Logic (or mathematics if you want) is my only tool to explain (at least it tries) experimental results (an experiment without logic is magic).

Currently QM is formally a statistical tool that describes the experiments [outcomes] statistics and not the outcomes. Each time people want to interpret a quantum state as a classical outcome (i.e. the state is the object, or whatever we want), we have paradoxes.
Having introduced my reductive and very personal position on current QM (I am not trying to say more than I can get from its postulates), I just can’t understand your assertions without a formal logically consistent context (i.e. what is the most difficult to achieve in the resolution of a problem).
I understand, partly, that you want to model deterministically the outcomes of QM experiments (i.e. a deterministic description): Fine, currently this is not addressed by the current QM theory. It would be a real pleasure to have the logical description of such a self consistent deterministic theory of QM outcomes (without loopholes) (i.e. we can therefore compare the benefits of a deterministic versus the statistical description of QM phenomena as we can do with Newtonian mechanics).

When you say “your description would not be enough to explain the observations”. You are right and I have already said that current QM deals only with statistics not with the outcomes. It does not explain the outcomes, just its statistics.

When you say “ I'm afraid you are deceiving yourselves if you think the solution to entanglement is that easy”, you are right: physics is not easy, even the simplest problem is a headache to understand and a nightmare when we add the interpretation (especially when we want to convince other people, like in politics :). Entanglement statistics is a relatively new subject in physics history. However, formally, the entanglement state is well described by mathematics. That’s what I use when I speak about entanglement. In this very *reductive context*, I can say I “understand” entanglement (that’s my claim).

Seratend.

 

[vanesch]

While DrChinese is working on our posts to make its constructive comments (and may be, to detect also errors :), I will try to give my point of view.

Continued in the "decoherence" thread...

cheers,
patrick.