A How Does MWI explain Type I PDC Photon Polarization Entanglement?

[DrChinese]

Calling MWI knowledgeable members and advocates!

My question includes a deep dive into how Type I parametric down conversion (PDC) entanglement is created, and specifically mapping the narrative for the Many Worlds Interpretation (MWI). Here is a seminal reference on Type I PDC: Ultra-bright source of polarization-entangled photons (1998) Kwiat et al

a. The crystals used for Type I PDC are cut such that for a Diagonally oriented (45 degree) laser beam (wavelength 351 nm): A Vertical axis crystal occasionally (and randomly) down-converts a single Vertical photon into two Horizontally polarized photons (wavelength 702 nm). The two output photons have twice the wavelength of the input photons, but half the energy - so total energy is conserved. Horizontal photons are not affected and go straight through the crystal, as do unaffected Vertical photons. So the Vertical crystal has the effect of changing all Diagonal photons into either Vertical photons or Horizontal photons.

In my example, I will assume that conversion occurs at the rate of 1:1 million, and I will assume the laser provides a beam of 200 million photons per second. This is just for ease of discussion and calculation, and all estimates will be normalized to an ideal average. So in one second: there are 200 million Diagonal input photons; 99,999,900 Vertical output photons; 200 down-converted Horizontal output photons; and 100,000,000 Horizontal output photons.

Importantly: The HH> output pairs that result from the Vertical crystal (we'll call that PDC#1) are NOT polarization entangled.

b. Likewise: we add a Horizontal crystal (which is simply a Vertical crystal rotated 90 degrees), placed directly next in the path after PDC #1 - we'll call this PDC#2. The 100,000,000 Horizontal 351 nm output photons from PDC#1 will mostly pass through PDC#2 unaffected, but about 100 of them should randomly down-convert to 2 Vertical 702 nm photons.

So far, there is no entanglement anywhere. That requires yet another step, closely overlapping the H-polarized and V-polarized output cones as pictured.

What does MWI say has happened so far? Tracing out a single Diagonal input 351 nm photon from the laser:

When the photon goes through PDC#1: there are 3 possible outcome branches.

• i) 1 Vertical 351 nm photon, MWI weight 999,999:2 million;
• ii) 2 Horizontal down-converted 702 nm photons, MWI weight 1:2 million;
• iii) 1 Horizontal 351 nm photon, MWI weight 1 million:2 million.

There is no polarization entanglement, all 3 branches contain photons in a specific polarization state.

When each of the 3 branches' photons go through PDC#2: The i) and ii) MWI branches contain photons that will pass through unaffected, so they remain as before. (A 702 nm photon is not materially affected, and a Vertical photon is not affected by a Horizontal PDC crystal.) The iii) MWI branch contains a photon that is eligible to be down-converted into 2 Vertical photons. The iii) branch now drops in weight to 1:999,999; and a new branch iv) is created with 2 Vertical down-converted 702 nm photons, MWI weight 1:2 million. We are now at the far edge of PDC#2.

Branches i) and iii) can be ignored for our purposes, as no entangled pairs can result from these. So we have left two equally weighted branches:

• ii) MWI branch with 2 down-converted 702 nm photons in state HH>;
• iv) MWI branch with 2 down-converted 702 nm photons in state VV>.

There is no entanglement in either, no superposition of any kind. As far as I know, MWI branches are completely independent and don't interact in any manner. If so, then the above branches are mutually exclusive and I am unaware of any method of later blending them. And there seems to be no particular reason that overlapping the future output cones should change anything, since that should simply lead to a stream of unentangled 702 nm photon pairs going to the same places. [EDIT: I have learned that the above is not strictly correct. Under certain circumstances, recoupling of branches is possible. From hedweb.com/manworld.htm "it only requires that the divergent paths of the ... particle overlap again at some space-time point for an interference pattern to form, because only the single particle has been split." However, I would still like to see in detail how this would work in my example.]

In orthodox QM, the entanglement occurs when the sources physically overlap to become indistinguishable. But here, by definition, we have independent branches that are ignorant of each other's quite different evolution according to the deterministic Schrödinger equation. And neither branch has any way to know if they will overlap in the future, or not.

