Can one photon interfere with another photon?

[carrz]

Between emission and absorption, the wavefunction and associated probability distribution continually evolve, taking into account where the photon could potentially have been absorbed but wasn't.

Look at it from the practical point of view. There is only one way we measure the interference. We measure it relative to photons position at the detector which depends on photons exit angle from the slits.

Since frequency and wavelength do not change it's the amplitude. And so we are left with this direct relation between the amplitude and the exit angle. Therefore, photon trajectory is a function of photon amplitude. It must be, there is nothing else.

 

[stevendaryl]

In any double slit setup, it is SELF interference. That is not usually obvious because there are so many photons present. It is much more obvious when something with a rest mass is diffracted through the slits. Self interference is the quantum effect being observed. Please note that interference is also a classical wave property, which makes it somewhat confusing to distinguish one from the other.

I'm confused about the relationship between classical and quantum interference.

If you do a classical analysis of the double slit experiment, Maxwell's equations predict a characteristic interference pattern, with light and dark regions. Classically, it doesn't matter whether the light sent through the slits is from the same source, or two different sources, as long as the two sources are in-phase. So imagine using two in-phase light sources to produce the double-slit diffraction pattern. You see characteristic light and dark regions. Now, dial down the intensity from each of the sources until you start seeing individual photons. I would expect that the "classical" diffraction pattern would gradually give way to a probabilistic treatment, where the light region would become an area with high probability of a photon appearing, and the dark region an area of low probability. But in this case, there are two different sources. So the interference pattern of the photons are not due to self-interference.

So would the interference pattern disappear in the case of separate sources, in the low-intensity limit?

 

[DrChinese]

I'm confused about the relationship between classical and quantum interference.

If you do a classical analysis of the double slit experiment, Maxwell's equations predict a characteristic interference pattern, with light and dark regions. Classically, it doesn't matter whether the light sent through the slits is from the same source, or two different sources, as long as the two sources are in-phase. So imagine using two in-phase light sources to produce the double-slit diffraction pattern. You see characteristic light and dark regions. Now, dial down the intensity from each of the sources until you start seeing individual photons. I would expect that the "classical" diffraction pattern would gradually give way to a probabilistic treatment, where the light region would become an area with high probability of a photon appearing, and the dark region an area of low probability. But in this case, there are two different sources. So the interference pattern of the photons are not due to self-interference.

So would the interference pattern disappear in the case of separate sources, in the low-intensity limit?

I wouldn't call it that way. The separate sources would still be building the pattern up. There gets to be a tricky spot as we drop the intensity, but when it is low enough, it should be clear that the interference is always from self-interference.

So to be specific: I can prepare a source of pairs of entangled photons called X, and another source of similar pairs called Y. I use the Alice of each pair to "herald" the arrival of the other (Bob) via a X detector and a Y detector, each with a precise timing mechanism. I route the Bob of the X and Y stream through fiber so that they are both precisely directed* on the same course to the double slit. Barring nothing else, there will be interference and it will be clear it is self interference because the Alice arrival times are clearly labeled on the X and Y detectors. The times should be far enough apart most of the time so the source is clear.

Any variation of the intensity should give the same result.

*And this would have a few special requirements, namely that the photon is no longer entangled as to position/momentum.

 

[carrz]

So would the interference pattern disappear in the case of separate sources, in the low-intensity limit?

I don't know of any experiment that produced interference pattern with two separate light sources. If photons can not interact with other photons that should actually be impossible. The real puzzle seems to be what the slits are doing to photons rather than what photons are doing themselves.

 

[DrChinese]

Look at it from the practical point of view. There is only one way we measure the interference. We measure it relative to photons position at the detector which depends on photons exit angle from the slits.

Since frequency and wavelength do not change it's the amplitude. And so we are left with this direct relation between the amplitude and the exit angle. Therefore, photon trajectory is a function of photon amplitude. It must be, there is nothing else.

