  We introduce specific solutions to the linear harmonic oscillator, named
bubbles. They form resonant families of invariant tori of the linear dynamics,
with arbitrarily large Sobolev norms. We use these modulated bubbles of energy
to construct a class of potentials which are real, smooth, time dependent and
uniformly decaying to zero with respect to time, such that the corresponding
perturbed quantum harmonic oscillator admits solutions which exhibit a
logarithmic growth of Sobolev norms. The resonance mechanism is explicit in
space variables and produces highly oscillatory solutions. We then give several
recipes to construct similar examples using more specific tools based on the
continuous resonant (CR) equation in dimension two.
