Decoherence in the double slit experiment

[Agrippa]

I have two questions about the following type of scenario:

We have a laser sending photons through the usual double slit apparatus giving us the usual interference pattern, except that now we introduce some physical matter (that are not photons) that the photons will interact with before going through the double slit apparatus. The only assumption I want to make about this introduced physical matter and the nature of the interactions is that the photons will become entangled with that physical matter before those photons make it to the double slit apparatus.

Question (1): will the fact that the photons entangled with that matter destroy the interference pattern? I suspect it might, since I think (think) the reason why large molecules (larger than buckyballs) don't exhibit interference effects in the double slit experiment, is because they entangle with air molecules. However, I'm not sure: is this a matter of degree? Does just "a little bit of entanglement" (for each photon that makes it to the apparatus) only make a little bit of difference to the pattern? If so, is there a rigorous definition of "a little bit of entanglement"? Is there a measure of "how much" entanglement with the environment destroys the interference completely?

Question (2): if the answer to (1) is that we still get (some amount of) the interference pattern, then is it the case that affecting the introduced physical matter can affect the shape of the interference pattern? Here, by "affecting the introduced matter", I mean changing their states in some way without collapsing them.

Any insights here would be greatly appreciated!

 

[lekh2003]

Question (1): will the fact that the photons entangled with that matter destroy the interference pattern? I suspect it might, since I think (think) the reason why large molecules (larger than buckyballs) don't exhibit interference effects in the double slit experiment, is because they entangle with air molecules. However, I'm not sure: is this a matter of degree? Does just "a little bit of entanglement" (for each photon that makes it to the apparatus) only make a little bit of difference to the pattern? If so, is there a rigorous definition of "a little bit of entanglement"? Is there a measure of "how much" entanglement with the environment destroys the interference completely?

No, no, no. Larger molecules do not demonstrate the interference pattern because their wavelengths are infinitesimally small, in accordance with the De Broglie equation. Entanglement has little to do with large molecules not demonstrating interference effects, forget about entangling with air molecules.

Question (2): if the answer to (1) is that we still get (some amount of) the interference pattern, then is it the case that affecting the introduced physical matter can affect the shape of the interference pattern? Here, by "affecting the introduced matter", I mean changing their states in some way without collapsing them.

The only way I can expect the interference pattern to change (if we do mysteriously achieve this entanglement) might be because we now have knowledge of the positions of the photons, which would collapse the wave functions.

 

[Agrippa]

No, no, no. Larger molecules do not demonstrate the interference pattern because their wavelengths are infinitesimally small, in accordance with the De Broglie equation. Entanglement has little to do with large molecules not demonstrating interference effects, forget about entangling with air molecules.

What you say contradicts just about everything I have read on this topic.

Firstly, larger molecules do not have infinitesimally small wavelengths, they just have very small wavelengths, and here the experiment just needs to be designed to cater for those small wavelengths - so this is no explanation of lack of interference pattern.

Secondly, interference patterns may disappear even for electrons if some other systems (say, sufficiently many stray cosmic particles scattering off the electron) suitably interact with the wave between the slits and the screen. Here the interference pattern is lost because the electron has become entangled with the stray particles! The phase relation between the two components of the wave function, which is responsible for interference, is well-defined only at the level of the larger system composed of electron and stray particles, and can produce interference only in a suitable experiment including the larger system [REF].

My question was not about electrons interacting with some matter after the slits and before the screen, my question was about photons interacting with some matter before the slits. That question still remains. I hope someone can answer it!

 

[lekh2003]

Firstly, larger molecules do not have infinitesimally small wavelengths, they just have very small wavelengths, and here the experiment just needs to be designed to cater for those small wavelengths - so this is no explanation of lack of interference.

One thing I know for sure is that you don't usually see interference for large molecules is because of the size of their wavelengths. I was simply trying to explain to you that the molecules you are discussing do not demonstrate interference, because of their wavelength, not entanglement.

Here interference is lost because the electron has become entangled with the stray particles!

Interference is not lost. Interference happened, but you couldn't see it on the screen. It doesn't mean it didn't happen, the results were being tampered with by stray particles.

My question was not about electrons interacting with some matter after the slits and before the screen, my question was about photons interacting with some matter before the slits. That question still remains. I hope someone can answer it!

I answered the question. The photons might not exhibit an interference pattern since you might be able to know which slit the photon goes through. You know which slit it goes through, because before it went through the slits, it became entangled and began to give you information. This collapses the wave function.

 

[Agrippa]

One thing I know for sure is that you don't usually see interference for large molecules is because of the size of their wavelengths. I was simply trying to explain to you that the molecules you are discussing do not demonstrate interference, because of their wavelength, not entanglement.