You cannot say what the trajectory was. The momentum is smeared. A photon is indivisible in terms of detection (no half photons) and yet the probability at spots varies. But only one detection. So you cannot link trajectory (momentum) and amplitude in the manner you describe. Interference patterns are due to self-interference, and the path cannot be known for there to be such interference.

 

[DrChinese]

I don't know of any experiment that produced interference pattern with two separate light sources. If photons can not interact with other photons that should actually be impossible. The real puzzle seems to be what the slits are doing to photons rather than what photons are doing themselves.

See my post #33 above.

The slits do not have anything to do with interference other than to create the shape of the bar. Diamond shaped slits create (roughly) diamond shape bars, for example.

 

[tim1608]

The real puzzle seems to be what the slits are doing to photons rather than what photons are doing themselves.

Hi Carrz

As I see it, the slits split the probability distribution of a single photon into two probability sub-distributions which both belong to the same photon. These two sub-distributions then interfere with each other on the wavefunction level. (The probability value of any point in each sub-distribution is proportional to the square of the sub-wavefunction value at that point.) The two sub-wavefunctions are added together to create the interference pattern of constructive and destructive interference. The probability of finding a photon at any point in the resulting interference pattern is then proportional to the square of the resulting wavefunction value at that point.

 

[carrz]

The probability of finding a photon at any point in the resulting interference pattern is then proportional to the square of the resulting wavefunction value at that point.

Do you think that two identical photons emitted in exactly the same direction, will not end up detected at exactly the same position?

 

[carrz]

See my post #33 above.

The slits do not have anything to do with interference other than to create the shape of the bar. Diamond shaped slits create (roughly) diamond shape bars, for example.

Does polarization have any impact on the interference pattern? That is, do we get the same pattern regardless of whether the light source is not polarized or polarized in any arbitrary plane?

 

[DrChinese]

Do you think that two identical photons emitted in exactly the same direction, will not end up detected at exactly the same position?

Of course they don't in a double slit experiment. That is the point. These are quantum particles and they do not have classical-only attributes as you describe them.

Please keep in mind that the double slit experiment has been done with large molecules too. These are easily controlled individually. See for example:

http://arxiv.org/abs/quant-ph/0110012

It should be clear that quantum particles can exhibit the ability to self interfere. There are plenty of studies on this point.

 

[DrChinese]

Does polarization have any impact on the interference pattern? That is, do we get the same pattern regardless of whether the light source is not polarized or polarized in any arbitrary plane?

Generally, no differences due to polarization.

 

[tim1608]

Do you think that two identical photons emitted in exactly the same direction, will not end up detected at exactly the same position?

Hi Carrz

As I understand it, the emitter does not emit the photon as a particle that already knows its location. Instead, all that the emitter emits is a wavefunction which corresponds to an associated probability distribution which describes where the photon might be. The precise information about where the photon actually is and its direction, does not yet exist in any form at this point. The photon only obtains an exact location at the point of absorption.

 

[DrChinese]

As I understand it, the emitter does not emit the photon as a particle that already knows its location. Instead, all that the emitter emits is a wavefunction which corresponds to an associated probability distribution which describes where the photon might be. The precise information about where the photon actually is and its direction, does not yet exist in any form at this point. The photon only obtains an exact location at the point of absorption.

In a double slit setup, it could have arrived there by at least 2 paths (corresponding to the 2 slits). You could say that there are any number of paths >1 actually.

 

[Cthugha]

I don't know of any experiment that produced interference pattern with two separate light sources. If photons can not interact with other photons that should actually be impossible. The real puzzle seems to be what the slits are doing to photons rather than what photons are doing themselves.

These experiments are possible. See for example: Am. J. Phys. 68, 245 (2000) ("Interference fringes from stabilized diode lasers").

