What if the path difference between two routes is different?

[the_gravelator]

From my understanding photonic (and other) interference effects arise because of a fundamental uncertainty in the path taken by a single photon. So in young's slits the photon could go through either slit and we can't know which one so interference occurs. The moment we do know the wavefunction collapes and two-slit interference is lost.

What if the path difference between the two routes is different (which it will be in any physical experiment)?? As two-slit interference is observable in the lab this implies we still have no knowledge of the route a photon takes, despite different path lengths. What if we made the path difference really big, like 1 metre? or a million miles? Let's take a million miles. So we set out to recreate Taylor's beautiful experiment which constructs an interference pattern even though there is only one photon in the apparatus at any time. But we introduce this massive path difference with a couple of mirrors after one of the slits.

So it's all set up in a 1/2 million mile lab (we'll need the world to be big and flat for this:) ), we turn on our extremely weak light source, uncover the photographic plate, and then 1 second later cover the plate up and walk away. We do this a million times. Each time we do this we can expect, let's say, one photon to have been emitted. But if we detect a photon we know which route it took because it would be impossible for a photon to have traveled the long route in 1 second. So when we develop each of the million films and map the position of any photons detected what pattern do we get? A Young's slits pattern? I think not but I can't see how taking the film away after 1 second necessitates a collape of wavefunction, as it's clear that simply introducing a path difference does not destroy the diffraction pattern.

It's a complete head trip, cos if you left the plate long enough for a photon to have taken either route then diffraction should totally occur, so it's like you have to wait for the photon to check out the whole of either route before it makes its mind up on the probability distribution it will adopt. Crazy. But then in theory 50% of photons in this case would still take the short route AND still create a diffraction pattern, despite NOT having had time to check out the whole situation. What the f**k?!

So what do you think? There's no real question here but any enlightening input would be really cool.

 

[caribou]

I'd been thinking a little about a similar problem involving a vast two-slit experiment and timing the photon since it travels at a constant speed and the two possible paths it could take would obviously have two different times. Which time the photon took from emission to detection would suggest a definite path, and that can't be right.

I suspect the answer may be to the experiment in your post is that if one path takes 1 second and another path takes 5 seconds, which path did the photon take if it arrives after 3 seconds? Same for the experiment in the paragraph above, in that if one path takes 1 second and the other takes 1.2 seconds, which path did the photon take if it arrives in 1.1 seconds?

Exactly. It's imposible to tell. I suspect the photon takes a time between that of the individual paths when there is interference. A sort of average time for the different path lengths that equals the time something traveling at the speed of light would take to cover the average path length. I believe these to be faster-than-light and slower-than-light paths ("histories") as in those in Richard Feynman's sum-over-histories.

The two possible histories have a faster-than-light speed for the longer path and slower-than-light speed for the shorter path, and the histories interfere with each other and cause an effect like there was a "middle" path for the photon with an average speed, the speed of light.

So that's my guess at the answer, that the actual time is between either time for a single path.

 

[Billy T]

...What if the path difference between the two routes is different ... As two-slit interference is observable in the lab ...

The interference is destroyed even with many photons from line source when the path difference exceeds the "coherence length of the photon" - typically less than one meter of path difference. I have actually watched the interference pattern fade out as I increased the path difference (using a two path wave front division set up - two half silvered mirrors and two full mirrors. I am not sure of the second name, but this arrangement is usually called a Mach-Zelner interferometer.)

You can understand what is going on if you think in Fourier transforms. Many cycles of the optical wave lengh exist if the line is very sharp (well defined frequency) and then each photon might be several meters long, but most lab sources produce wider half width lines and have photon with less cycles, less well defined frequency, and they are shorter. The interference is only observed if the photon wave traveling both paths recombines with itself with high overlap of its "parts" - it is really to possible for humans to under this aspect of the quantum world we live in.

If your interest is in the time of flight, rather than in interference patterns, I am not completely sure what to say. -I never did anything in this area. Certainly very fast electro-optical shutters now exist and one could measure the flight time by gating the "weak source" single photons thru that then could travel by two different paths to the detector (very good time resolution is also possible in detectors now).

I strongly suspect that the distribution of times observed would be bimodal (two peaks) not an average of the flight times by the two different paths - that is timing of the wave / photon would exhibit its particle characteristics, not it wave characteristics and then you could infer which path it took. Research google to be sure.