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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.)

Mark_Bell
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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.

Steven Sagona
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  • When I think about photon number states $|n\rangle$ I think about discrete mode that is in a cavity. Otherwise the description is made with wavepackets $|n_\xi \rangle$, where I think about traveling waves. Am I wrong? If photon number states are in a cavity, how can i use beam splitter? Or better to say: is it really the fock states that are measured or something related that can give me indirect information? I don't know if I am clear regarding what I am saying – Mark_Bell Mar 30 '21 at 20:52
  • @Mark_Bell, here is a paper that measures a $|1\rangle$ from a single photon traveling in free-space. While it's technically true that a photon has a traveling wavepacket, you understand that just describes that there is a probability distribution with measuring that single photon as a function of time, right? A "wavepacket" of $|1\rangle$ will really be something like $\sum_n c_n |1_n\rangle$, but you're never going to measure anything like $|2\rangle$ and you can distinguish between all the different Fock states. – Steven Sagona Mar 30 '21 at 22:25
  • This notation $\sum_n c_n |1_n\rangle$ is shorthand for something like $|1\rangle_1 |0\rangle_2 |0\rangle_3...|0\rangle_N$ + ...+ c_n|0\rangle_1 |0\rangle_2 |0\rangle_3.....|0\rangle_n...|0\rangle_N$ – Steven Sagona Mar 30 '21 at 22:27
  • @Mark_Bell, "When I think about photon number states |n⟩ I think about discrete mode that is in a cavity. Otherwise the description is made with wavepackets |nξ⟩, where I think about traveling waves. Am I wrong?" Certainly you're familiar with entangled photon pairs, right? These are not confined in a cavity and are usually written as "single mode" photons. – Steven Sagona Mar 30 '21 at 22:48
  • I am not sure that i completely understand this: "it's technically true that a photon has a traveling wavepacket, you understand that just describes that there is a probability distribution with measuring that single photon as a function of time". Maybe i am missing something. When say, for example, a two level system emits a photon, in spontaneous emission, is this $|1\rangle$ or a wavepacket? I think there must be a wavepacket because bandwidth is inversely proportional to lifetime. So Fock states seems just the basis of other states, and are approximately produced – Mark_Bell Mar 31 '21 at 00:08
  • In the paper you have linked, they use parametric down conversion for producing a state $|\psi \rangle = N( |0, 0 \rangle + \int dk_s dk_t \phi(k_s, k_t) | 1_{k_s} 1_{k_t} \rangle) $ and filtering spectrally and spatially the trigger channel so to approach single photon state. If I understand correctly, it just like they are filtering the wavepacket to reduce the bandwidth and approach the limit case of "single" frequency. So Fock state is here, as i said before a limiting case. Is it impossible to have a perfect single mode without frequency spreading? Or maybe I missing something? – Mark_Bell Mar 31 '21 at 00:24
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    @Mark_Bell, even a cavity will have a frequency spread of the form you have written, it's just a matter of how "narrow" this spread is. A cavity can be used to shrink this frequency spread, but the spread is still there. – Steven Sagona Mar 31 '21 at 00:38