You're just wrong about this. Physicists will always take De Broglie Wavelengths into account when designing their experiments. Even when they correctly do so for larger molecules, interference patterns are still lost. The standard explanation for this concerns entanglement with the environment.

Interference is not lost. Interference happened, but you couldn't see it on the screen. It doesn't mean it didn't happen, the results were being tampered with by stray particles.

The question (again) is why the interference pattern is lost. "The results were tampered with" is not an explanation.

I answered the question. The photons might not exhibit an interference pattern since you might be able to know which slit the photon goes through. You know which slit it goes through, because before it went through the slits, it became entangled and began to give you information. This collapses the wave function.

The scenario I described did not involve anyone obtaining which-path information. I would appreciate it if you let someone else try to answer my questions.

 

[lekh2003]

The scenario I described did not involve anyone obtaining which-path information. I would appreciate it if you let someone else try to answer my questions.

Ok sure, let someone else explain to you.

 

[Agrippa]

Ok sure, let someone else explain to you.

Thanks.

 

[Demystifier]

Question (1): will the fact that the photons entangled with that matter destroy the interference pattern? I suspect it might,

As you suspect, it may.

(larger than buckyballs) don't exhibit interference effects in the double slit experiment,

Actually they do. See Ref. [1] in https://arxiv.org/abs/1406.3221 .

However, I'm not sure: is this a matter of degree? Does just "a little bit of entanglement" (for each photon that makes it to the apparatus) only make a little bit of difference to the pattern?

Yes, it is a matter of degree.

If so, is there a rigorous definition of "a little bit of entanglement"? Is there a measure of "how much" entanglement with the environment destroys the interference completely?

Yes, see e.g. Eqs. (35), (36) and (38) in https://arxiv.org/abs/1406.3221 . In the case (38), the interference is destroyed completely.

Question (2): if the answer to (1) is that we still get (some amount of) the interference pattern, then is it the case that affecting the introduced physical matter can affect the shape of the interference pattern? Here, by "affecting the introduced matter", I mean changing their states in some way without collapsing them.

Yes, it may affect.

 

[lekh2003]

As you suspect, it may.Actually they do. See Ref. [1] in https://arxiv.org/abs/1406.3221 .Yes, it is a matter of degree.Yes, see e.g. Eqs. (35), (36) and (38) in https://arxiv.org/abs/1406.3221 . In the case (38), the interference is destroyed completely.Yes, it may affect.

You do have a much better explanation which is welled backed up. I retrieve my explanation.

 

[Agrippa]

As you suspect, it may.
Actually they do. See Ref. [1] in https://arxiv.org/abs/1406.3221 .
Yes, it is a matter of degree.
Yes, see e.g. Eqs. (35), (36) and (38) in https://arxiv.org/abs/1406.3221 . In the case (38), the interference is destroyed completely.
Yes, it may affect.

Thanks for the link, it's very useful. So going back to my example in light of (35)-(38), it seems that my photons will yield an interference pattern if their reduced density matrix (taken by tracing out my introduced physical matter) has non-diagonal terms close to one, meanwhile, my photons won't yield an interference pattern if their reduced density matrix has non-diagonal terms close zero. If the non-diagonals are in between, then the photons will create a partial interference pattern.

So we do have a measure of "amount of entanglement" in terms of closeness to one in the non-diagonals. I'm still puzzled about the physical meaning of this, which is important since I want to know what conditions must obtain for my introduced matter, for the photon interference pattern to be gone or to just be partial. Unfortunately I did not find helpful the talk of the (internal or external) environment "having the capacity for a record of which-path info", since whether or not a bunch of particles contains a record of some event depends on who is interpreting them.

My guess is this: Let system A be entangled to B, and let system C be entangled to D. Assume that measuring (thereby collapsing) A almost fully collapses B, whereas measuring (thereby collapsing) C only partially collapses D. If the measurements are of position, then "partial collapse" means "partial localization" i.e. B localizes to a greater degree than D. In that case, AB has a greater "amount" of entanglement than CD (as measured by the relevant non-diagonals).

 

[vanhees71]

Just one more thought: The problem to get interference phenomena with large objects is not only due to their large mass and the corresponding small deBroglie wavelength, $\lambda=\hbar/(2 \pi p)$ but also due to the high energy-level density of their intrinsic state. For the Buckyballs there were very nice diffraction experiments by Zeilinger et al.

https://www.univie.ac.at/qfp/research/matterwave/c60/
http://dx.doi.org/10.1038/44348

A systematic study of decoherence by heating the Buckyballs up:

https://www.uni-due.de/~hp0198/research/decoherence/thermodeco/index.html
https://arxiv.org/abs/quant-ph/0402146

 

[bhobba]

No, no, no. Larger molecules do not demonstrate the interference pattern because their wavelengths are infinitesimally small, in accordance with the De Broglie equation. Entanglement has little to do with large molecules not demonstrating interference effects, forget about entangling with air molecules.

