Interference with short and Long path?

[QuantumUnbound]

If I set up an interference experiment and make one path much longer than the other would I still see an interference pattern? If so why? It would seem to me that having one path very long would give which-way information because you could easily time the photons to see which path it went through. So I'm thinking you wouldn't see an interference pattern.

 

[vanesch]

If I set up an interference experiment and make one path much longer than the other would I still see an interference pattern? If so why? It would seem to me that having one path very long would give which-way information because you could easily time the photons to see which path it went through. So I'm thinking you wouldn't see an interference pattern.

The thing you're looking at, classically, is coherence length: it is the difference in path length of a split beam which still allows you to obtain interference, and is classically related to the steadiness of the phase relationship between the field in the beam at x, and at x - L. This converts of course to a coherence time, T = L / c.
Now, if you want to have a coherence time bigger than a certain T_min (so that you do see interference), that means that the spectral distribution should be narrower than a certain delta-f ~ 1/T_min. But that implies that the envelope of the signal cannot vary strongly in a time T_min.
As such, you do not have a precise timing of your photon, so that you cannot know exactly when it was emitted (and hence, whether it took the long or the short road).
From the moment that you can know this timing (for instance, with a fast shutter), you introduce an amplitude enveloppe of your signal, which will enlarge the spectral width beyond delta-f, meaning that you have no strict phase relation anymore between the field at x and the field at x-L (in other words, the coherence length of your light is shorter than L) ; you won't have interference anymore. But, using the timing, you CAN now know which path the photon took.

Always the same...