  We study products of functions evaluated at self-adjoint polynomials in
deterministic matrices and independent Wigner matrices; we compute the
deterministic approximations of such products and control the fluctuations. We
focus on minimizing the assumption of smoothness on those functions while
optimizing the error term with respect to $N$, the size of the matrices. As an
application, we build on the idea that the long-time Heisenberg evolution
associated to Wigner matrices generates asymptotic freeness as first shown in
$[9]$. More precisely given $P$ a self-adjoint non-commutative polynomial and
$Y^N$ a $d$-tuple of independent Wigner matrices, we prove that the quantum
evolution associated to the operator $P(Y^N)$ yields asymptotic freeness for
large times.
