  We characterize quantum limits and semi-classical measures corresponding to
sequences of eigenfunctions for systems of coupled quantum harmonic oscillators
with arbitrary frequencies. The structure of the set of semi-classical measures
turns out to depend strongly on the arithmetic relations between frequencies of
each decoupled oscillator. In particular, we show that as soon as these
frequencies are not rational multiples of a fixed fundamental frequency, the
set of semi-classical measures is not convex and therefore, infinitely many
measures that are invariant under the classical harmonic oscillator are not
semi-classical measures.
