  We consider the long time statistics of a one-dimensional stochastic
Ginzburg-Landau equation with cubic nonlinearity while being subjected to
random perturbations via an additive Gaussian noise. Under the assumption that
sufficiently many directions of the phase space are stochastically forced, we
find that the dynamics is attractive toward the unique invariant probability
measure with a polynomial rate that is independent of the vanishing viscosity.
This relies on a coupling technique exploiting a Foias-Prodi argument
specifically tailored to the system. Then, in the inviscid regime, we show that
the sequence of invariant measures converges toward the invariant measure of
the stochastic Schr\"odinger equation in a suitable Wasserstein distance.
Together with the uniform polynomial mixing, we obtain the validity of the
inviscid limit for the solutions on the infinite time horizon with a log log
rate.
