Determination of entanglement by observing only one photon

[fluidfcs]

TL;DR Summary

Is it possible to infer entanglement by observing only one path output from BBO?

Hi everyone, background for my question is here:

In a commonly used SPDC apparatus design, a strong laser beam, termed the "pump" beam, is directed at a BBO (beta-barium borate) or lithium niobate crystal. Most of the photons continue straight through the crystal. However, occasionally, some of the photons undergo spontaneous down-conversion...

and https://www.researchgate.net/publication/45424433_Direct_generation_of_photon_triplets_using_cascaded_photon-pair_sources:

[T]he cut of the BBO ... sets some of the properties of the outcoming photons, like ... whether they exit the crystal in the same direction as the original photon or at different angles

My question is whether it's possible to determine if two photons are entangled without using a coincidence counter but rather by looking for a single photon along a specific path. What I want to do is take a measurement against entangled photon B but not against photon A, while still being able to know that photon A must exist and must be entangled with photon B. Could I use a BBO crystal such that photon B only exists, or only exists at a certain angle, if a corresponding entangled photon A also exists?

 

[Nugatory]

Summary:: Is it possible to infer entanglement by observing only one path output from BBO?

My question is whether it's possible to determine if two photons are entangled without using a coincidence counter but rather by looking for a single photon along a specific path.

Not possible. A single photon that is part of an entangled photon pair is identical in every measurable respect from one that is not.

Even measuring both, it's not possible to determine that the two photons in a pair were entangled. Say I measure one photon on one axis and the other on an axis perpendicular to the first. It turns out that their polarizations were perpendicular; one is H and one is V. But that will happen 50% of the time with any random uncorrelated unentangled photon pair so I haven't learned anything.

Instead I have to satisfy myself that my photon source is reliably generating entangled pairs. To do this I measure a large number of pairs; if I find that every pair consists of one H and V no matter how I orient my polarizers (as long as I keep the perpendicular to one another) then I can be reasonably confident that I'm looking at entangled pairs and not just uncorrelated pairs of photons that happen to be wandering through my detectors at the same time.

 

[fluidfcs]

Not possible. A single photon that is part of an entangled photon pair is identical in every measurable respect from one that is not.

Even measuring both, it's not possible to determine that the two photons in a pair were entangled. Say I measure one photon on one axis and the other on an axis perpendicular to the first. It turns out that their polarizations were perpendicular; one is H and one is V. But that will happen 50% of the time with any random uncorrelated unentangled photon pair so I haven't learned anything.

Instead I have to satisfy myself that my photon source is reliably generating entangled pairs. To do this I measure a large number of pairs; if I find that every pair consists of one H and V no matter how I orient my polarizers (as long as I keep the perpendicular to one another) then I can be reasonably confident that I'm looking at entangled pairs and not just uncorrelated pairs of photons that happen to be wandering through my detectors at the same time.

Thanks for the reply. I'd like to confirm that I understand what you're saying. Let's assume a single proton being fired for the sake of simplicity.

A single photon traveling through a BBO crystal has three possible outcomes:
1. Travel path A
2. Travel path B
3. Travel both paths as en entangled pair

Subsequently, confirming the existence of a photon on either path A or path B doesn't tell you anything about whether there happens to be an entangled proton on the other path.

Is that all correct?

The way I had been thinking about this, which seems to be incorrect if I'm understanding you correctly, was that in the picture below the photon would hit APD A if and only if it also hit APD B, and vice versa, (in which case the two are necessarily entangled) else it hits the beam stop.

 

[Nugatory]

Ah - Sorry, I misunderstood the question

 

[fluidfcs]

Hi, anyone that can offer some guidance here please?

