Almost reducibility and oscillatory growth of Sobolev norms

For 1D quantum harmonic oscillator perturbed by a time quasi-periodic quadratic form of $(x,-\mathrm{i}{\mathrm{\partial }}_x)$, we show its almost reducibility. The growth of Sobolev norms of solution is described based on the scheme of almost reducibility. In particular, an $o(t^s)-$upper bound is shown for the ${\text{CH}}^s-$norm if the equation is non-reducible. Moreover, by Anosov-Katok construction, we also show the optimality of this upper bound, i.e., the existence of quasi-periodic quadratic perturbation for which the growth of ${\mathcal{H}}^s-$norm of the solution is $o(t^s)$ as $t\to \mathrm{\infty }$ but arbitrarily ``close" to $t^s$ in an oscillatory way.