Wrong an many counts. They show interference, even in the macro world they show interference, it's just so damn small its beyond our technological ability at the moment to detect it. That technological ability is a moving line. We are seeing more and more strange effects macro wise as technology progresses.

Entangling with air molecules? That goes on all the time. And models show interference effects quickly reduce to zero - but never quite reaches it. Again its technology that limits showing and detecting it - not physical principles. How then does decoherence explain we generally do not see interference patterns - its simply a technological matter - we currently can't do it - but that is not to say we never will. This makes it a rather subtle area that some get a bit confused about, not understanding what For All Practical Purposes means.

Thanks
Bill

 

[Cthugha]

Question (1): will the fact that the photons entangled with that matter destroy the interference pattern?

That of course depends on what property you want to entangle. The double slit is a simple experiment, where the pattern is a function of the relative phases (and intensities, but let us assume these are the same at the two slits) of the light fields at the two slit positions. This makes the double slit an experiment that depends on photon momentum. Photons with a different momentum will hit the slits at some angle. This creates some path length variation to the two slits, which in turn results in a phase difference and thus leads to a different interference pattern.

As a consequence, if you shoot single photons at the double slit and somehow create momentum entanglement with some piece of matter, such that the photons will arrive at some random angle, the interference pattern will vanish as you will effectively get a superposition of many interference patterns which cancel out. If you instead shoot single photons at the double slit and somehow create for example only polarization entanglement with some piece of matter without changing anything else, the interference pattern will not be influenced by this.

The influence of separability on coherence and entanglement is briefly discussed in this article:
https://arxiv.org/abs/quant-ph/0112065
There, photons are not entangled with matter, but momentum entangled photon pairs are used, but with respect to the underlying physics it does not really matter much whether you entangle the photons with matter, other photons or whatever.

 

[DrChinese]

Thanks for the link, it's very useful. So going back to my example in light of (35)-(38), it seems that my photons will yield an interference pattern if their reduced density matrix (taken by tracing out my introduced physical matter) has non-diagonal terms close to one, meanwhile, my photons won't yield an interference pattern if their reduced density matrix has non-diagonal terms close zero. If the non-diagonals are in between, then the photons will create a partial interference pattern.

So we do have a measure of "amount of entanglement" in terms of closeness to one in the non-diagonals. I'm still puzzled about the physical meaning of this, which is important since I want to know what conditions must obtain for my introduced matter, for the photon interference pattern to be gone or to just be partial. Unfortunately I did not find helpful the talk of the (internal or external) environment "having the capacity for a record of which-path info", since whether or not a bunch of particles contains a record of some event depends on who is interpreting them.

My guess is this: Let system A be entangled to B, and let system C be entangled to D. Assume that measuring (thereby collapsing) A almost fully collapses B, whereas measuring (thereby collapsing) C only partially collapses D. If the measurements are of position, then "partial collapse" means "partial localization" i.e. B localizes to a greater degree than D. In that case, AB has a greater "amount" of entanglement than CD (as measured by the relevant non-diagonals).

There are a lot of issues/assumptions in what you are setting up for your question that makes it difficult to nail down an exact answer. For example:

1. Entangled particles do not exhibit interference effects as long as they are entangled. The easiest way to understand this point is that entangled particles are not coherent. Cthugha alludes to this above.
2. I cannot even conceive of a method of entangling a single photon with an apparatus *and* it still existing (as in continuing on its merry way). Photon entanglement is possible in many ways, but the most common is PDC which creates 2 photons from 1.
3. In the double slit: "having the capacity for a record of which-path info" does not require an observer or any interpretation of same, as you imply. You are correct that interference effects are variable between 0 and 100%, according to the setup. Anything that serves to provide which-slit information will eliminate interference to some degree, even if that information is completely lost in the environment. Imagine placing polarizers in front of each slit. When parallel, there IS interference. When crossed, NO interference. And they can be varied in between as desired for partial interference. This is because polarization information is being encoding to the chosen amount, which could give information about which-slit. Such information need not be read for the results to be seen.

 

[vanhees71]

Ad 1) What do you mean by "engangled particles are not coherent"? It depends on the prepared state whether its coherent or not but not necessarily with whether it's an entangled state. It's of course a bit empty to say the system is in an entangled state. You have to say which observables are entangled (e.g., many Bell-test experiments with photons use polarization-entangled photon pairs).