 

[vanhees71]

As I understand it, you describe the usual creation of entangled photon pairs by parametric down-conversion. Here a photon from the laser beam is absorbed by the crystal and splits into two photons running in two direction due to birefringence. If you select a pair at the intersections of the corresponding cones you get an entangled pair of photons, one traveling with momentum $\vec{p}_1$ and the other with momentum $\vec{p}_2$ and being either both with equal polarizations (type I conversion) or with opposite polarizations (type II conversion). An example for type-2 conversion is to have the singlet-polarization state. The full two-photon state is then described by
$$|\Psi \rangle=\frac{1}{\sqrt{2}} (\hat{a}^{\dagger}(\vec{p}_1,\lambda_1=1) \hat{a}^{\dagger}(\vec{p}_2,\lambda_2=-1) - \hat{a}^{\dagger}(\vec{p}_1,\lambda_1=-1) \hat{a}^{\dagger}(\vec{p}_2,\lambda_2=1)|\Omega \rangle.$$
Here $|\Omega \rangle$ is the vacuum state. Which of the two possibilities occurs when a measurement is done described by this state is completely random: The single photons are ideally unpolarized but there are 100% correlations when measuring the polarizations of the photons in the same polarization direction you always get perfectly opposite results, i.e., if the photon with momentum $\vec{p}_1$ is H-polarized (in the defined direction) that with the momentum $\vec{p}_2$ is necessarilly V-polarized and vice versa.

 

[DrChinese]

My question is whether it's possible to determine if two photons are entangled without using a coincidence counter but rather by looking for a single photon along a specific path. What I want to do is take a measurement against entangled photon B but not against photon A, while still being able to know that photon A must exist and must be entangled with photon B. Could I use a BBO crystal such that photon B only exists, or only exists at a certain angle, if a corresponding entangled photon A also exists?

Yes and no. As Nugatory correctly pointed out, a photon cannot be specifically determined to be entangled (or not) by any direct test. However - and I think this is what you are asking: can you assume that the photons that exit the PDC apparatus along paths A and/or B are entangled?

Again, the answer is yes and no. Most will emerge in entangled pairs (assuming the BBo crystal was cut for entanglement). In a hypothetical test, entanglement is "assumed" if the photon goes along path A. In other words, if a photon is detected at A, then there must be an entangled partner headed to B. So in this case, the answer to your question would be YES.

In actual tests, some of the photons are not actually entangled (for various reasons). I could not give you a guess of the percentage, but within a typical lab it is not very high. To make sure the pairs should be considered as "entangled" for experimental purposes, the detection times for each A/B pair are usually required to be within a small window of time T=|T(A)-T(B)| - a coincidence window - of perhaps 25 nanoseconds or so. Of course, T is adjusted for each photon's path length to the respective detector - in approximate terms, a photon travels about a foot in a nanosecond. Such a requirement is not strictly necessary, this is done to distill the results to a subset of high quality entangled pairs. Note that there is definitely no requirement that pair members leave the crystal at precisely the same time, and in fact they won't. There is always a degree of uncertainty, and the precise origination time and initial starting point of each entangled photon should not be considered as sharply defined values.

So for your case, you would simply drop the coincidence requirement T. Each time you have a detection at A, you assume there was a photon at B that you route in whatever direction you like. That won't be true 100% of the time, but it will happen frequently enough that you can probably move to the next step of your idea.

 

[vanhees71]

Yes and no. As Nugatory correctly pointed out, a photon cannot be specifically determined to be entangled (or not) by any direct test. However - and I think this is what you are asking: can you assume that the photons that exit the PDC apparatus along paths A and/or B are entangled?

I don't understand what both of you mean by "photons that exit the PDC apparatus along paths A and/or B". As I tried to explain above, there's always one photon of the pair leaving the apparatus with momentum $\vec{p}_1$ and the other with $\vec{p}_2$. For parametric downconversion the setup (crystal, incoming "pump beam") are adjusted such that the phase-matching condition is fulfilled and that you use only specific photon pairs that then are entangled (in type-I conversion with equal polarization in type-II with opposite polarization). That means you don't know which polarization photon "A" (defined as the one hat has momentum $\vec{p}_1$) and photon "B" (defined as the one that has momentum $\vec{p}_2$) have, but you have 100% correlation when measuring polarization in the same direction described by the entangled state.

 

[DrChinese]

I don't understand what both of you mean by "photons that exit the PDC apparatus along paths A and/or B".

I would think that it is obvious what is intended, especially since it is so labeled by the OP in post #3: There are 2 output mechanisms that route photons towards detection apparatus A (usually tended by Alice) and detection apparatus B (usually tended by Bob). So path A is the routing to detector A, etc.

