Photon interference and path differences

[kurt101]

TL;DR Summary

I would like to understand how photon interference is affected by path differences.

Most sources I come across on the topic of photon interference focus on the phase differences, but neglect discussion on the wave amplitude. Wave amplitude diminishes with distance; a single photon's energy does not.

So in a double slit experiment with photons being emitted one at a time, if the path from the photon source to one of the slits is longer in time/distance than the path to the other slit, will the interference pattern be reduced compared to if the paths were equal time/distance but with the same phase?

Would the shorter path slit have a greater influence on the pattern on the screen? (presumably because it has a greater wave amplitude)

Primarily this is of interest to me because I want to think it should be simple to say whether a photon's wave and particle aspects are separate entities or not; answering the age old question is a photon a particle or a wave. I expect to get the answer it is not so simple, but hopefully I can come away with a better understanding on why.

So my second question on this: If you only look at the photons that could have reached the detector screen by the shorter path, which you could know if you had a periodic shutter at the photon source, would you see any interference?

 

[kurt101]

After asking my question, the related threads section had links that answered most of my questions:
https://www.physicsforums.com/threads/destructive-interference-with-different-path-lengths.741790/
https://www.physicsforums.com/threads/interference-with-short-and-long-path.109974/

Sometimes I search forever on google looking for an explanation, but forget to search explicitly on the Physics forum.

So the answers to my questions are:
1) Yes, the interference pattern would be reduced.
2) No you would not see any interference if you only looked at the photons with knowledge of what path they could have taken.

I did not see a definitive answer on this question though:
Would the shorter path slit have a greater influence on the pattern on the screen? (presumably because it has a greater wave amplitude)

 

[PeterDonis]

I want to think it should be simple to say whether a photon's wave and particle aspects are separate entities or not

They're not "entities" at all. They're descriptions we use to help us try to understand what we observe. What you are looking for here doesn't even make sense.

 

[PeterDonis]

Wave amplitude diminishes with distance; a single photon's energy does not.

You are ignoring the crucial fact that, in quantum electrodynamics, which is what you are trying to use here, terms like "wave amplitude" and "photon" don't have simple classical meanings the way you are assuming. In fact they might not even be applicable at all, depending on the state the quantum field is in.

in a double slit experiment with photons being emitted one at a time

Sources for experiments like this do not emit "photons" in the sense you are using the term here. They emit coherent states of the quantum EM field, which are not eigenstates of photon number and are not properly described as "one photon being emitted at a time". The best you can do is that the expectation value of photon number is approximately one, but that is only an expectation value and doesn't tell you that you will observe exactly one photon inside the apparatus if you try to measure the photon number observable.

 

[kurt101]

They're not "entities" at all. They're descriptions we use to help us try to understand what we observe. What you are looking for here doesn't even make sense.

I don't know what angle you are coming from. I would think most anyone would know exactly what I am saying when I use the term entity. I can not think of a much more generic term to describe something we observe or measure.

The definition of entity: a thing with distinct and independent existence

A photon is not a "thing"? A wave is not a "thing"?

 

[kurt101]

You are ignoring the crucial fact that, in quantum electrodynamics, which is what you are trying to use here, terms like "wave amplitude" and "photon" don't have simple classical meanings the way you are assuming. In fact they might not even be applicable at all, depending on the state the quantum field is in.

Sources for experiments like this do not emit "photons" in the sense you are using the term here. They emit coherent states of the quantum EM field, which are not eigenstates of photon number and are not properly described as "one photon being emitted at a time". The best you can do is that the expectation value of photon number is approximately one, but that is only an expectation value and doesn't tell you that you will observe exactly one photon inside the apparatus if you try to measure the photon number observable.

I am primarily interested in what we observe in experiments. However let me ask my question using the Feynman path integral approach which maybe you would prefer I use.

If I was to sum up the contributions from the random zig zag paths from the EM field source to the detection screen in the experiment I described, would I find that the contributions of lines going through the shorter path slit contribute more than the lines going through the longer path slit?

And a secondary question:
When you sum up the paths using the Feynman path integral approach for photons, is the contribution reduced in anyway based on the length of one of these Feynman paths/lines? I did not think it was and that the algorithm naturally would lose line density with distance from the source, but I don't know and it has been a question that I have tried to find an answer for and have not been able to so far.

 

[PeroK]

Primarily this is of interest to me because I want to think it should be simple to say whether a photon's wave and particle aspects are separate entities or not; answering the age old question is a photon a particle or a wave. I expect to get the answer it is not so simple, but hopefully I can come away with a better understanding on why.

"Particle-like" and "wave-like" are names given to describe behaviour. They don't necessarily represent a duality in the theory of QM. This is one difference between the physics you encounter on the Internet and the physics you encounter in university textbooks. I have two textbooks on QM. The first mentions "wave-particle duality" once, as a historical footnote. The second doesn't mention it at all!