To see why it is simple to see interference using a double slit, but hard using two lasers can be found by looking at the math. Whether you will see constructive or destructive interference at some position depends on the relative phase of the two light fields in question. For lasers, the phase will drift away quite quickly. Even for extremely good lasers, that will happen on a scale of milliseconds. As a consequence there will always be an interference pattern, but it will change every few milliseconds. For common lasers, this will rather happen on a scale of nanoseconds. Our eyes are slow and will just average over the many patterns and their sum is no pattern at all. If you have a fast detector, you will be able to see it.
In a double slit, the phase difference is determined only by geometry. There will be a well defined path length difference between the two slits and any point on the detection screen. This path length difference corresponds to a phase difference which determines the shape of the interference pattern. Now, if the phase of the light beam changes, it changes the same way at both slits. As you take the difference, this phase drift just cancels out. Therefore, it is much easier to see an interference pattern using a double slit.

Am I correct in thinking that the delay line would be applied to one of the two paths to ensure that two probability sub-distributions of the same photon cannot arrive at the beam splitter at the same time?

You usually use pulsed excitation to make the system emit a single photon. The delay between two consecutive photons is typically something like 13 ns. Now you just use the delay line to compensate this temporal offset.

I am not entirely sure of what exactly you mean by the "modes". Do you mean the areas on the beam splitter where the probability distributions of the photons will land? Am I correct that by using fibers or some other method, these areas will be made as small as possible?

Just have the system emit many single photons and have a look at their averaged distribution in real space. You want this distribution to be the same for both beams. However, you do not want to make it too small. Adjustment becomes a nightmare if you do.

Not exactly. Varying a part of an individual photon's wavefunction and associated probability distribution changes the probability of finding the photon in the changed part of its probability distribution.

A bit of nitpicking: There are no wavefunctions for photons. Wavefunctions imply eigenstates which stay unchanged under the appropriate measurement. Measuring a photon will destroy it, so that concept does not work. You rather calculate probability amplitudes for different events.

Also, it is a good idea to rethink what interference really means. If you ask an authority in quantum optics, Nobel prize winner Roy Glauber, then he will tell you that "interference of photons" is a really bad terminology. He stated in "Quantum Optics and Heavy Ion Physics" (http://arxiv.org/abs/nucl-th/0604021): "First of all, the things that interfere are not the photons themselves, they are the probability amplitudes associated with different possible histories. You can obviously have
different histories that involve more than one photon at a time."

It is a good thing to follow this advice. In quantum optics one uses the same approach as elsewhere: Consider the initial state, consider the final state and all the indistinguishable ways to get from one to the other. Add them, get the square and you are done. One can imagine that simply for a single photons. However, one can also imagine that there may be situations where the physics is more complex and the single photon level is not sufficient. Consider a light source which will always emit two photons simultaneously, but in a completely random direction. A look at the single photon level will give you the mean intensity at each position, but there is no way to tell, whether the source emits pairs or not. A look at the two-photon level will show you that whenever a photon is detected at some point, the second photon will show up ath the same point. All these phenomena include coincidence counts involving more than one photon and include what is loosely called multi-photon interference. However, I strongly suggest to follow Glauber's advice and simply consider these effects inside a framework of interfering probability amplitudes for complicated final states.

By the way, the paper above and Glauber's Nobel lecture "100 years of light quanta" both contain some great insights. The one I cited is a bit more radical and provoking.

 

[atyy]

Also, it is a good idea to rethink what interference really means. If you ask an authority in quantum optics, Nobel prize winner Roy Glauber, then he will tell you that "interference of photons" is a really bad terminology. He stated in "Quantum Optics and Heavy Ion Physics" (http://arxiv.org/abs/nucl-th/0604021): "First of all, the things that interfere are not the photons themselves, they are the probability amplitudes associated with different possible histories. You can obviously have different histories that involve more than one photon at a time."

This seems to involve the path integral picture? If so, is it really correct, given that the path integral must usually be rotated into imaginary time?

 

[stevendaryl]

These experiments are possible. See for example: Am. J. Phys. 68, 245 (2000) ("Interference fringes from stabilized diode lasers").