You can label them whatever you like, I'll continue to use A and B as I always do.

 

[vanhees71]

I thought the setup is parametric down conversion, where two photons are emitted one you label with A, the other with B. Physically what is meant is, however, that there's one photon with momentum $\vec{p}_1$ and one photon with momentum $\vec{p}_2$. The places A and B of course determine the direction of the photon's momentum detected at these places. The photons themselves don't travel on specific paths. I think a lot of confusion about photons (particularly entangled ones as discussed here) is in the misconception of "photon paths". The Wikipedia article has it right though it's not very explicit, but when you carefully think about the first and the last picture, you get a pretty accurate picture. At least that's what's described by the two-photon state in my posting #6, corresponding to the type-ii-conversion case.

 

[DrChinese]

I thought the setup is parametric down conversion, where two photons are emitted one you label with A, the other with B. Physically what is meant is, however, that there's one photon with momentum $\vec{p}_1$ and one photon with momentum $\vec{p}_2$. The places A and B of course determine the direction of the photon's momentum detected at these places. The photons themselves don't travel on specific paths. I think a lot of confusion about photons (particularly entangled ones as discussed here) is in the misconception of "photon paths". The Wikipedia article has it right though it's not very explicit, but when you carefully think about the first and the last picture, you get a pretty accurate picture. At least that's what's described by the two-photon state in my posting #6, corresponding to the type-ii-conversion case.

I think the relevant path is the one to the detectors, and nothing more. You certainly can route an entangled photon just like you can route any photon, and you can do so while it is still part of an entangled system. That is not to say that it has one specific path. And in fact, in this case, the relevance of the time window is that there is not a specific path for the entangled system.

I said in post #7: "Note that there is definitely no requirement that pair members leave the crystal at precisely the same time, and in fact they won't. There is always a degree of uncertainty, and the precise origination time and initial starting point of each entangled photon should not be considered as sharply defined values." Although I didn't say so explicitly, that means the path of the photon is not sharply defined either.

 

[fluidfcs]

Yes and no. As Nugatory correctly pointed out, a photon cannot be specifically determined to be entangled (or not) by any direct test. However - and I think this is what you are asking: can you assume that the photons that exit the PDC apparatus along paths A and/or B are entangled?

Again, the answer is yes and no. Most will emerge in entangled pairs (assuming the BBo crystal was cut for entanglement). In a hypothetical test, entanglement is "assumed" if the photon goes along path A. In other words, if a photon is detected at A, then there must be an entangled partner headed to B. So in this case, the answer to your question would be YES.

In actual tests, some of the photons are not actually entangled (for various reasons). I could not give you a guess of the percentage, but within a typical lab it is not very high. To make sure the pairs should be considered as "entangled" for experimental purposes, the detection times for each A/B pair are usually required to be within a small window of time T=|T(A)-T(B)| - a coincidence window - of perhaps 25 nanoseconds or so. Of course, T is adjusted for each photon's path length to the respective detector - in approximate terms, a photon travels about a foot in a nanosecond. Such a requirement is not strictly necessary, this is done to distill the results to a subset of high quality entangled pairs. Note that there is definitely no requirement that pair members leave the crystal at precisely the same time, and in fact they won't. There is always a degree of uncertainty, and the precise origination time and initial starting point of each entangled photon should not be considered as sharply defined values.

So for your case, you would simply drop the coincidence requirement T. Each time you have a detection at A, you assume there was a photon at B that you route in whatever direction you like. That won't be true 100% of the time, but it will happen frequently enough that you can probably move to the next step of your idea.

I appreciate the thorough responses from both of you. The detail above seems to be in-line with what I'm after. One follow-up question. In the actual test, could an unentangled photon occasionally travel both via path A and separately also via path B (I do not mean at the same time, assume separate photons here), or would it always travel a specific one of the two routes (e.g. always path A in the exceptional case that it doesn't go straight through). I'm assuming based on your response that a photon could travel path A or Path B, though in most cases would travel neither path and instead would pass-through to the "beam stop" in my previous diagram. Do I have it right?

