How to produce and detect Fock or photon number states? I would like to build, if possible, an intuition of the physical methods on how photon number states $|n\rangle$ are experimentally produced and how are measured. We can focus on single mode.
It would be helpful a schematic description of one kind of these experiments (the set up and a little bit of mathematics can help), that can perform this task, emphasizing the key logic behind and the physical intuition, avoiding unnecessary details.
(I read some papers that touch this subject, but they are not pedagogical, and i sometimes lost my self in the details.)
 A: The most common way to experimentally measure photon number states is by measuring the coincidence information of hong-ou-mandel interference. If you take two single photons and you interfere them on a beam splitter, the photons that come out will always come out in pairs. This uniquely happens for single photon that are idential, and therefore, if you see a reduction in coincidences, you know you have single photons.
If you want to generally reconstruct the quantum state in the Fock state basis, you need to do "quantum state estimation."  If you use just single photon detectors, you'll never be able to distinguish between different superposition states, like $|0\rangle + |1\rangle$ and $|0\rangle - |1\rangle$. Homodyne detection makes measurements of a different operator that can see the difference between these states. Essentially, a homodyne detector measures the electric field of a photon, as opposed to intensity (E^2), and this at a quantum level lets us measure the statistical information of the amplitude of the electric field. We can then use this information to figure out which state is the most likely to reconstruct that statistics, which you can find here.
