Time measurement in a double slit experiment with single photons

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Main Question or Discussion Point

Assumption: Screen detector is much closer to the slits than in "standard experiment" and the small angle approximation can't be used to determine the interference fringe maxima, but the interference pattern still occurs.
Is it possible to measure the time of detection in such setup accurately? If so, how would the time distribution for fringe maxima look like? Could it be calculated from path-integral formulation? Would it correspond to the averaged path length divided by speed? "Time" is the period of time between the emission and the detection.

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PeroK
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Assumption: Screen detector is much closer to the slits than in "standard experiment" and the small angle approximation can't be used to determine the interference fringe maxima, but the interference pattern still occurs.
Is it possible to measure the time of detection in such setup accurately? If so, how would the time distribution for fringe maxima look like? Could it be calculated from path-integral formulation? Would it correspond to the averaged path length divided by speed? "Time" is the period of time between the emission and the detection.
What's the relevance of the screen being much closer? And, what's the relevance of the time of detection?

You could certainly measure the time that an electron hits the screen, but how do you measure the time it started? Or, the time it passed through the slits?

Note descriptions of such experiments often include a mixture of QM and classical concepts. In some ways this is fair enough: an electron, for the most part, follows a near-classical trajectory except where it interacts with a narrow slit or slits (or, at least, can be modelled as such). But, you cannot then be arbitarily precise about this and pretend that QM and the UP do not apply to the electron at these times.

It might be useful for you to try to decsribe the double-slit experiment in purely QM terms, without the concept of a classical trajectory at all.

What's the relevance of the screen being much closer?
The point is to make the difference between path lengths (from slits to the same point on screen) relatively greater so they could not be approximated as parallel, to make this difference more significant in comparison to the distance between the slits, so the inequality: slitsDist<<pathLength is no longer true.
And, what's the relevance of the time of detection?
(...)
It might be useful for you to try to decsribe the double-slit experiment in purely QM terms, without the concept of a classical trajectory at all.
I want to know as much as I can about the difference between QM wave function propagation of photon interfering with itself and CM paths of photon considered as a classical wave or a particle. I think that travel time is an important aspect in this respect.
You could certainly measure the time that an electron hits the screen, but how do you measure the time it started? Or, the time it passed through the slits?
As I reckon, any measurement at the slits would destroy the interference pattern, so I want to measure the time of photon leaving the gun, so I need the detector right there.
Note descriptions of such experiments often include a mixture of QM and classical concepts. In some ways this is fair enough: an electron, for the most part, follows a near-classical trajectory except where it interacts with a narrow slit or slits (or, at least, can be modelled as such). But, you cannot then be arbitarily precise about this and pretend that QM and the UP do not apply to the electron at these times.
If such setup would make the time measurement inaccurate due to the uncertainty principle or because of any other reason, I still want to know if time calculation is possible and what are the results.

PeroK
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The point is to make the difference between path lengths (from slits to the same point on screen) relatively greater so they could not be approximated as parallel, to make this difference more significant in comparison to the distance between the slits, so the inequality: slitsDist<<pathLength is no longer true.

I want to know as much as I can about the difference between QM wave function propagation of photon interfering with itself and CM paths of photon considered as a classical wave or a particle. I think that travel time is an important aspect in this respect.

As I reckon, any measurement at the slits would destroy the interference pattern, so I want to measure the time of photon leaving the gun, so I need the detector right there.

If such setup would make the time measurement inaccurate due to the uncertainty principle or because of any other reason, I still want to know if time calculation is possible and what are the results.
If we are talking about light (photons) then there is no way to detect a photon that does not effectively destroy it. The exception is where pairs of entangled photons are created and you can measure (gain information about) a photon by detecting its entangled partner.

To be precise the photon is, without doubt, a relativistic particle. And, standard QM is strictly non-relativistic. The description of the double-slit experiment usually skirts round this issue and assumes that certain principles of QM must also hold for photons. Which indeed can be shown experimentally.

To make any genuine comparison (in terms of experimental results matching theory) between light as an EM wave and light in QM, you would need QED or the full-blown QFT.