 

[DrChinese]

In the actual test, could an unentangled photon occasionally travel both via path A and separately also via path B (I do not mean at the same time, assume separate photons here), or would it always travel a specific one of the two routes (e.g. always path A in the exceptional case that it doesn't go straight through).

A couple of background notes that may help you, apologies if this is already clear to you:

An entangled photon pair (from PDC via a nonlinear crystal) is a system consisting of 2 indistinguishable photons. Each of those entangled photons will be routed to paths A and B, i.e. towards detectors A and B. They remain as a non-separable system of 2 photons until they are observed, even though they are spatially separated at all times. This is the characteristic that many people find "strange" - they appear to be "somehow" connected regardless of their distance apart in spacetime. Tests of entangled pairs have verified their entanglement to distances in well excess of 100 kilometers (there is no theoretical limit).

The photons that go straight through are not entangled, and will never take a path to the A or B detectors. That is because the entangled photon pairs are "bent" a couple of degrees from the straight line as they go through the PDC crystal, while the vast majority of photons are unaffected. Perhaps 1 entangled pair emerges for every 10 million photons that go straight through.

 

[fluidfcs]

A couple of background notes that may help you, apologies if this is already clear to you:

An entangled photon pair (from PDC via a nonlinear crystal) is a system consisting of 2 indistinguishable photons. Each of those entangled photons will be routed to paths A and B, i.e. towards detectors A and B. They remain as a non-separable system of 2 photons until they are observed, even though they are spatially separated at all times. This is the characteristic that many people find "strange" - they appear to be "somehow" connected regardless of their distance apart in spacetime. Tests of entangled pairs have verified their entanglement to distances in well excess of 100 kilometers (there is no theoretical limit).

The photons that go straight through are not entangled, and will never take a path to the A or B detectors. That is because the entangled photon pairs are "bent" a couple of degrees from the straight line as they go through the PDC crystal, while the vast majority of photons are unaffected. Perhaps 1 entangled pair emerges for every 10 million photons that go straight through.

Thank you, yes, that is a core thing I hoped to confirm with this post and appreciate you doing so. I have two follow-up questions if you don't mind please.

First, per post #7, it sounds like you're saying that unentangled photons can also in very rare cases take either bent path A or B and do so while not being entangled, at a rate much lower than 1 / 10M. Just want to confirm I understood you correctly there.

Second, to extend my original question a bit, suppose path A contained a double-slit and detector. Are we able to modify the manifestation of an interference pattern from entangled photon A by performing measurement on entangled photon B?

Thanks again for your replies, they are greatly appreciated.

 

[vanhees71]

Forget about paths here! It's really crucial to understand in this context that there are no paths and that photons are indistinguishable bosons!

 

[DrChinese]

Forget about paths here! It's really crucial to understand in this context that there are no paths and that photons are indistinguishable bosons!

Your (non)use of the word "path" is different than everyone else's! And ignoring that word is not "crucial" to anything. I thought we settled this.

Scientists in labs route photons on "paths" in every similar experiment. The being from one locus (source) to another (detector) with all kinds of things (fiber, mirrors, filters, splitters, polarizers) in between. We call that a path in normal language, and there is no generally accepted alternate word for this - although maybe you have one?

No one is saying there is one well-defined specific path, from one specific source to one specific detection point. There isn't in the quantum world. But if the photons don't travel the general path the experiment dictates, there will be no detections, and no results. Further, if pathway is not exactly calibrated, the experiment fails. And that very precise path is defined EXACTLY as would be expected for classical light in all respects.

Conclusion: It's a path! If you feel better about it, call it a "quantum path". I am not going to debate this further, as it has absolutely nothing to do with this thread and what @fluidfcs is asking.

 

[vanhees71]

All there is are two photons, one with momentum $\vec{p}_1$ and one with momentum $\vec{p}_2$ and polarizations H and V (for the example of type-2 conversion). You cannot adequately describe it in words though. The only precise way to say what you mean by this entangled two-photon state indeed is
$$|\Psi \rangle=\frac{1}{\sqrt{2}} \left [\hat{a}^{\dagger}(\vec{p}_1,H) \hat{a}^{\dagger}(\vec{p}_2, V) - \hat{a}^{\dagger}(\vec{p}_1,V) \hat{a}^{\dagger}(\vec{p}_2,H) \right]|\Omega \rangle.$$
#14 shows that the OP is still confused about the basic concept of entangled photons, and it's indeed a complicated concept. To talk again and again about paths is obviously not helpful!

