Constructing the Wigner Function from Measurements of Entangled Photons

[neils]

TL;DR

How do you measure the Wigner Function of a quantum optical state?

Suppose i measure the phase and amplitude of some radiation, this might be a coherent state of an entangled state, how would i construct the Wigner function from these measurements?

 

[atyy]

Does an optical state have a Wigner function? I know of the Wigner function in quantum mechanics, but not sure what it would be in quantum field theory.

https://arxiv.org/abs/2108.06268
Functional Quantum Field Theory in Phase Space
José A. R. Cembranos, Marcos Skowronek

http://webhome.phy.duke.edu/~muller/Talks/YITP_080813.pdf
Wigner Functional Method in Quantum Field Theory
Berndt Müller

 

[Demystifier]

Does an optical state have a Wigner function?

Of course. https://www.amazon.com/dp/352729435X/?tag=pfamazon01-20

 

[neils]

Many thanks for the good links and works in this area. From looking at these i can now see that what I'm measuring is the Q-function. I can now appreciate both the Q and the Wigner function are both convolutions with the P-function.

However, i looking at the texts of the above i didnt see any method of generating the Wigner function from the Q-function. I now see this would require some kind of deconvolution, but it there a tried and tested method for this?

A further question arising is what might be the best function to analyse the quantum state, Q, P, or Wigner function, given that I'd be interested confirming measured entangled photon pair states?

thank you for any help.
Neil

 

[Cthugha]

However, i looking at the texts of the above i didnt see any method of generating the Wigner function from the Q-function. I now see this would require some kind of deconvolution, but it there a tried and tested method for this?

You can do so, but as any deconvolution process doing so may introduce errors and may involve ambiguities. For example, having negative values in the Wigner function is a flag for nonclassicality, while the Husimi Q function is necessarily non-negative. So reconstructing the negative parts via deconvolution does not really work well. The information content of all functions is the same, anyway. The Husimi Q function, the density matrix, the Wigner function and the Glauber-Sudarshan P distribution contain the same amount of information about the light field. However, in some instances it is easier to identify non-classicality using the Wigner function.

If you want to measure the Wigner function, the way to do it is balanced homodyne detection. You do not measure amplitude or phase, but a quadrature distribution which is the projection of the Wigner function along one axis. You may now repeat this for several axes by changing the relaitve phase between your signal and your local oscillator and can then perform quantum state tomography, where you reconstruct the Wigner function (or more typically the density matrix as it is easier) from this set of projections. A standard overview can be found here:
Reviews of Modern Physics paper by Lvovsky

A further question arising is what might be the best function to analyse the quantum state, Q, P, or Wigner function, given that I'd be interested confirming measured entangled photon pair states?

Entangled in what? The Wigner function is a phase space description, so it is well suited for looking at continuous degrees of freedom. If you consider an optical Schrödinger cat state to be entangled, the Wigner function would be a good way to look at it. However, if you really want to have a look at pair states, you need correlations between different modes, so you need to measure a two-mode Wigner function. This works well for twin beams in well defined modes, where you can measure pairs of quadratures (e.g. a joint measurement of the same quadrature for both beams followed by a measurement of the joint orthogonal quadratures in both beams), but if the entanglement includes many modes it is tedious and not really helpful to do homodyne detection due to its inherent mode sensitivity.

 

[neils]

Many thanks for the very helpful comments.

In terms of what is entangled, it would be two photons created by spontaneous parametric down-conversion, entangled in polarisation and momentum. I'd like to be able to measure something that would confirm that they are indeed entangled. So as you say, the two-mode Wigner function might be the thing to measure.

I'm also curious about what happens if two pairs of entangled photons (spaced very close together in time) are measured in a single sample? I heard if photons of similar characters are close together they bunch, so is the effect of this just that it makes a bigger 'blip' on the detector output, so you don't really know if it is a single pair of entangled photons, or two pairs very close together.

Thank you for any help.