 

[DrChinese]

1. Would the shorter path slit have a greater influence on the pattern on the screen? (presumably because it has a greater wave amplitude)

...

2. So my second question on this: If you only look at the photons that could have reached the detector screen by the shorter path, which you could know if you had a periodic shutter at the photon source, would you see any interference?

As PeterDonis has correctly noted, some of your comments cannot be considered good descriptions. But there are a couple of things I can add that might help.

1. You couldn't say that either slit has more or less influence. The shorter path (assuming the slits are the same size but one is positioned closer) will allow more light to go through it. Most (if not all) will effectively contain which-slit marking (as it is not matched to the light going the other way). Thus: little or no interference pattern will form. You will just see 2 bars, one larger than the other.

2. There is no interference if you know which path the light traverses. That is the fundamental point of the experiment.

 

[PeterDonis]

A photon is not a "thing"? A wave is not a "thing"?

Not the way you're using those terms in this discussion. The "thing" is the quantum electromagnetic field. "Photon" and "wave" are descriptions of particular kinds of states of that field.

 

[PeterDonis]

let me ask my question using the Feynman path integral approach which maybe you would prefer I use.

If you have which path information, you don't have one path integral, you have two: one for each of the two paths (shorter and longer). A single Feynman path integral only contains contributions from alternatives that can't be distinguished in the results of the experiment. So in this case, since you can tell which slit the light went through/which path (shorter or longer) it took, you have separate path integrals for the two cases.

 

[kurt101]

1. You couldn't say that either slit has more or less influence. The shorter path (assuming the slits are the same size but one is positioned closer) will allow more light to go through it. Most (if not all) will effectively contain which-slit marking (as it is not matched to the light going the other way). Thus: little or no interference pattern will form. You will just see 2 bars, one larger than the other.

Thanks, but let's say the number of photons that took either the short or the long path is statistically even and no photons are lost on either path. Let's say you only compare the photons that could have arrived over either path without knowing which path they took.

In a loose classical sense you might think:
1) when the photon actually took the short path it will be interfering with the ghost photon that took the long path.
2) when the photon actually took the long path it will be interfering with the ghost photon that took the short path.

In this scenario will you see the same pattern of interference as the scenario where the two paths are the same?

 

[PeroK]

In a loose classical sense you might think:
1) when the photon actually took the short path it will be interfering with the ghost photon that took the long path.
2) when the photon actually took the long path it will be interfering with the ghost photon that took the short path.

In this scenario will you see the same pattern of interference as the scenario where the two paths are the same?

There is no such thing as a "ghost" photon. If no attempt is made to detect the photon before it impacts the final screen, then there is only an evolving wave-function for a single photon. The evolution of the wave-function will be affected by the paths available: constrained by one or more slits in this case. The position the photon impacts the screen will be determined probabilitistically based on the wave-function.

If there are two slits there is always "interference", in the sense that what appears on the screen may not be the sum of two single-slit patterns. This is because the wave-function in this case is a superposition of two distinct paths.

If you identify "which way" the photon went, then the pattern on the screen will represent a wave-function associated with a single path.

 

[Cthugha]

Thanks, but let's say the number of photons that took either the short or the long path is statistically even and no photons are lost on either path. Let's say you only compare the photons that could have arrived over either path without knowing which path they took.

In a loose classical sense you might think:
1) when the photon actually took the short path it will be interfering with the ghost photon that took the long path.
2) when the photon actually took the long path it will be interfering with the ghost photon that took the short path.

In this scenario will you see the same pattern of interference as the scenario where the two paths are the same?

First, as a disclaimer: The standard double slit is a fully classical experiment. In order to determine the interference pattern that results, it is fully sufficient to consider the emitted fields. It does not matter, whether they are emitted in the form of single photons, laser light or light from a light bulb.

Second, in order to understand the results when considering different path lengths, it may be helpful to remember that for some time the double slit was not primarily used to perform experiments about the foundations of physics, but to measure the physical properties of the light source. Accordingly, the answer to what happens when the path lengths are different is: it depends on the source.

A version of the double slit was used as a stellar interferometer to measure the angular diameter of stars. Why does this work? No light source will emit a perfectly sinusoidal wave, where the phase is perfectly determined at all times. Interactions within the light source or with its surroundings will randomize the phase of the emitted light field over time. Now, there are two extreme cases in the double slit: if these random phase changes occur rarely, the phase difference of the two beams traveling via the two slits and arriving at the detector plane at some point will solely depend on the path length difference of the two paths. This is a constant given only by geometry and will always be the same. In the other extreme case, the random phase changes occur very frequently. In this case, the phase difference of the two beams traveling via the two slits and arriving at the detector plane at some point will depend only on the random phase changes that occurred within the emitter, but not on the geometry of the double slit. This phase difference will be random as well and change quickly with time. In the first case you will see an interference pattern. In the second case you will not see a pattern.

