As a specialist told me, given a smooth affine Poisson algebra $S$ over $\mathbb{C}$, up to a choice of certain characteristic class, one can find one of the deformation quantizations of $S$, say $A$, such that the Poisson cohomology of $S$, suitably tensored with $\mathbb{C}[[h]]$, is isomorphic to the Hochschild cohomology of $A$. (Maybe my memory is not very good, the above is what I remember now. )

My question is, given $S$, how to compute explicitely $A$.

For example, if $S=\mathbb{C}[X, Y]$ with $\{X, Y\}=XY$, how to compute the "right" deformation quantization? I guess that $A=\mathbb{C}[[h]]<X, Y>/(XY-(h+1)YX)$

What about polynomial Poisson algebras in general?