So how does MWI explain polarization entanglement emerging from either of these worlds?
Note: This question was identically asked in StackExchange.

 

[PeterDonis]

What does MWI say has happened so far?

Nothing much. Since the H and V branches can be made to interfere with each other by a subsequent action, no branching has occurred as far as the MWI is concerned. (No decoherence has occurred.)

This is a key point that you seem to be overlooking: if there is any possibility of having two alternatives interfere, then as far as the MWI is concerned, there's no branching yet.

 

[PeterDonis]

As far as I know, MWI branches are completely independent and don't interact in any manner.

You're looking at this backwards. You don't ask the MWI to tell you when branching occurs, and then deduce from what it tells you what things can or can't interfere with each other. You first ask the actual physics involved what things can or can't interfere with each other (when decoherence doesn't or does occur), and then you tell the MWI, okay, here's where the branching occurs--where the decoherence occurs. Then the MWI tells you what it says that means.

 

[DrChinese]

In my estimation, decoherence occurs too often to explain the results. There won’t be enough entangled pairs produced. Due to weighting, it’s off by a large factor.

 

[syed]

Entanglement arises not from separate branches merging, but from the global wavefunction maintaining coherence before any branching occurs. In the Type I PDC setup you're talking about, when the down-conversion paths from the two crystals are made indistinguishable, the outcomes HH⟩ and VV⟩ do not correspond to separate branches. They just form a superposition within a single world: (HH⟩ + VV⟩)/√2.

Branching in MWI happens only after decoherence, so as long as the experimental setup prevents which-path information, the entangled state exists prior to branching.

 

[PeterDonis]

In my estimation, decoherence occurs too often to explain the results. There won’t be enough entangled pairs produced.

What is your estimation based on?

Note also that any such estimation would apply to any interpretation, since whether or not decoherence occurs is not interpretation dependent. So if your estimation were correct (which I doubt it is), it would rule out any explanation of the results.

 

[DrChinese]

Nothing much. Since the H and V branches can be made to interfere with each other by a subsequent action, no branching has occurred as far as the MWI is concerned. (No decoherence has occurred.)

This is a key point that you seem to be overlooking: if there is any possibility of having two alternatives interfere, then as far as the MWI is concerned, there's no branching yet.

I would certainly expect that a V> photon interaction with a V crystal produces decoherence, since the resulting branches are quite different: a single 351 nm V> photon; versus two 702 nm H> photons.

How would the evolving Schrödinger wave "know" to hold off on decoherence/branching? Doesn't this defeat the entire purpose of MWI - since the branches wouldn't materialize until future events unfold?

I admit I'm confused...

 

[DrChinese]

Entanglement arises not from separate branches merging, but from the global wavefunction maintaining coherence before any branching occurs.

Branching in MWI happens only after decoherence...

1. I follow this idea. But as I said to PeterDonis: I would certainly expect that a V> photon interaction with a V crystal produces decoherence, since the resulting branches are quite different: a single 351 nm V> photon; versus two 702 nm H> photons.

How has decoherence not occurred? Neither of these outcomes themselves are affected by the second crystal. Vice versa, if the original photon ended up H> instead (a 50-50 possibility) I would think its evolution would be along a different branch. The down-converted pairs definitely are on distinguishable paths at this point. Only later are they overlapped.

I admit I'm confused...

 

[PeterDonis]

the resulting branches are quite different

By calling them "branches", you're just assuming decoherence has occurred. But, since the experimental fact is that the H and V beams can be put into a second crystal to produce an entangled pair of photons, decoherence can't have occurred at the first crystal. It's certainly not intuitively clear how that can be possible, but experimental facts trump intuition. And we know there are things like beam splitters, which are also "crystals" of a sort, but which can clearly produce two output beams that are not decohered, since Mach-Zehnder interferometers work.

 

[PeroK]

I would certainly expect that a V> photon interaction with a V crystal produces decoherence, since the resulting branches are quite different: a single 351 nm V> photon; versus two 702 nm H> photons.

How would the evolving Schrödinger wave "know" to hold off on decoherence/branching? Doesn't this defeat the entire purpose of MWI - since the branches wouldn't materialize until future events unfold?