The double-slit is not the be-all and end-all of QM experiments. One key difference in using QED is not what happens in the double-slit, but what happens when light interacts with matter.

PeroK
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I want to know as much as I can about the difference between QM wave function propagation of photon interfering with itself and CM paths of photon considered as a classical wave or a particle. I think that travel time is an important aspect in this respect.
To answer this question as best I can. If you study the theory of EM, by applying Maxwell;s equations you get transmission and reflection coefficients for the behaviour of light at a boundary (where the refractive index changes).

If you study QM, by solving the SDE (Schrodinger equation) you get transmission and reflection coefficients for a particle at a potential barrier. And, remarkably these coefficients take the same mathematical form as you get in the theory of EM.

This leaves you with two ways to model EM radiation: as an EM wave obeying Maxwell's equations; and, as a quantum particle behaving probabilistically. With the assumption that the same QM behaviour you can show for non-relativistic particles extends to photons.

I'm personally not aware of how QED would be used to predict what happens in the double-slit. Maybe someone else can help you there.

Marcin
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The point is to make the difference between path lengths (from slits to the same point on screen) relatively greater so they could not be approximated as parallel
You can produce the same effect by making the slits farther apart. There will be no interference when the difference in the travel times is greater than the coherence time; the position of the dot on the screen provides a which-way measurement.

The double-slit is not the be-all and end-all of QM experiments. One key difference in using QED is not what happens in the double-slit, but what happens when light interacts with matter.
Wikipedia quote:
Richard Feynman called it "a phenomenon which is impossible […] to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery [of quantum mechanics]."
That's why I'm sticking to it :)

To answer this question as best I can. If you study the theory of EM, by applying Maxwell;s equations you get transmission and reflection coefficients for the behaviour of light at a boundary (where the refractive index changes).

If you study QM, by solving the SDE (Schrodinger equation) you get transmission and reflection coefficients for a particle at a potential barrier. And, remarkably these coefficients take the same mathematical form as you get in the theory of EM.

This leaves you with two ways to model EM radiation: as an EM wave obeying Maxwell's equations; and, as a quantum particle behaving probabilistically. With the assumption that the same QM behaviour you can show for non-relativistic particles extends to photons.
I'm personally not aware of how QED would be used to predict what happens in the double-slit. Maybe someone else can help you there.
Thank you for all the answers.

Nugatory
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Wikipedia quote:
Careful here - Feynman was speaking metaphorically here, not making a rigorous statement about QM. You’ll find a lot more than just the double slit experiment in his QED textbook.

You can produce the same effect by making the slits farther apart. There will be no interference when the difference in the travel times is greater than the coherence time; the position of the dot on the screen provides a which-way measurement.
The point is to keep the interference. The difference in travel times must still be less than the coherence time - using your words.

Careful here - Feynman was speaking metaphorically here, not making a rigorous statement about QM. You’ll find a lot more than just the double slit experiment in his QED textbook.
I sincerely believe you, but I'm still not afraid of quoting wikipedia. This is a full quote taken from the context specifically about the double slits, with a reference to the source.

I'd like to know if this is a reasonable assumption: Travel time of SINGLE photon from slits to the fringe maxima on screen has to be equal or greater than the longer path divided by speed (c for photon). If the time was shorter, the interference pattern could not occur.

PeterDonis
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Travel time of SINGLE photon
If you aren't measuring the travel time, you can't assume it has any definite value. To measure the travel time, you would need to measure the time the photon passes the slits, and the time the photon reaches the detector. But, as has been pointed out, the first time measurement would destroy the photon.

If you aren't measuring the travel time, you can't assume it has any definite value. To measure the travel time, you would need to measure the time the photon passes the slits, and the time the photon reaches the detector. But, as has been pointed out, the first time measurement would destroy the photon.
I've decided to skip the drawing of the part of the path from the gun to the slits, to make it more clear. Imagine, that the first detector is at the outlet of the gun. I'm also asking is it a reasonable assumption regardless of the measurement.

PeterDonis
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Imagine, that the first detector is at the outlet of the gun.
Doesn't matter, since, as noted, if you try to detect the photon at the slits--which you need to if you want to measure the time from the slits to the detector--you will destroy the photon.

