We study a nonlinear Schrödinger equation with a linear damping, i.e. a zero order dissipation, and an additive noise. Working in $\mathbb R^d$ with $d\le 3$, we prove the uniqueness of the invariant measure when the damping coefficient is sufficiently large. The talk is based on a joint work with Zdzis{\l}aw Brze{\'z}niak and Benedetta Ferrario.