Single photon interference In any interference experiment, whether it be an unequal arm length interferometer or the classical double slit set up, we have two unequal distances from source (or slits) to screen/detector. Single photon interference in the regime of quantum mechanics is explained by saying that every photon interferes with itself. This generally gives rise to the following misconception that this interference seems impossible since the photon going through the shorter path always gets absorbed at the detector before the one at the longer path makes it.
I have gone through this question which seems to be asking a similar question to this, but allow me to explain.
In the double slit experiment, is it correct to say that just after the photon is ejected from our source, the wavefunction, which may have a speed greater than the speed of light or have no speed at all and is present at all points in space, from the moment the photon was released, has already interfered at the detector before the photon would make it to the detector at velocity $c$ and it's only after time $t = (\text{path length})/c$ that we get the measurement/click at the detector?
I also understand that the wavefunction cannot be given a physical meaning but I've seen some sources mention the wavefunction propagating as a spherical wavefront. I'm not sure what to make of this. I find it easier to not even try to visualise the wavefunction at all!
 A: The confusion usually arises from trying to treat the double slit experiment from a particle point of view, where photons/electrons are injected on one side of the screen and detected on the other side. If we consider photons/electrons as waves, this problem does not arise: we solve the wave equation (Maxwell or Schrödinger) in all the space, and its eigenmodes already contain the interference, due to the boundary conditions on the screen. We can then think of a photon/electron injected at one point as a wave packet, which we expand in this eigenmodes. The narrower this packet is, the more modes it includes - thus, in this extreme particle view, the photon/electron is present on the other side of the screen from the very beginning, but we have to wait till its amplitude on the other side becomes detectable (i.e., results in the finite probability of detection).
Finally, one could explicitly construct a wave packet localized on one side of the screen. In this case we will have a wave front propagating in the direction of the detector with the speed less or equal the speed of light, but interference is already present in it.
A: The wavefunction of a particle is a representation of the relative likelihood of detecting the particle at any point in space and time.
If a particle is emitted from a source at a given instant, and a portion of the leading edge wavefunction arrives at a detector through the right slit earlier than through the left slit because of path length distances, then the two portions of the wavefunction do not overlap at the detector until the leading edge of the wavefunction also arrives at the detector.  Because there is no overlap during that brief interval, we know the wavefunction arrived from the right slit and we also know there is no interference during the interval.
At the trailing edge of the wavefunction, the wavefunction will arrive at the detector only via the left slit for a brief interval, so again there will be no interference during that interval and we will know that the wavefunction arrived only via the left slit.
However, during the time portions of the wavefunction arrive via both slits, there is interference at the detector and the particle has an indeterminate trajectory: it goes through both slits.
A: I believe you are placing too much emphasis on stating that in any interference experiment the slits define two different distances to the screen/detector.  The wave properties associated with a single photon experiment are based on the wavefunction reaching both slits at exactly the same time.  Once the photon has passed through the slits (pictured as having passed through both slits) there may indeed be a small difference in the distance traveled from the two slits to the detector/screen (depending on where on the screen the detection occurs).  However, this difference in distance is much less than the wavelength of the photon.    Therefore, it is not accurate to say that the one traveling the shorter distance got there first because you must take into account the uncertainty within the experiment.  The slits are always set to be only a tiny fraction of the wavelength of the incident particle.  The exact time when the detection occurs at the screen is uncertain within the wavelength of the particle. Therefore, you cannot say with certainty when the particle will arrive at the detector for either of the two possible path. If fact, you can argue that the photon traveling either path will arrive at the detection screen at the same time (within the uncertainly of the experiment).
A: The problem is with describing the photon as a particle which it isn't. The right description would be the presence of electromagnetic fields interacting with charges. In quantum field theory with Fermi's golden rules there is a probably a photon quanta is removed or created but it has nothing to do with particle interfering with itself. The field obeys rules like the superposition principle which leads to interference. The concept of a photon interfering with itself does not make sense and does not need to be worried about.