 

[fluidfcs]

All there is are two photons, one with momentum $\vec{p}_1$ and one with momentum $\vec{p}_2$ and polarizations H and V (for the example of type-2 conversion). You cannot adequately describe it in words though. The only precise way to say what you mean by this entangled two-photon state indeed is
$$|\Psi \rangle=\frac{1}{\sqrt{2}} \left [\hat{a}^{\dagger}(\vec{p}_1,H) \hat{a}^{\dagger}(\vec{p}_2, V) - \hat{a}^{\dagger}(\vec{p}_1,V) \hat{a}^{\dagger}(\vec{p}_2,H) \right]|\Omega \rangle.$$
#14 shows that the OP is still confused about the basic concept of entangled photons, and it's indeed a complicated concept. To talk again and again about paths is obviously not helpful!

Hi, you're probably right that I am not understanding this properly. Can you help me? If the photons are not taking different physical paths then how does this experiment work?

@DrChinese would you possibly be able to respond to post #14 please? Many thanks.

 

[vanhees71]

The Walborn quantum eraser experiment I've chosen as my habilitation lecture. Here is the English translation of the presentation:

https://itp.uni-frankfurt.de/~hees/publ/habil-coll-talk-en.pdf

 

[DrChinese]

Hi, you're probably right that I am not understanding this properly. Can you help me? If the photons are not taking different physical paths then how does this experiment work?

@DrChinese would you possibly be able to respond to post #14 please? Many thanks.

Sorry for the confusion. You can safely ignore @vanhees71's comments about "photon not having paths". His reasoning is hypertechnical (and correct in the sense he means it), but does nothing to address your questions. Quantum particles do not travel in classical paths. However, they do travel! And where they travel "from" and "to" is called their path (that's the definition!). This quantum path is essentially the same as the path of a classical particle, except in certain cases which don't apply to your example on entanglement. Please note that where a particle travels "from" and "to" is similarly a quantum source and a quantum destination - which do not correspond precisely to exact locations in spacetime. And while vanhees71 speaks of the momentum of entangled particles, they have no more exact momentum than they do exact paths. (Or energy, or frequency.)

So back to your questions from post #14.

1. First, per post #7, it sounds like you're saying that unentangled photons can also in very rare cases take either bent path A or B and do so while not being entangled, at a rate much lower than 1 / 10M. Just want to confirm I understood you correctly there.

Unentangled photons - the vast majority 99.999999% - go straight through, essentially unaffected by the presence of the crystal. When pairs are produced, they are "bent" as mentioned by an amount. This amount varies. What happens is that the experimenter collects pairs from a "sweet spot" in which those pairs emerge. For a variety of reasons, a very small portion of those pairs are not entangled. Keep in mind that unintended environmental factors can cause this, it is simply a result common to most entanglement creation that there is some "noise" present that degrades the efficiency.

2. Second, to extend my original question a bit, suppose path A contained a double-slit and detector. Are we able to modify the manifestation of an interference pattern from entangled photon A by performing measurement on entangled photon B?

There is no detectible manifestation of any change on photon A based on anything you do to B, and vice versa. Were that possible, you could send FTL (faster than light) messages - as you probably already wondered. When you perform certain kinds of coincidence counting, you can "see" an interference pattern. But because you have to bring information from 2 separated detectors together first - which requires classical communication channels - there is nothing to be gained. vanhees71 mentions a seminal experiment which explains this in his post #19, and below is a reference to the full paper itself. That paper mentions photon "path" 38 times.

https://arxiv.org/abs/quant-ph/0106078

Also, please note that there are some so-called interpretations of quantum mechanics in which photons move in precise paths (trajectories). Those are *not* to be discussed in this forum, there is a separate subforum on Quantum Interpretations in which those subjects can be discussed.

 

[PeterDonis]

Moderator's note: A subthread on the usage of the term "path" as it relates to photons has been moved to a separate thread: https://www.physicsforums.com/threa...an-when-they-say-photons-have-a-path.1012960/