Accordingly, whether you will see an interference pattern when there is a path difference, depends solely on the source. If the phase of the emitted light randomizes during the time it takes the light to travel the length difference between the two paths, no interference pattern will be seen. Otherwise you will see a pattern of full or partial visibility. To go back to the beginning: for light from distant stars, the time scale of phase randomization of the part of light that actually reaches us is determined by the angular diameter of that star. This is why one may use a version of the double slit to determine the angular diameter of stars in astronomy.

 

[kurt101]

Thanks for answering my questions so far. In particular it was very helpful to understand that the consistency of the source phase is very important when there is a difference in paths. However I am still looking for a more definitive answer.

In an experiment using a beam splitter and some kind of detector that can measure interference like shown in the picture below. Will these two experiments give the same pattern of interference when the source phase is consistent?

In the picture below B/S is the beam splitter; the green and red are the 2 paths; and D is the detector. In the experiment on the right, for the red path, the numbers (1-6) illustrate the path taken.

For both experiments the paths chosen are intended to make the phases the same at the detector.

Assuming you run the experiment for a sufficiently long time, will these 2 experiments give the same interference pattern?

 

[Cthugha]

For both experiments the paths chosen are intended to make the phases the same at the detector.

Assuming you run the experiment for a sufficiently long time, will these 2 experiments give the same interference pattern?

I am not sure how this can be more definitive as the answer for all of these scenarios is the same. As long as the red and the green beam are cw beams that have the same intensity and spot diameter and you do not change the polarization of any of the beams and the phase difference is fixed and determined only by the geometry of the setup and not by some random interactions that may take place along the way or inside the source, you will see the full interference pattern regardless of which path you choose at the red side.

This is by no means different from the standard way of describing the scenario that tells us that the presence of which-way information reduces the visibility of the interference pattern. The interactions that potentially provide us with which-way information are the same that change the phase, polarization or intensity of the light fields involved.

 

[kurt101]

I am not sure how this can be more definitive as the answer for all of these scenarios is the same. As long as the red and the green beam are cw beams that have the same intensity and spot diameter and you do not change the polarization of any of the beams and the phase difference is fixed and determined only by the geometry of the setup and not by some random interactions that may take place along the way or inside the source, you will see the full interference pattern regardless of which path you choose at the red side.

This is by no means different from the standard way of describing the scenario that tells us that the presence of which-way information reduces the visibility of the interference pattern. The interactions that potentially provide us with which-way information are the same that change the phase, polarization or intensity of the light fields involved.

Thanks! Does this experiment illustrate why there can't be a classical like wave that guides the photon?

If there really was a classical like wave that guided the photon in some way, the wave would diminish in amplitude, correct? In this experiment, I would have expected a different result for the experiment on the right, because the classical wave would interfere with very different classical amplitude than the experiment on the left. At least 50% of the time for the experiment on the right you would have a weaker classical wave interfering with a stronger classical wave.

Is my reasoning seem ok here?

 

[Cthugha]

Thanks! Does this experiment illustrate why there can't be a classical like wave that guides the photon?

No, and I do not really see, how you arrive at this conclusion. Let me emphasize one thing I said earlier again: The double slit is a classical experiment. It tells you absolutely nothing about single photons and the predictions are exactly the same for light fields with the same field properties, no matter, whether they are single photons or laser fields of the same mean intensity.

As I said in the last response, you will see a perfect interference pattern if the intensities of both beams are the same and you will not if they are not.

Why should the wave diminish in amplitude? I assume that you use collimated beams, where diffraction of the beam does not play a significant role for the distances considered here. If you beams that spread out, of course the visibility of the interference pattern will be reduced by an amount that is proportional to the difference in intensities. However, that is again true for any light field. Any light field will show the same spreading, no matter whether you send single photons or coherent laser beams.

Just in case this is not clear: single photons are not localized balls emitted at some well-determined time. If you remember the discussion about how it matters whether the phase of the light field is changed via interactions or not, it is important to note that anything that clearly indicates an emission process is such a phase-changing interaction. In other words: any single photon is just as delocalized as the classical wave and its duration also equals the classical time over which the phase randomizes. There really is absolutely no difference between single photons and other light beams of the same mean intensity with a standard double slit using just a single detector.

 

[DrChinese]

The interactions that potentially provide us with which-way information are the same that change the phase, polarization or intensity of the light fields involved.

As I understood his diagram on the right, the path lengths are different. I.e. the red one is longer than the green. That won't produce interference, because the red/green paths are distinguishable in principle.

If some of the light can take the path 1-6 without going through the 2-3-4-5 loop, then that portion would produce interference.

 

[PeterDonis]

Does this experiment illustrate why there can't be a classical like wave that guides the photon?

If you are proposing "a classical like wave that guides the photon" as a description of what standard QM says, it is wrong; that's nothing like what standard QM says.

If you are proposing it as an alternative explanation, it's personal theory and is off limits for discussion here.

In either case, this thread is closed.