Ad 2) Measuring means to have an interaction between the system to be measured and the measurement device precisely such that the "pointer state" of the device gets entangled with respect to the measured observable. E.g., in the Stern-Gerlach experiment you sort particles according to their spin-z component by entangling this spin-z component with the position of the particle and then measure the position. Then you have a (nearly) one-to-one connection (through entanglement) between the location of the particle and the spin-z component.

Ad 3) If you gain certain which-way information you loose your interference pattern. If you have no which-way information at all and a sufficiently coherent matter wave (over the distance of the slits) you get 100% contrast of the interference pattern. If you gain only partial which-way information you still have more or less contrast of a residual interference pattern.

 

[Demystifier]

Ad 1) What do you mean by "engangled particles are not coherent"?

He means that any of the individual particles is not coherent, even if the system of all entangled particles together is coherent.

 

[Cthugha]

He means that any of the individual particles is not coherent, even if the system of all entangled particles together is coherent.

I suppose the point here is simply that you can have coherence with respect to many different properties and entanglement with respect to different properties as well and there is only a trade-off between complementary properties. Momentum entanglement and first-order spatial coherence cannot coexist up to arbitrarily high amounts. The same goes for entanglement in energy and first-order temporal coherence. But you cannot take any arbitrary kind of entanglement and any arbitrary kind of coherence and hope that one destroys the other.

 

[DrChinese]

Ad 1) ... It's of course a bit empty to say the system is in an entangled state. You have to say which observables are entangled (e.g., many Bell-test experiments with photons use polarization-entangled photon pairs).

Ad 2) Measuring means to have an interaction between the system to be measured and the measurement device precisely such that the "pointer state" of the device gets entangled with respect to the measured observable. E.g., in the Stern-Gerlach experiment you sort particles according to their spin-z component by entangling this spin-z component with the position of the particle and then measure the position. Then you have a (nearly) one-to-one connection (through entanglement) between the location of the particle and the spin-z component.

Ad 3) If you gain certain which-way information you loose your interference pattern. If you have no which-way information at all and a sufficiently coherent matter wave (over the distance of the slits) you get 100% contrast of the interference pattern. If you gain only partial which-way information you still have more or less contrast of a residual interference pattern.

1) Yes, entangled bases can vary as you say. Nonetheless, I'm not aware of any preparation possible where photons would be entangled on some basis *and* be coherent as regards to a double slit setup (i.e. a typical DS interference pattern). They can't be momentum entangled and coherent, right? And I don't believe they can be spin entangled *and* coherent. But perhaps some esoteric preparation is possible, or I am confused on this point.

2) This description doesn't seem to address my comment, which was directed at the OP's specification: "the photons will become entangled with that physical matter before those photons make it to the double slit apparatus". I may be short-sighted, I just can't imagine how this is possible. If it is unphysical, I don't know how we can answer such a question.

3) I thought I said that.

 

[DrChinese]

But you cannot take any arbitrary kind of entanglement and any arbitrary kind of coherence and hope that one destroys the other.

I see your point, which I see now is what vanhees71 was telling me as well.

However: in the context of the OP's questions, I'm not sure we need to consider this. Or we could ask the OP: "On what basis do you expect the photon to be entangled, and with what exactly?" I'm saying no answer could lead to an interference pattern and be physical in the context of a double slit setup. But I could be wrong.

 

[Cthugha]

However: in the context of the OP's questions, I'm not sure we need to consider this. Or we could ask the OP: "On what basis do you expect the photon to be entangled, and with what exactly?" I'm saying no answer could lead to an interference pattern and be physical in the context of a double slit setup. But I could be wrong.

Indeed I am not sure what the OP is after, I am just not a friend of people needing to "unlearn" something. I am aware that to most of us it is clear what we talk about, when we consider the double slit and entanglement, but the layman might have different scenarios in mind.

For example if you have single photons that are only polarization entangled with something else and nothing else is changed and you shoot them at a simple double slit, you will still see the typical double slit pattern. There simply is no which-way information present as the path taken is independent of the polarization unless you place some additional optical elements such as calcite crystals. If I remember correctly, Walborn showed such a pattern obtained using a Bell state in his double slit quantum eraser paper for comparison.

 

[Agrippa]

However: in the context of the OP's questions, I'm not sure we need to consider this. Or we could ask the OP: "On what basis do you expect the photon to be entangled, and with what exactly?" I'm saying no answer could lead to an interference pattern and be physical in the context of a double slit setup. But I could be wrong.

