Destructive interference with different path lengths?

[fleem]

A laser's beam is split and then recombined, and the two path delays are adjusted so there is destructive interference at a detector. Let's also say one of the two paths has a measurable delay over the other.

Classically speaking, when the laser is first turned on, the detector would be illuminated as soon as the beam traverses the short path but before the long path is traversed. Once the long path is also traversed, destructive interference occurs and the detector is no longer illuminated (and then the setup is a dielectric mirror).

However, quantum mechanically speaking every photon must experience destructive interference in this configuration because there is nothing to decohere the photons.

Both descriptions have problems. The classical description says that even very sparse photons somehow know how long its been since the laser's "on" switch was flipped. The quantum description allows FTL communication along the long path (one could block the long path some distance away and the detector would show an immediate change in illumination).

So my question is, which is it?

 

[Cthugha]

However, quantum mechanically speaking every photon must experience destructive interference in this configuration because there is nothing to decohere the photons.

Of course not! The probability amplitude for a photon arriving at the detector position is of course zero in that early time regime before the long path is traversed and there can be no interference during this time.

Both descriptions have problems. The classical description says that even very sparse photons somehow know how long its been since the laser's "on" switch was flipped. The quantum description allows FTL communication along the long path (one could block the long path some distance away and the detector would show an immediate change in illumination).

The quantum and the classical description both give the same prediction. No interference while light arriving at the detector has only one possible path to go there. Destructive interference if both paths are possible. The quantum description does not allow FTL communication. If you block the long path, the detector will show a change in illumination some time afterwards. The exact time depends on how long light needs to travel from where you block the beam to the detector.

 

[fleem]

Thanks, Cthugha. I agree that photon behavior will depend only on the configuration within its light cone and that there will be the classical delays as noted. But what I don't understand is how a single photon will decide to self-interfere according to when the laser "on" switch was flipped. Let's say the laser power is turned down so we get very sparse recoils in the laser, each indicating a photon was emitted. Why does a sparse photon emitted some measurable time after the laser is turned on, still decide to self-interfere according to how long it has been since the "on" switch was flipped? Is it that the detector doesn't let the photon self-interfere until the configuration (including the "on" switch) is within the light cone of the detector? I do believe the "on" switch does play a notable role: energy must be conserved. But its certainly not clear to me how this all works.

 

[Cthugha]

But what I don't understand is how a single photon will decide to self-interfere according to when the laser "on" switch was flipped. Let's say the laser power is turned down so we get very sparse recoils in the laser, each indicating a photon was emitted. Why does a sparse photon emitted some measurable time after the laser is turned on, still decide to self-interfere according to how long it has been since the "on" switch was flipped?

If you get a recoil which allows you to know when a photon was emitted, the emitted light will more or less never show interference because it becomes as incoherent as it gets. You need coherent light in order to see interference. Moving to the language of quantum optics, this means that all photons inside a coherence volume are indistinguishable and you can only nail down the "exact emission time" (which is a bad concept anyway) with better accuracy than the coherence time. If you get single photon interference with different path lengths, that automatically means that you can in principle only know so little about the photon that it could have taken both paths.

In a nutshell, thinking about photons as tiny bullets is usually a bad idea and leads to seemingly misleading conclusions.

 

[craigi]

Thanks, Cthugha. I agree that photon behavior will depend only on the configuration within its light cone and that there will be the classical delays as noted. But what I don't understand is how a single photon will decide to self-interfere according to when the laser "on" switch was flipped. Let's say the laser power is turned down so we get very sparse recoils in the laser, each indicating a photon was emitted. Why does a sparse photon emitted some measurable time after the laser is turned on, still decide to self-interfere according to how long it has been since the "on" switch was flipped? Is it that the detector doesn't let the photon self-interfere until the configuration (including the "on" switch) is within the light cone of the detector? I do believe the "on" switch does play a notable role: energy must be conserved. But its certainly not clear to me how this all works.

For single photon interference the wave function must overlap in space and time. If the two paths have a large enough difference in length then you won't seen any interference. Shorter differences in length will exhibit interference, but the probability amplitudes for the 2 different paths will differ.

If you remember that the photon is traveling at the speed of light, but there is some finite uncertainty on the time that it was emitted, then the superposition should make sense.

 

[fleem]

@Cthugha, are you saying that the destructive interference can be disabled at any time simply by observing the recoil of the laser? I'm not sure I understand that.

 

[Cthugha]

No, I am saying that usually there is no measurable laser recoil that could tell you something about photon emission. You would need a recoil that puts the resonator into a state which is orthogonal to the initial state (no overlap between the states). For common resonators the recoil just drowns in uncertainty. In cavity optomechanics with resonator mirrors mounted on cantilevers you might get different results, though. This is more or less the same discussion you get when asking why the reflection from mirrors encountered in typical interferometers does not allow you to identify the photon path via mirror recoil.

If the recoil is so strong that it actually puts the resonator into a state orthogonal to the initial state, you will also not find destructive interference. The emission time is known now and there is no temporal overlap of the two probability amplitudes associated with the two paths.

 

[fleem]

I think I see what you are saying. From a classical perspective I could say that if we devise a very tiny mirror that has measurable per-photon recoil, then that movement in the mirror sufficiently scrambles the photon phase and frequency to negate interference, and it does so on theoretical grounds. Energy transfers from photon to mirror, and we can't know everything about that transfer. I agree that quantum mechanically we should simply talk about coherence and decoherence, and not think of the photon ever at one spot, I'm just being the devil's advocate to understand this better. Thanks alot!.

 

[Nugatory]

I agree that quantum mechanically we should simply talk about coherence and decoherence, and not think of the photon ever at one spot

Then don't

Cthuga's advice above ("thinking about photons as tiny bullets is usually a bad idea and leads to seemingly misleading conclusions") is spot on, but I think you're trying to ask the natural next question: "OK, then how SHOULD I think about the photon?".

Here's an intuitive picture which works pretty well for most of these light/photon problems.

Light always propagates according to the wave equations of classical E&M. Interference, diffraction, coherence, all that stuff, just use the classical wave picture. Photons only come into play and things only get quantumish when the electromagnetic radiation interacts with something; we observe that the energy in electromagnetic radiation is always transferred in discrete amounts delivered to a single point, with probability proportional to the intensity of the light at that point. We can call that point "the place where a photon hits", but it's better to think of it as the place where a photon appears as part of the interaction with the incoming electromagnetic radiation.