To see why it is simple to see interference using a double slit, but hard using two lasers can be found by looking at the math. Whether you will see constructive or destructive interference at some position depends on the relative phase of the two light fields in question. For lasers, the phase will drift away quite quickly. Even for extremely good lasers, that will happen on a scale of milliseconds. As a consequence there will always be an interference pattern, but it will change every few milliseconds. For common lasers, this will rather happen on a scale of nanoseconds. Our eyes are slow and will just average over the many patterns and their sum is no pattern at all. If you have a fast detector, you will be able to see it.

That's what I suspected was true, but that seems to contradict claims that a photon "only interferes with itself".

 

[DrChinese]

... Also, it is a good idea to rethink what interference really means. If you ask an authority in quantum optics, Nobel prize winner Roy Glauber, then he will tell you that "interference of photons" is a really bad terminology. He stated in "Quantum Optics and Heavy Ion Physics" (http://arxiv.org/abs/nucl-th/0604021): "First of all, the things that interfere are not the photons themselves, they are the probability amplitudes associated with different possible histories. You can obviously have
different histories that involve more than one photon at a time."

It is a good thing to follow this advice. In quantum optics one uses the same approach as elsewhere: Consider the initial state, consider the final state and all the indistinguishable ways to get from one to the other. Add them, get the square and you are done. One can imagine that simply for a single photons. However, one can also imagine that there may be situations where the physics is more complex and the single photon level is not sufficient. Consider a light source which will always emit two photons simultaneously, but in a completely random direction. A look at the single photon level will give you the mean intensity at each position, but there is no way to tell, whether the source emits pairs or not. A look at the two-photon level will show you that whenever a photon is detected at some point, the second photon will show up ath the same point. All these phenomena include coincidence counts involving more than one photon and include what is loosely called multi-photon interference. However, I strongly suggest to follow Glauber's advice and simply consider these effects inside a framework of interfering probability amplitudes for complicated final states.

Great points! Can't go too wrong quoting Glauber.

 

[DrChinese]

That's what I suspected was true, but that seems to contradict claims that a photon "only interferes with itself".

In a double slit setup, I think the "only interferes with itself" view is fine. It is in other places that view is dubious.

But most of those setups are over my head.

 

[Cthugha]

This seems to involve the path integral picture? If so, is it really correct, given that the path integral must usually be rotated into imaginary time?

Well, you can take the approach via path integrals or via canonical quantization. "Really correct" is a complicated wording. If you ask an experimentalist, whatever gives predictions in accordance with the experiment (and does not make millions of assumptions) is correct.

That's what I suspected was true, but that seems to contradict claims that a photon "only interferes with itself".

Glauber has an opinion on that, too. The passage comes directly before the one I quoted earlier:
"When you read the first chapter of Dirac’s famous textbook in quantum mechanics [8], however, you are confronted with a very clear statement that rings in everyone’s memory. Dirac is talking about the intensity fringes in the Michelson interferometer, and he says,

Every photon then interferes only with itself. Interference between two different photons never occurs.

Now that simple statement, which has been treated as scripture, is absolute nonsense."

I guess you need a Nobel prize to be able to say it that way. As DrChinese already said correctly, in a common double slit self-interference is all you need.

 

[atyy]

Well, you can take the approach via path integrals or via canonical quantization. "Really correct" is a complicated wording. If you ask an experimentalist, whatever gives predictions in accordance with the experiment (and does not make millions of assumptions) is correct.

Looking at http://en.wikipedia.org/wiki/Hong–Ou–Mandel_effect, the final state is:

(c⁺ + d⁺)(c⁺ - d⁺)|0,0>_cd = (c⁺c⁺ + d⁺d⁺ - c⁺d⁺ + d⁺c⁺ )|0,0>_cd

And the interference is due to (- c⁺d⁺ + d⁺c⁺ )|0,0>_cd = 0 ?

So whereas this is commonly stated as intereference between two photons, Glauber's complaint is that it should be seen as intereference between two states, each with two photons? I do see probability amplitudes here, but not histories. So maybe histories has to refer to the path integral, but not the canonical formalism?

 

[carrz]

These experiments are possible. See for example: Am. J. Phys. 68, 245 (2000) ("Interference fringes from stabilized diode lasers").