I deliberately left out those details because I was hoping to learn what kinds of entanglements would leave the interference pattern as is, what kinds what destroy it partially, and what kinds would destroy it completely.

But let me say a little more. And hopefully what I say here allows you to conceive of a method of entangling a single photon with an apparatus before the photon goes towards the slits. Let's imagine we have a laser that sends photons through a box. The box is such that photons easily penetrate it, yet it stores matter that the photons reflect/scatter from. The reflecting/scattering part is what creates the entanglements (right?). Photons exiting the box are then directed towards a single slit (with a width of one wavelength of the photon) by mirrors. The photons then head towards the two slits.

I would love to learn more about what kinds of material in the box (and therefore, what kinds of entanglements) would allow, destroy, or partially destroy the interference pattern that would have been there if the photons had not gone through the box.

But maybe it's good to start with an extreme case, where the box contains a very warm and complex environment (maybe a brain!). Is this a situation that completely destroys the interference pattern in virtue of the entanglements? If so, how to "bring down the complexity" of the box contents, so that the photons yield a partial interference pattern?

 

[Agrippa]

That of course depends on what property you want to entangle. The double slit is a simple experiment, where the pattern is a function of the relative phases (and intensities, but let us assume these are the same at the two slits) of the light fields at the two slit positions. This makes the double slit an experiment that depends on photon momentum. Photons with a different momentum will hit the slits at some angle. This creates some path length variation to the two slits, which in turn results in a phase difference and thus leads to a different interference pattern.

As a consequence, if you shoot single photons at the double slit and somehow create momentum entanglement with some piece of matter, such that the photons will arrive at some random angle, the interference pattern will vanish as you will effectively get a superposition of many interference patterns which cancel out. If you instead shoot single photons at the double slit and somehow create for example only polarization entanglement with some piece of matter without changing anything else, the interference pattern will not be influenced by this.

The influence of separability on coherence and entanglement is briefly discussed in this article:
https://arxiv.org/abs/quant-ph/0112065
There, photons are not entangled with matter, but momentum entangled photon pairs are used, but with respect to the underlying physics it does not really matter much whether you entangle the photons with matter, other photons or whatever.

Thanks, that's very helpful. I don't want the photons to hit the slits at random angles. If you look above to my response to Dr. Chinese, I have more fully specified the set-up so that (hopefully, correct me if I'm wrong) this complication is removed. In this case, do we still get the kind of momentum entanglement that would completely destroy the interference pattern? Or would it just partially destroy it? Thanks!

 

[vanhees71]

1) Yes, entangled bases can vary as you say. Nonetheless, I'm not aware of any preparation possible where photons would be entangled on some basis *and* be coherent as regards to a double slit setup (i.e. a typical DS interference pattern). They can't be momentum entangled and coherent, right? And I don't believe they can be spin entangled *and* coherent. But perhaps some esoteric preparation is possible, or I am confused on this point.

I still don't get what you want to say. It sounds as if you think that coherence and entanglement are somehow excluding each other, but that's obviously not the case. Take my favorite quantum-eraser experiment by Walborn et al: They use polarization-entangled Bell states of photon pairs and they get interference patterns with quite good contrast, i.e., the single photons are pretty coherent (of course not perfectly coherent). The entire experiment cannot work with incoherent states (e.g., thermal photons).

 

[Cthugha]

Thanks, that's very helpful. I don't want the photons to hit the slits at random angles. If you look above to my response to Dr. Chinese, I have more fully specified the set-up so that (hopefully, correct me if I'm wrong) this complication is removed. In this case, do we still get the kind of momentum entanglement that would completely destroy the interference pattern? Or would it just partially destroy it? Thanks!

Maybe it is better to answer this in a rather general manner. If you entangle some property of your photons with something else, you will find that this property will vary strongly each time you measure it and in principle a measurement on the entangled partner could at any time set your photon into a state, where your property of choice takes one of these values at random in each measurement run. Now the double slit is at its heart a pretty classical experiment, which is not sensitive to whether the light used is classical or nonclassical at all on its own. One can see that easily from the definition of first-order coherence in terms of photon creation and annihilation operators, if you already know some advanced quantum mechanics.

Now the question is, why the interference pattern disappears for some entangled states. For the typical double slit, there are two ways of destroying the pattern.

1) The two paths become distinguishable. You get this for example by placing optics that change the path of photons depending on their polarization. Or you may get this if you introduce a polarization shift that depends on which slit the photon takes. You may introduce optics that create some time delay for photons going through a certain slit and so on and so forth. Stuff like that is used in many complicated versions of the double slit. However, in the "vanilla" double slit, there are no markers like that. Unless your intended version of entanglement creates some initial asymmetry in the double slit experiment (such as left-circularly polarized photons go to the left slit, right-circularly polarized photons go to the right slit) or enables you to nail down the exact time of photon emission so precisely that the photon travel time differences from the slits to the screen positions become larger than this window, the entanglement will not have any influence with respect to this point.

