In Example 12.1, we introduced the Hong-Ou-Mandel interferometer and presented an analysis of thinking about the photons produced by the laser as classical electromagnetic waves. In this exercise, we will instead consider the photons as indistinguishable, quantum mechanical particles, to a rather surprising conclusion.

(a) Let's analyze this experiment quantum mechanically, with individual, identical photons emitted from the laser. Let's call the state \(|20angle\) when two photons hit the upper detector and none on the lower detector, the state \(|02angle\) the opposite, and the state \(|11angle\) when one photon hits each detector.

\footnotetext{23 M. Koashi and A. Winter, "Monogamy of quantum entanglement and other correlations," Phys. Rev. A} 69(2), 022309 (2004).

For the state \(|11angle\), draw the possible photon trajectories from the laser to the detectors. Quantum mechanically, we have to sum together the probability amplitudes of all possible trajectories to determine the net probability amplitude of the observed state \(|11angle\). What do you find? Do you ever detect the state \(|11angle\) quantum mechanically? Don't forget to include the relative change in phase between the two photons at the beam splitter.
(b) Now, draw the photon trajectories for the observed states \(|20angle\) and \(|02angle\). Are these observed quantum mechanically?