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.
The following picture shows the four possible outcome categories:
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.
Best Answer
The classical image is not quantum-mechanical, therefore you are not considering two photons, rather two beams:
You can see that in the wikipedia image for the phase relation between the incoming waves and the output ports.
https://www.cs.princeton.edu/courses/archive/fall06/cos576/papers/zetie_et_al_mach_zehnder00.pdf
In the above document you can find a description of the classical beamsplitter experiment, which is further explained by the vectorial description of the fields interacting at the boundary of the beamsplitter:
here:
$\vec{E_i} = \vec{E_{oi}} e^{i \vec{k_i}\cdot \vec{r}}$
$\vec{E_r} = \vec{E_{or}} e^{i \vec{k_r}\cdot \vec{r}}$
$\vec{E_t} = \vec{E_{ot}} e^{i \vec{k_t}\cdot \vec{r}}$
I omit the temporal description of the wave for simplicity's sakes but there is an additional temporal phase term in the exponential $\omega_{i,r,t}t$.
Continuity at the boundaries require that the phase is matched at the boundary (for energy conservation purposes):
$\hat{n} \times \vec{E_i} + \hat{n} \times \vec{E_r} = \hat{n} \times \vec{E_t}$
This is generally derived from Maxwell's equations (Here I don't provide proof besides the fundamental equations but this explanation can be found in fundamental optics and photonics books e.g. Saleh/Teich, Born/Wolf, etc.).
Hopefully this clears why the quantum-mechanical description would violate classical rules: For the "non-classical" cases, the phase relation is not retained from the classical pictures, but as could be seen in HOM experiments, it is possible to detect each photon at a different detector but only when dealing with single photons.