2) There is some varying initial relative phase shift at the slits. This does not give you which-way information, but it will change the interference pattern. The maximum of the interference pattern willl always be where the light fields from the two slits add up in phase. If you add a relative phase shift to the double slit, the positions of maxima and minima will move accordingly. If this phase variation is large, you end up with the superposition of many interference patterns, which sum up to no pattern at all. You can easily see this effect just by using the same light source and putting it close to a double slit and placing it far away afterwards. Putting it close will destroy the pattern. Putting it far away will preserve the pattern. This shows that the coherence measured is usually not a property of the light source. So how can you use entanglement to destroy the interference pattern in this way? If your entangled property results in such a relative phase shift between the paths to the slits, this will of course destroy your pattern. This is the case for momentum entanglement as different momenta result in different emission angles and therefore different path lengths between your point of emission and the two slits. Unless you have a similar kind of entanglement that results in such a phase shift, your double slit interference pattern will not change at all.

By the way, with respect to the double slit, there is also no difference between preparing a deterministic set of photons and a set of photons with the same ensemble-averaged distribution of properties via entanglement. There is nothing special in using entangled light for the double slit. The special thing is rather the other way around on the entanglement side: You cannot prepare a set of fully momentum entangled photons that will show first-order coherence in a double slit.

The entire experiment cannot work with incoherent states (e.g., thermal photons).

Do you mean the simple double slit? It works well with thermal light (filtered of course). Thermal light is second-order incoherent, but first-order coherence does not depend on the kind of light source used at all. It is one of the really unfortunate developments of language in physics that both of these things have been termed coherence.

 

[DrChinese]

I still don't get what you want to say. It sounds as if you think that coherence and entanglement are somehow excluding each other, but that's obviously not the case. .

It's pretty simple. Photons entangled via PDC do not produce interference patterns in a double slit setup. That's what the OP is asking about, and I am trying to deliver an answer that addresses that. Here's a reference from Zeilinger, see Fig. 1 on p. 290.

http://www.hep.yorku.ca/menary/courses/phys2040/misc/foundations.pdf

I said originally there are considerations of coherence, and other complicating factors involved in explaining that result. Which there are.

 

[DrChinese]

I deliberately left out those details because I was hoping to learn what kinds of entanglements would leave the interference pattern as is, what kinds what destroy it partially, and what kinds would destroy it completely.

But let me say a little more. And hopefully what I say here allows you to conceive of a method of entangling a single photon with an apparatus before the photon goes towards the slits. Let's imagine we have a laser that sends photons through a box. The box is such that photons easily penetrate it, yet it stores matter that the photons reflect/scatter from. The reflecting/scattering part is what creates the entanglements (right?). Photons exiting the box are then directed towards a single slit (with a width of one wavelength of the photon) by mirrors. The photons then head towards the two slits.

I would love to learn more about what kinds of material in the box (and therefore, what kinds of entanglements) would allow, destroy, or partially destroy the interference pattern that would have been there if the photons had not gone through the box.

But maybe it's good to start with an extreme case, where the box contains a very warm and complex environment (maybe a brain!). Is this a situation that completely destroys the interference pattern in virtue of the entanglements? If so, how to "bring down the complexity" of the box contents, so that the photons yield a partial interference pattern?

Generally, the types of interactions you describe (reflection/scattering/etc.) do not produce entanglement of light (or other particles). I would recommend reading up of entanglement of light via PDC (parametric down conversion). Understanding that will undoubtedly help you to frame a good question. Observable entanglement requires very specific setups, and it is not something you can meaningfully discuss outside of such contexts (as you are attempting to). People frequently say that *any* interaction leads to some kind of entanglement. Unfortunately, that is such a gross oversimplification that it is meaningless. In these references, you'll see that entangled light is manipulated in many ways within an experimental setup that do NOT involve any entanglement with the manipulating apparatus (which is the opposite of what you were thinking):

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

My comments in this thread have been intended to explain that *generally* entangled photons do NOT produce interference patterns at all. For them to produce an interference pattern, some/all of their entanglement must be terminated. That can be accomplished by placing a single point for the light to be diffracted through. After that, the photons will act per normal and display an interference pattern.

