That is a mere practical detail and it can be solved by using a denser filter and waiting longer for the result. It is already known that diffraction patterns and interference fringes are seen at the output even when the photon flux is so low that only one of them is inside the apparatus at a time.
Royal Society childrens Xmas lectures actually did it at low flux as a demo sometime in the past decade or so. Since Boksenbergs Image Photon Counting System it has been relatively easy to demonstrate in real time.
OK since you ask. Take T1 < T2 and let the coherence length of the photon be TC >> T2 then a rough description of what is observed with increasing time is :
0 < t < T1 nothing only system noise T1 < t < T2 diffraction pattern characteristic of slot 1 alone T2 < t < T1+TC diffraction pattern of slots with interference fringes T1+TC < t < T2+TC diffraction pattern characteristic of slot 2 alone t > T2+TC nothing only system noiseIn reality it would be a smooth transition between photons arriving belonging to the various patterns in proportions that depend critically on the value of t. So yes you will be able to see photons that travel through in time T1 but they will not show fringes.
In a photon you have a tricky contradiction in that to have a precisely defined monochromatic frequency it must be long (many cycles of oscillation) in the time domain but to be detected by the photoelectric effect it is necessarily compacted at a point to zap a single electron.
But as soon as you do that you destroy the interference pattern that you are trying to measure by forcing the system into a different eigenstate. Once you know which slit the photon passes though there is no longer a choice of indistinguishable paths for the photon to take to the screen and the interference fringes of its wavefunction vanish.
Exactly. Although AFAIK it has never been done with time gated measurements I am pretty sure what the theory predicts.
If you are to see interference effects in the Youngs slit experiment then the photon must be given access to two or more indistinguishable paths. The attempts to test Bohr's Complementarity Principle go by the name of Welcher-Weg (which way) experiments. A modern paper online at arXiv discusses some of the issues without too much mathematics and with reference to recent experiments that try to probe this morass.
I don't know the author but his descriptions seem OK.
Regards, Martin Brown