I don't have that book. Are you saying those experiments do not use any slits, they just cross the beams and get interference pattern?

"First of all, the things that interfere are not the photons themselves, they are the probability amplitudes associated with different possible histories..."

Possible histories? That's weird even as a mathematical concept, how can actual photons interact with an abstract idea? It's surreal.

 

[Cthugha]

So whereas this is commonly stated as intereference between two photons, Glauber's complaint is that it should be seen as intereference between two states, each with two photons? I do see probability amplitudes here, but not histories. So maybe histories has to refer to the path integral, but not the canonical formalism?

Well, there are two possible ways for two photons to end up at different detectors: Both are reflected or both are transmitted. These are the two "histories" and their probability amplitudes interfere destructively.

I don't have that book. Are you saying those experiments do not use any slits, they just cross the beams and get interference pattern?

Yes. They see the pattern for simple crossed beams if the detector integrates for less than 1 ms. For longer integration times the pattern loses contrast quickly.

Possible histories? That's weird even as a mathematical concept, how can actual photons interact with an abstract idea? It's surreal.

Is it that abstract? Consider a double slit. If you detect a photon somewhere behind the double slit, you cannot tell which slit it passed through. So you get two possible histories: the photon going through slit 1 and the photon going through slit 2.

In classical physics you just add the two fields and calculate the intensity. Here you add the probability amplitudes for the indistinguishable possible ways from A to B and calculate the probability density. The math does not change.

 

[carrz]

Instead, all that the emitter emits is a wavefunction which corresponds to an associated probability distribution which describes where the photon might be. The precise information about where the photon actually is and its direction, does not yet exist in any form at this point. The photon only obtains an exact location at the point of absorption.

That's a strange way to say we can't see photons unless we absorb them. Wave function is an abstract concept, it can not be materialized in the real world as such, it can not be "emitted" actually, only metaphorically.

It is the real photons that are emitted, at specific location, with specific direction, with actual electric and magnetic fields oscillating in a precisely defined geometrical plane, around actual axis that is their velocity vector, with actual frequency and wavelength.

Just because we can't see photons before the detection and don't know exact locations where they are emitted, reflected or deflected from, it does not mean they do not actually exist along their way following the shortest path between any two such interaction points.

 

[carrz]

Is it that abstract? Consider a double slit. If you detect a photon somewhere behind the double slit, you cannot tell which slit it passed through. So you get two possible histories: the photon going through slit 1 and the photon going through slit 2.

Shouldn't I be able to tell you which slit it passed through if I knew exact direction and location it was emitted from? Photons have to travel in a straight line to obey constancy of the speed of light, don't they?

 

[bhobba]

If photons cannot interfere with photons, then what in the world are they interfering with in a double slit experiment? There is nothing supposed to be there but the slit and the photons, and if it is not the photons, it must be the slit. Would edge-diffraction not produce the same pattern even without any interference?

Indeed it is the interaction of the slit and the photon. But as Dr Chinese points out its not the slit per se - its the fact it in effect is like a position measurement. I say like because photons, traveling at the speed of light do not have a frame where they are at rest hence to not have a position. A better way of looking at it is probably Feynmans path integral approach - the slits limit the paths the photon can take to reach the screen. Although it is called interference by analogy to wave experiments that's not the real reason - the real reason is the principles of QM:
http://arxiv.org/ftp/quant-ph/papers/0703/0703126.pdf

They start talking about interference and end up talking about entanglement. What does quantum entanglement have anything to do with wave interference?

Nothing. But, strictly speaking, quantum mechanically the double slit has nothing to do with interference either - as explained in the link I gave.

If two photons are entangled then they are partly in each others state so that means effects that would normally apply to only one photon are possible.

Thanks
Bill

 

[bhobba]

Shouldn't I be able to tell you which slit it passed through if I knew exact direction and location it was emitted from? Photons have to travel in a straight line to obey constancy of the speed of light, don't they?

No they don't - remember the path integral approach - they travel by all paths - not a single one.

