# How to interpret single photon interference when the two possible paths are different in length?

Here is my question.

I struggle with the definition of single photon interference. Let’s assume we have a Michelson interferometer and the interference pattern we observe is a single photon result, than if we have and we almost always have different length of the two possible photon paths after the beam splitter than how it could be possible to have the same time arrival of the two photon path when they are traveling different distances? Maybe my assumption that the beam splitter generate the two path in the moment the photon is interacting with the beam splitter is false, but I can’t explain the obtained interference unless I accept that there is a time delay in the two possible paths. This is not a paradox only for this type of experiment but for all the experiments that involve some kind of interference. It seems for me that we can not use the usual assumption of simultaneousness of events.

Have a look at my answer to Slit screen and wave-particle duality because this covers a lot of topics relevant to your question.

You're correct that if we imagine the photon as a little ball then if the arms of the interferometer are different lengths the two "halves" of the little ball cannot arrive at the detector at the same time. But this is not how the interference works. It does not work by the photon splitting and its two halves then interfering with each other.

To imagine the light as a little ball (the photon) ricocheting around your interferometer is highly misleading. The behaviour of the light is best explained by quantum field theory, but sticking to regular quantum mechanics we'd have to say that until we interact with the light (e.g. by it hitting a CCD or photographic plate) the light is delocalised over your whole interferometer.

This doesn't mean the photon has a position but we don't know it, it means the light simply has no position in the classical sense. The wavefunction that describes it covers your whole experimental equipment. The probability of detecting the photon at some point in your kit is given by the magnitude squared of its wavefunction at that point. If you change the geometry of your interferometer then you will change the wavefunction and therefore change the probability of detecting the photon at any particular point.

• On the scale of orthodox quantum mechanics interpretation it is totally wrong to thing of a photon as a small energy ball and that it is somehow delocalized over the entire interferometer. This view of course does not help to see the phenomenon more clear because it only explain what is happening on the macro scale and not what can potentially happen on the micro scale. It is difficult for me to agree that a photon could spread over an interferometer that may be not meters but light years long and that this is corresponding to an entity that upon measurement is localized on micro scale. – Kolyo Apr 18 '15 at 16:55
• My personal preferences are on some explanation combining bohemian mechanics and MWI and stating that the photon wave function is interfering with a wave function of the same photon but moving through the other path in a “parallel world” or other reality. This is of course a pure speculation but this is the way that we can continue to think of a photon like something that have physical meaning. However unfortunately with this interpretation it is not possible to explain the same time arrival unless we assume non-simultaneous movement of the two parts of the wave function. – Kolyo Apr 18 '15 at 16:57
• Even if we assume the orthodox view that the photon is delocalized over the entire instrument, it still lead to problems how it can make an interference interaction between two different lengths interferometer arms? Can somebody give idea how big can be this difference and still producing visible interference. For example is it possible that the one arm of the interferometer is 1000 longer than the other and we still have interference? – Kolyo Apr 18 '15 at 16:58

When one says "photon" one is in the quantum mechanical frame. Quantum mechanics does not follow the rules of classical mechanics if one tries to consider the photon one classical entity, like a bag of energy flowing. The photon is a point like elementary particle in the standard model, it has no extent and when it hits the detector it registers at a point of space (x,y,z) at time t. One point. There is no interference pattern from one photon in the energy given up. It is all at one point.

Single photon interferences appear cumulatively, from the random accumulation of individual photons:see page7 here..

Maybe my assumption that the beam splitter generate the two path in the moment the photon is interacting with the beam splitter is false,

Yes , it is wrong. At the individual photon frame the beam splitter affects the probability of which path the single photon will take, it does not split the photon. In quantum mechanics it is the probabilities that control the behavior of elementary particles, and the accumulation of individual hits of different photons will give the interference patterns, which are probability histograms as far as an individual photon's behavior can be gauged.

The probabilities in quantum mechanics are given by the square of the wavefunction's solution for the specific problem with its boundary conditions.

As far as the electromagnetic waves go, it is simpler mathematically to treat the behavior classically. The quantum mechanical wavefunction for the photon is derived by a special form of Maxwell's equation and the classical field emerges naturally form the accumulation of zillions of photons, but that is another story.

• "...the beam splitter affects the probability of which path the single photon will take, it does not split the photon." is importent for understanding every diffraction phenomenon and slit experiments too. – HolgerFiedler Apr 18 '15 at 11:47