  Given small initial solutions of the nonlinear quantum harmonic oscillator on
$\mathbb{R}$, we are interested in their long time behavior in the energy space
which is an adapted Sobolev space. We perturbate the linear part by $V$ taken
as multiplicative potentials, in a way that the linear frequencies satisfy a
non-resonance condition. More precisely, we prove that for almost all
potentials $V$, the low modes of the solution are almost preserved for very
long times.