They are emitted in a certain state, and that state, by the fact its emitted by a conceptual point source, means its position (loosely - as I said photons do not have position - its path going through that point is probably a better way of looking at it) is known at that time, hence its momentum is completely unknown. Since photons all travel at the speed of light that means its direction is unknown.

Again see the link on the QM analysis of the double slit experiment I gave that explains it entirely in these terms.

Thanks
Bill

 

[carrz]

No they don't - remember the path integral approach - they travel by all paths - not a single one.

There is only one path which is the shortest distance between two points, it's a straight line. If a photon doesn't interact with anything on the way and if it doesn't want to go faster than the speed of light it has no choice but to take that one path. We never measured otherwise.

Since photons all travel at the speed of light that means its direction is unknown.

Just because it is unknown doesn't mean the actual path was not precisely defined single direction.

Again see the link on the QM analysis of the double slit experiment I gave that explains it entirely in these terms.

I have, it does not contradict that photons, in real world actuality, always travel one specific path which is a straight line between any two interaction points.

 

[PeterDonis]

Since photons all travel at the speed of light

Strictly speaking, this isn't actually true, is it? Photons have a nonzero amplitude to travel faster or slower than light. More precisely, photons emitted from a given event in spacetime have a nonzero amplitude to be detected at events that are timelike or spacelike, rather than null, separated from the emission event. It's just that the amplitudes for timelike or spacelike separations die off very quickly with "distance", so for many experiments you can ignore them and treat the photons as traveling only on null paths.

 

[atyy]

Glauber has an opinion on that, too. The passage comes directly before the one I quoted earlier:
"When you read the first chapter of Dirac’s famous textbook in quantum mechanics [8], however, you are confronted with a very clear statement that rings in everyone’s memory. Dirac is talking about the intensity fringes in the Michelson interferometer, and he says,

Every photon then interferes only with itself. Interference between two different photons never occurs.

Now that simple statement, which has been treated as scripture, is absolute nonsense."

I guess you need a Nobel prize to be able to say it that way. As DrChinese already said correctly, in a common double slit self-interference is all you need.

Here is an argument against Glauber's claim that Dirac was in error. In the two photon case, the photons are identical, so one cannot talk about "each" photon. Therefore Dirac's statement that every photon interferes only with itself is true in cases in which "each" photon can be distinguished. In the case where there are two indistinguishable photons, it is the histories that interfere according to Glauber. If it is the histories rather than the photons that interfere, then it is true that "interference between two different photons never occurs".

 

[vanhees71]

Strictly speaking, this isn't actually true, is it? Photons have a nonzero amplitude to travel faster or slower than light. More precisely, photons emitted from a given event in spacetime have a nonzero amplitude to be detected at events that are timelike or spacelike, rather than null, separated from the emission event. It's just that the amplitudes for timelike or spacelike separations die off very quickly with "distance", so for many experiments you can ignore them and treat the photons as traveling only on null paths.

This is not correct. It doesn't make sense to say, "photons travel along a trajectory". One should not think about photons as miniature billard balls flying around at the speed of light. As a massless particle of spin 1 this picture makes never sense! A photon doesn't even have a clear definition for what's meant by "position" since there is no position operator in the strict sense.

The correct description of photons is QED and nothing else. An asymptotically free photon is defined as a single-particle Fock state. Here, "particle" must be read as a metaphor. As I stressed before, a photon cannot be viewed as a little particle in the classical sense. We "only" have a very abstract description about what's going on, and this is quantum field theory.

As for any "particle" you can calculate cross sections (S-matrix elements) for reactions. E.g., you can evaluate the probability for a detector (e.g., a photo plate) to find a photon at a certain location behind a double slit. Supposed you have prepared the photons such that they are pretty monochromatic you find an interference pattern as for macroscopic electromagnetic waves. It is clear from the formalism that for a single photon you get interference of the possibilities to move through one or the other slit, as long as you do not make it possible to decide through which slit the photon has gone. If you do so the interference pattern vanishes. This is a typical case of quantum behavior. It does not mean that one photon interferes with itself. It's not even clear to me what you mean by that.