Defining Wigner for particle bases

[DhruvaSambrani]

Given a state in the position/momentum bases, we can calulcate the wigner function as (black text)

as defined in wiki.

We should implement this function for states on the particle bases.

[DhruvaSambrani]

A starting point can be

function mywigner(psi)
    map(Iterators.product(samplepoints(psi.basis), samplepoints(MomentumBasis(psi.basis)))) do (x, p)
        xmax = psi.basis.xmax
        xmin = psi.basis.xmin
        N = psi.basis.N
        dy = (xmax - xmin)/N
        getloc(x) = round(Int, (x - xmin)/N) + 1
        ys = collect(filter(y->(xmin<x+y<xmax)&&(xmin<x-y<xmax), samplepoints(psi.basis)))
        1/(π*ħ) * sum(ys, init=0) do y
            conj(psi[getloc(x+y)]) * psi[getloc(x-y)] * cis(2*p*y/ħ) * dy
        end
    end
end

However, a better approach may be to actually perform a FT instead of running a sum

[Krastanov]

Starting with a simple implementation is perfectly reasonable. If you have the time, please submit a PR and the rest of us will try to help with review.

[david-pl]

FYI, there is already a transform implemented that changes from position to Fock basis and vice versa (see https://docs.qojulia.org/api/#QuantumOpticsBase.transform). You could also use that to transform your state to a FockBasis and compute the Wigner function using the one already implemented.
I'm not sure if that aligns with the linked definition though and finding the right xvec and yvec to pass into wigner may be tricky.

[DhruvaSambrani]

This would probably not be ideal since there are approximation steps (an upper limit to n) at each representation. This is probably also why this is not default behavior?

I may however be wrong, so please take with salt.

[david-pl]

Well that cutoff (and potentially offset) should essentially just be the same as the xmin, xmax, N you choose to represent your PositionBasis. But yes, these need to be chosen such that they represent the same basis, which is an additional pitfall and probably not ideal. You could try it out, though. Might be that the choice turns out to be obvious.