A couple of the other posters have pointed out that there are multiple bases that light can be entangled on, which is absolutely correct. The usual bases are position/momentum and spin/polarization. It is *possible* there is a scenario in which a photon can be entangled on some basis and still display an interference pattern. I have not seen any experiments with that specifically though, and even were it the case, it really wouldn't matter as it pertains to your question. That's because it would NOT be entangled on any basis relevant for a double slit experiment.

 

[Cthugha]

My comments were intended to say that this statement:

It's pretty simple. Photons entangled via PDC do not produce interference patterns in a double slit setup. That's what the OP is asking about, and I am trying to deliver an answer that addresses that.

is way too general. Photons entangled solely in polarization created in a SPDC process may produce interference patterns in double slit experiments in general.
See figure 2 in Walborn's standard double slit quantum eraser paper, where photons out of the BBO crystal are directly sent to the double slit without using any pinholes or additional optics for diffraction and the single-photon interference pattern can be seen clearly:
https://arxiv.org/abs/quant-ph/0106078

The discussion is on page 2 and 3, up to equation 7. It is only the special case of momentum entanglement that destroys coherence in the double slit.

However, I fully agree with you that it makes sense for the OP to read up on details of PDC and understanding the double slit experiment in detail. There is little to be gained by investigating complicated versions of entanglement in the double slit before understanding the basics. And most likely there is also little to be gained after understanding the basics unless one is interested in a very special scenario.

 

[vanhees71]

It's pretty simple. Photons entangled via PDC do not produce interference patterns in a double slit setup. That's what the OP is asking about, and I am trying to deliver an answer that addresses that. Here's a reference from Zeilinger, see Fig. 1 on p. 290.

http://www.hep.yorku.ca/menary/courses/phys2040/misc/foundations.pdf

I said originally there are considerations of coherence, and other complicating factors involved in explaining that result. Which there are.

You are right for this setup (momentum-entangled two-photon states through type-1 PDC). The signal photons are here indeed only made coherent through filtering to a good momentum eigenstate by coincidently registering the idler photons in the focal plane of the Heisenberg lense: This is obtained by projecting to well-defined momentum of the signal photon (going through the slit) by measuring the momentum of the idler photon by registering it in the focal plane of the Heisenberg lens. It's indeed always good to make explicit the specific situation one has in mind. BTW: It should be, at least in principle, be possible to set this which-path erasing up in the delayed-choice way.

I had the Walborn experiment in mind, where type-2 PDC was used to create polarization-entangled pairs. There you get double-slit interference for the complete ensemble of the signal photon and only gain which-way information by putting quarter-wave plates into the slits, oriented in 90-degree relative orientation (see also @Cthugha 's posting #27):

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

The putative "coherence" of accordingly selected photons is only limited by the uncertainty of the PDC process (i.e., the phase-matching conditions) itself.

 

[DrChinese]

Photons entangled solely in polarization created in a SPDC process may produce interference patterns in double slit experiments in general. See figure 2 in Walborn's standard double slit quantum eraser paper, where photons out of the BBO crystal are directly sent to the double slit without using any pinholes or additional optics for diffraction and the single-photon interference pattern can be seen clearly:
https://arxiv.org/abs/quant-ph/0106078

The discussion is on page 2 and 3, up to equation 7. It is only the special case of momentum entanglement that destroys coherence in the double slit.

I am familiar with this excellent reference provided by both you and vanhees71, but not quite familiar enough it seems. Thanks for correcting me about the polarization entanglement side.

 

[drl]

Dr.Chinese Am not very informed on Q.M. but it would seem to me that in the use of polarization crystals changing the polarization of photons alters their wave functions from each other so that their interference pattern would be affected to the degree in which their orientation was altered. Still, I am at sea that when detectors are placed at both slits we lose the interference pattern. It would seem to me that when we alter the wave function in the same way of what ever goes thru both slits it should restore the interference pattern. Thanking you for your reply.

 

[drl]

Dr. Chinese. Not very knowlegeable about Q.M. But in the use of polarized crystals the change in polarization of the crystals when they are crossed would also result in a change in their wave function to the degree in which they were turned, in turn, altering their interference pattern. I am still at sea using this logic that if detectors are placed at both slits, I would expect that the wave functions remain the same and the interference pattern should result which I was informed is not the case. Please Exlain

 

[Agrippa]

Generally, the types of interactions you describe (reflection/scattering/etc.) do not produce entanglement of light (or other particles).

Are you sure about this? Imagine my box contains a particle that's in a superposition of being on the left side of the box and of being on the right side of the box. Then I send my photon through. Does the photon not then enter into a superposition of reflecting from the left side of the box and reflecting from the right side of the box?

I would recommend reading up of entanglement of light via PDC (parametric down conversion).

