Spatial wave-function of a single photon and its measurement In the last decade there were several papers claiming that they've measured a "transverse quantum state" / "quantum wave-function" / "spatial Wigner function" of a single photon:


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*Measurement of the transverse spatial quantum state of light at the single-photon level. B.J. Smith et al. Opt. Lett. 30, 3365 (2005), arXiv:quant-ph/0507142

*Direct measurement of the quantum wavefunction. J.S. Lundeen et al. Nature 474, 188 (2011), arXiv:1112.3575

*Compressive Direct Measurement of the Quantum Wave Function. M. Mirhosseini, et al. Phys. Rev. Lett. 113, 090402 (2014), arXiv:1404.2680

*Hologram of a single photon. R. Chrapkiewicz et al. Nature Photon. 10, 576 (2016), arXiv:1509.02890
Most of them refer to a Iwo Bialynicki-Birula's paper "Photon wave function" [Prog. Opt. 36, 245 (1996), arXiv:quant-ph/0508202] when describe the measured object (or don't refer to anything). Having read these papers and some other literature discussion, as well as this forum (see links below), I still cannot really understand of what exactly was measured by the authors of the experimental papers and weather it makes sense to call that a "photon wavefunction", so I assume I am missing something important. 
I wonder if


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*they've measured the electric field / amplitude of Maxwell mode with a single-photon in it (then what's so quantum about that, other than you have to do long counting)?

*they've measured the spatial momentum quantum state, but by the means of Fourier optics transformed it into distribution over spatial points (they why to call it the way they do)

*there is a reason to introduce a real quantum spatial wave-function of a single photon (than how it lines up with the absence of position operator for a photon and other problems discussed in the topics below?) 
and I would greatly appreciate if someone can help.
 A: Here is a paper giving the maxwell wavefunction of a photon.

(I am aware that a similar derivation using the A potential exists, but do not remember where I saw it.)
So it is a wavefunction, which is complex.
It has a chapter on measuring the wave function.

If  a  single-photon  state  of  the  field  is  created,  then  to  know  its  quantum  state  means  to know  its  electric  and  magnetic field distributions in space and time. Such a state is a single-photon wave-packet state, and its generation is an important goal in quantum-information researc
Recently,  a technique has been developed to measure  the transverse spatial quantum state of an ensemble of identically prepared photons . The single-photon light beam is sent into  an all-reflecting, out-of-plane Sagnac  interferometer, which performs a relative rotation of 180° and a mirror inversion on the wave fronts of the counter-propagating beams. The  Sagnac  performs  a  two-dimensional  parity  operation  on  one  of  the  beams  relative  to  the  other.  The  fields  are recombined at the output beam splitter and are interfered on a photon-counting photomultiplier tube (PMT), allowing the emerging  beams  to  be  detected  at  the  single-photon  level.  The  mean  photo-count  rate  is  directly  proportional  to  the transverse spatial Wigner function at a phase-space point that is set by the tilt and translation of a mirror external to the interferometer.

So as all probability distributions, it needs an ensemble at the end.(mean photo count rate)
A: The questions seems to boil down to "what is so quantum about a single photon?" In the absence of interactions, a single photon is described by the same mathematics as a classical electromagnetic (EM) field. (Helmholtz equation for the electric field for instance when one considers the propagating field without charges and currents.) The only difference is the interpretation of the quantity in the equations. (The modulus square of the wave functions gives a probability density while the modulus square of the electric field gives the intensity.) So, if one can have a classical EM field (bright laser light) that has a Laguerre-Gaussian mode, then one can also have a single photon with that mode describing its spatial wave function.
Those experiments that measure single photon wave functions are therefore almost the same as those that would measure the classical EM fields, with the only differences being the source that would produce (effectively) single photons and the physical detection process. For the single photon detection, one would typically use avalanche photo diodes that are sensitive to single photons.
The issue with the lack of a position operator for the photon does not imply that one cannot represent its wave function in the spacetime domain. All one needs for the latter is a coordinate basis. This basis does not have to be eigenstates of some position operator. In fact, it would not make sense for such a coordinate basis to be eigenstates of a position operator, because the coordinate basis needs to be four-dimensional whereas the set of eigenstates would only be three-dimensional.