I admit I'm confused...

All of MWI is essentially standard QM. Let's take a different scenario of two electrons. As the electrons are indistinguishable, it's technically not possible to describe them as separate electrons. They must be treated as a system of two indistinguishable electrons. And the interaction is described by a joint wave-function of a system of two particles.

If the two electrons are far enough apart, however, they behave like two independent systems. Nothing knows what's going on. Instead, it's just that the wavefunction evolves in the second case as though it were two separate wave-functions. It isn't an either/or scenario. The electrons technically always interact.

Branches in MWI are similar. They always interact. But, a branch is loosely defined as when decoherence has rendered the wave-function into such a state that it evolves as though it were two independent systems. And the amplitude (I better avoid the word probability here!) of an unexpected interaction is vanishingly small.

In principle, it shouldn't matter whether you are considering MWI or not. In a collapse interpretation, the measurement collapses the wave-function into a single branch. But, in mathematically the same way, a measurement in MWI renders the wave-function into a state where all the possible branches are still there, but they are effectively evolving separately. The absolute irreverisbility of collapse has been replaced with the effective irreversibility of the branches subsequently recombining significantly.

I'm not sure what such an event would look like, because our brains might have been caught up in the melee, as it were!

 

[DrChinese]

And we know there are things like beam splitters, which are also "crystals" of a sort, but which can clearly produce two output beams that are not decohered, since Mach-Zehnder interferometers work.

Great point about beam splitters. But the reason I chose PDC crystals is because (unlike 50:50 splitters), the weighting is more like 1:1 million. The traditional Mach-Zehnder setup brings together those beams in a manner certain to lead to recombination. And in fact those beams originate in the same place.

On the other hand: the down-converted HH> pairs are created at a different place than the VV> pairs. And equally importantly, they are created at different times. There is no particular reason that the creation times in different crystals should be exactly synchronized such that they can join up later to create entanglement.

Specifically: the average distance between the 2 PDC crystals is 3 mm (their typical thickness), which requires about 10 picoseconds to traverse. Not a big difference, and I'm not sure how far apart is sufficient for decoherence to occur as I am imagining. Hopefully you see my point: The HH> pair is created 10 picoseconds before the matching VV> down-conversion must (also) occur in order for their future paths to line up and overlap correctly.

And because the likelihood of down conversion is so low in a any crystal, the likelihood of down-conversion occurring in both crystals with any particular photon would be infinitesimally low. I think.

 

[PeterDonis]

'm not sure how far apart is sufficient for decoherence to occur as I am imagining.

My understanding is that the decoherence time for light traveling in air is a lot longer than 10 picoseconds. But I'm not very familiar with the details of the literature on this.

3 mm is not very far; I'm pretty sure double slit experiments with light can be done with much longer distances and still show interference. Which would be an indication that light can travel coherently over such distances.

 

[DrChinese]

In principle, it shouldn't matter whether you are considering MWI or not. In a collapse interpretation, the measurement collapses the wave-function into a single branch. But, in mathematically the same way, a measurement in MWI renders the wave-function into a state where all the possible branches are still there, but they are effectively evolving separately. The absolute irreverisbility of collapse has been replaced with the effective irreversibility of the branches subsequently recombining significantly.

I agree that at first glance, it shouldn't matter. It's the second glance that is bothering me.

A single photon enters the 2 crystal apparatus. It must down-convert in both (a very rare event in either crystal), and must do so (give or take) 10 picoseconds apart. Then everything you say comes together nicely.

Does that seem reasonable to you? Because it doesn't to me. There is absolutely no reason that the down conversion in the second crystal would be more likely to occur when there is down conversion in the first. In standard QM, I don't worry about this because the full final context is considered. There is no branching ever to consider. So I don't have to ask whether decoherence has occurred. Instead, whatever result I witness is simply a random one from many possibilities. In a single world.

 

[PeterDonis]

Does that seem reasonable to you?

What seems reasonable to me is irrelevant. Sure, I don't have an intuitive picture of how this can happen. But as I said, experimental facts trump intuition. And decoherence has been studied experimentally to a considerable extent, and so far, from what I can gather, it often doesn't happen when our intuition says it should: quantum coherence can be maintained under conditions where our intuitions say it should break down.