PDC is method of entangling photon pairs. This is not what's happening in my set up. My box contains some matter (I was not thinking of photons). Photons are sent into the box, reflect off this matter, exit the box, and are then channeled toward a double slit apparatus. It is unclear to me how studying PDC is going to help me here.

 

[Agrippa]

Maybe it is better to answer this in a rather general manner. If you entangle some property of your photons with something else, you will find that this property will vary strongly each time you measure it and in principle a measurement on the entangled partner could at any time set your photon into a state, where your property of choice takes one of these values at random in each measurement run.

Sure, except that I'm not thinking of measuring the photon's entangled partners, since they're locked in a box that only the photons can penetrate. Instead, I'm using the double slit apparatus to measure the photons.

Now the question is, why the interference pattern disappears for some entangled states. For the typical double slit, there are two ways of destroying the pattern.

1) The two paths become distinguishable. You get this for example by placing optics that change the path of photons depending on their polarization. Or you may get this if you introduce a polarization shift that depends on which slit the photon takes. You may introduce optics that create some time delay for photons going through a certain slit and so on and so forth. Stuff like that is used in many complicated versions of the double slit. However, in the "vanilla" double slit, there are no markers like that. Unless your intended version of entanglement creates some initial asymmetry in the double slit experiment (such as left-circularly polarized photons go to the left slit, right-circularly polarized photons go to the right slit) or enables you to nail down the exact time of photon emission so precisely that the photon travel time differences from the slits to the screen positions become larger than this window, the entanglement will not have any influence with respect to this point.

I don't believe that such an asymmetry exists in my intended version. I've got photons (let's say infrared) that penetrate a box, reflect/scatter (and thereby entangle) with matter in that box, and then exit the box. After exiting the box I think no such asymmetries can exist since each exiting photon is then directed towards a single slit (with a width of one wave length of the photon), which then directs each photon toward the usual double-slit set up. So the two paths don't become distinguishable in the sense you describe (correct me if I'm wrong).

2) There is some varying initial relative phase shift at the slits. This does not give you which-way information, but it will change the interference pattern. The maximum of the interference pattern willl always be where the light fields from the two slits add up in phase. If you add a relative phase shift to the double slit, the positions of maxima and minima will move accordingly. If this phase variation is large, you end up with the superposition of many interference patterns, which sum up to no pattern at all. You can easily see this effect just by using the same light source and putting it close to a double slit and placing it far away afterwards. Putting it close will destroy the pattern. Putting it far away will preserve the pattern. This shows that the coherence measured is usually not a property of the light source. So how can you use entanglement to destroy the interference pattern in this way? If your entangled property results in such a relative phase shift between the paths to the slits, this will of course destroy your pattern. This is the case for momentum entanglement as different momenta result in different emission angles and therefore different path lengths between your point of emission and the two slits. Unless you have a similar kind of entanglement that results in such a phase shift, your double slit interference pattern will not change at all.

I'm puzzled as to why you are here talking about a phase shift that results from entanglement.In my example, I would have thought that the phase relation between the two components of the photon wave function, which is responsible for interference, is well-defined only at the level of the larger system composed of the photon and the entangled particle in the box, such that one can produce interference effects only in a suitable experiment including the larger system? For in this case, probabilities for results of measurements performed only on the photon are calculated as if the wave function had collapsed to one or the other of its two components/paths, but in fact the phase relations have merely been distributed over a larger system. Or is that what you mean by a phase shift?

 

[Michel Therrien]

I have two questions about the following type of scenario:

We have a laser sending photons through the usual double slit apparatus giving us the usual interference pattern, except that now we introduce some physical matter (that are not photons) that the photons will interact with before going through the double slit apparatus. The only assumption I want to make about this introduced physical matter and the nature of the interactions is that the photons will become entangled with that physical matter before those photons make it to the double slit apparatus.

Question (1): will the fact that the photons entangled with that matter destroy the interference pattern? I suspect it might, since I think (think) the reason why large molecules (larger than buckyballs) don't exhibit interference effects in the double slit experiment, is because they entangle with air molecules. However, I'm not sure: is this a matter of degree? Does just "a little bit of entanglement" (for each photon that makes it to the apparatus) only make a little bit of difference to the pattern? If so, is there a rigorous definition of "a little bit of entanglement"? Is there a measure of "how much" entanglement with the environment destroys the interference completely?

Question (2): if the answer to (1) is that we still get (some amount of) the interference pattern, then is it the case that affecting the introduced physical matter can affect the shape of the interference pattern? Here, by "affecting the introduced matter", I mean changing their states in some way without collapsing them.

Any insights here would be greatly appreciated!

What if you did the experiment in the dark?