The Hong–Ou–Mandel effect is a two-photon interference effect in quantum optics that was demonstrated in 1987 by three physicists from the University of Rochester: Chung Ki Hong, Zheyu Ou, and Leonard Mandel. The effect occurs when two identical single-photon waves enter a 1:1 beam splitter, one in each input port. When the temporal overlap of the photons on the beam splitter is perfect, the two photons will always exit the beam splitter together in the same output mode, meaning that there is zero chance that they will exit separately with one photon in each of the two outputs giving a coincidence event. The photons have a 50:50 chance of exiting (together) in either output mode. If they become more distinguishable (e.g. because they arrive at different times or with different wavelength), the probability of them each going to a different detector will increase. This effect can be used to test the degree of indistinguishability of the two photons and to implement logic gates in linear optical quantum computing.
If a detector is set up on each of the outputs (in bottom and top directions) then coincidences can never be observed, while both photons can appear together in either one of the two detectors with equal probability. While Quantum mechanics agree with empirical results, a classical prediction of the intensities of the output beams for the same beam splitter and identical coherent input beams would suggest that all of the light should go to one of the outputs (the one with the positive phase).
Why does classical prediction demand that only situation 1 is possible, even if the two photons are indistinguishable? I mean there cannot be a physical difference between the bottom side and the upside of the beam splitter if it is a symmetrical 50:50 beam splitter.