  This paper is devoted to proving the polynomial mixing for a weakly damped
stochastic nonlinear Schr\"{o}dinger equation with additive noise on a 1D
bounded domain. The noise is white in time and smooth in space. We consider
both focusing and defocusing nonlinearities, respectively, with exponents of
the nonlinearity $\sigma\in[0,2)$ and $\sigma\in[0,\infty)$ and prove the
polynomial mixing which implies the uniqueness of the invariant measure by
using a coupling method.