(An example would be doing double slit experiments with buckyballs. I don't know about your intuition, but mine says that buckyballs--which we can take a picture of with an electron microscope--ought not to be able to interfere with themselves that way. But experiment says they do.)

 

[DrChinese]

My understanding is that the decoherence time for light traveling in air is a lot longer than 10 picoseconds. But I'm not very familiar with the details of the literature on this.

3 mm is not very far; I'm pretty sure double slit experiments with light can be done with much longer distances and still show interference. Which would be an indication that light can travel coherently over such distances.

I'm not worried that decoherence might occur while traveling to the intersection point, as that is always possible to some small degree. In MWI, an interaction with a crystal should (this is what I am confused about) be a branching event; because in one branch you have 1 particle emerging (very likely); while in the other there are 2 (very unlikely). This part is irreversible. And yet the future of a branch irreversibly marked as a single photon must later down-convert in the second crystal into either a completely different photon pair, or remain on its path as a single photon - also irreversible.

Vaidman: "...macroscopic objects being in macroscopically distinguishable states A and B, 1/√2(|ΨA⟩+|ΨB⟩)|Φ⟩, splits immediately... into two worlds: the new “world+” and the “world−”." Certainly down-conversion qualifies by this criteria. Also Vaidman: "Probability Postulate: An observer should set his subjective probability of the outcome of a quantum experiment in proportion to the total measure of existence of all worlds with that outcome."

In our example, the orthodox QM prediction would be in proportion to 2:1 million since the chance of down-converting in a crystal (we have 2) is 1:1 million. But the predicted odds would be radically different if the down-conversion occurred in decohered branches.

 

[DrChinese]

Sure, I don't have an intuitive picture of how this can happen. But as I said, experimental facts trump intuition. And decoherence has been studied experimentally to a considerable extent, and so far, from what I can gather, it often doesn't happen when our intuition says it should: quantum coherence can be maintained under conditions where our intuitions say it should break down.

I'm fine with all this. The question is whether the rules of MWI impose an additional constraint (which of course advocates would likely deny). See for example the statements of Vaidman's Plato MWI in my post #15. Those are 2 specific claims. I am trying to apply them, and again having a time of it with my example.

Thanks for you ongoing assistance on this, and I am hoping both you and the others will continue challenging and/or correcting my understanding.

 

[PeterDonis]

Certainly down-conversion qualifies by this criteria

Not necessarily. The photons aren't macroscopic systems, and aren't measured when they come out of the first crystal. The first crystal itself is a macroscopic system, but there aren't two macroscopically distinguishable states of the crystal that correspond to the two output beams. There's just one state of the crystal after the beams come out.

 

[DrChinese]

Not necessarily. The photons aren't macroscopic systems, and aren't measured when they come out of the first crystal. The first crystal itself is a macroscopic system, but there aren't two macroscopically distinguishable states of the crystal that correspond to the two output beams. There's just one state of the crystal after the beams come out.

If there is down conversion, the photon count is 2. If not, count is one and those are different Fock states... right? That can be observed. By Vaidman's rule, those are 2 worlds with different proportional weights.

 

[PeterDonis]

If there is down conversion, the photon count is 2. If not, count is one and those are different Fock states... right?

In the math, sure. But if there is no measurement of the photons, then the actual state is a superposition of those two Fock states. And the experimental evidence shows that that superposition remains coherent at least up to the second crystal.

That can be observed.

Sure, by changing the experiment--put photon counters in between the first and second crystals. And if you do that, you'll get photon detections before the second crystal, and no possibility of producing entangled states.

But the actual experiment doesn't do that. In the actual experiment, between the first and second crystals, there is nothing done to make the two Fock states you describe macroscopically distinguishable. That means you can't reason as though they were, because in that particular experiment, they're not.

 

[Demystifier]

As far as I know, MWI branches are completely independent and don't interact in any manner.

I think that's the source of your confusion. The branches don't "interact" (or more precisely, don't interfere) with each other when they are macroscopic branches. By contrast, the microscopic branches can interfere with each other. Your branches HH> and VV> involve only two particles, hence they are microscopic, so they can interfere.