I'm also asking is it a reasonable assumption regardless of the measurement.

If you aren't measuring the travel time, you can't assume it has any definite value.

Doesn't matter, since, as noted, if you try to detect the photon at the slits--which you need to if you want to measure the time from the slits to the detector--you will destroy the photon.
I'm not trying to detect the photon at the slits. There are just two detectors: the first is at the outlet of the gun, the second is the screen.

PeterDonis
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I'm not trying to detect the photon at the slits.
Then you can't make any assumptions about the time it takes for the photon to travel from the slits to the detector. Which means you can't make any assumptions about whether or not that time is shorter than the coherence time, which was what you said you were interested in.

If you aren't measuring the travel time, you can't assume it has any definite value.
If I can't measure it, at least I want to calculate it and compare the results of CM calculation and QED or QM calculation.

PeterDonis
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If I can't measure it, at least I want to calculate it
If you aren't measuring it, you cannot make any assumptions about it. Which means you can't calculate anything meaningful either.

compare the result of CM calculation and QED or QM calculation.
What QED or QM calculation? There isn't one.

If you aren't measuring it, you cannot make any assumptions about it. Which means you can't calculate anything meaningful either.

What QED or QM calculation? There isn't one.
Is it impossible to determine the probability of detection in certain point and time from solving the Schrodinger equation or it's modification, or derive it from path-integral formulation? If so, I could say what time and detection position are the most probable.

PeterDonis
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Is it impossible to determine the probability of detection in certain point and time from solving the Schrodinger equation or it's modification, or derive it from path-integral formulation?
You could calculate probabilities for detection at the slits at particular times, but doing that would invalidate any calculations about probabilities at the detector, since detection at the slits would destroy the photon. So you cannot calculate any joint probabilities for detection at the slits at some time $t_1$ and detection at the detector at some later time $t_2$.

You could calculate probabilities for detection at the slits at particular times, but doing that would invalidate any calculations about probabilities at the detector, since detection at the slits would destroy the photon. So you cannot calculate any joint probabilities for detection at the slits at some time $t_1$ and detection at the detector at some later time $t_2$.
But the calculation (even at the slits) can't destroy the photon, right? :) But even if, I can skip it and calculate only the probabilities at the gun outlet and on the screen. That would satisfy me - I could make my comparison. Is it also impossible?

PeterDonis
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the calculation (even at the slits) can't destroy the photon, right?
How is this relevant to anything?

I can skip it and calculate only the probabilities at the gun outlet and on the screen.
Then you can't say anything about how long it takes the photon to travel from the slits to the screen.

That would satisfy me - I could make my comparison.
Comparison with what? You said you were interested in the difference in travel times of the photon from the two slits to the screen. The only way you can say anything meaningful about that is if you measure the time the photon passes the slits--which destroys the photon. If you do not measure the photon's time of arrival at the slits, you cannot say anything meaningful about the travel time from either slit to the screen. No calculation you can make will change that.

Then you can't say anything about how long it takes the photon to travel from the slits to the screen.
I can commit a horrible heresy and subtract the travel time from the gun to the slits (classical distance divided by speed) from the final result - the time of detection.
Comparison with what? You said you were interested in the difference in travel times of the photon from the two slits to the screen.
No. I want to compare a measured or calculated probabilistically time of detection and compare it with the travel time of classical paths through the both slits.
The only way you can say anything meaningful about that is if you measure the time the photon passes the slits--which destroys the photon.
This is my goal - to get addition information without the measurement at the slits.
If you do not measure the photon's time of arrival at the slits, you cannot say anything meaningful about the travel time from either slit to the screen. No calculation you can make will change that.
As above.

Nugatory
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I can commit a horrible heresy and subtract the travel time from the gun to the slits (classical distance divided by speed) from the final result - the time of detection.
It’s not a heresy, just an irrelevant calculation that gives you a number that doesn’t represent anything.

It’s not a heresy, just an irrelevant calculation that gives you a number that doesn’t represent anything.
If I repeated the experiment and this calculation multiple times for single photons, would the statistic also mean nothing?