Single photon interference

I guess this paper could be interesting for you:Nature Physics 11, 539–542 (2015) doi:10.1038/nphys3343. There is a physics world article available as well and which you might want to read first.

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.

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.

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).