  Motivated by the problem of long time stability vs. instability of KAM tori
of the Nonlinear cubic Schr\"odinger equation (NLS) on the two dimensional
torus $\mathbb T^2:= (\mathbb R/2\pi \mathbb Z)^2$, we consider a
quasi-periodically forced NLS equation on $\mathbb T^2$ arising from the
linearization of the NLS at a KAM torus. We prove a reducibility result as well
as long time stability of the origin. The main novelty is to obtain the precise
asymptotic expansion of the frequencies which allows us to impose Melnikov
conditions at arbitrary order.
