  Phenomenological models aiming to join gravity and quantum mechanics often
predict effects that are potentially measurable in refined low-energy
experiments. For instance, modified commutation relations between position and
momentum, that accounts for a minimal scale length, yield a dynamics that can
be codified in additional Hamiltonian terms. When applied to the paradigmatic
case of a mechanical oscillator, such terms, at the lowest order in the
deformation parameter, introduce a weak intrinsic nonlinearity and,
consequently, deviations from the classical trajectory. This point of view has
stimulated several experimental proposals and realizations, leading to
meaningful upper limits to the deformation parameter. All such experiments are
based on classical mechanical oscillators, i.e., excited from a thermal state.
We remark indeed that decoherence, that plays a major role in distinguishing
the classical from the quantum behavior of (macroscopic) systems, is not
usually included in phenomenological quantum gravity models. However, it would
not be surprising if peculiar features that are predicted by considering the
joined roles of gravity and quantum physics should manifest themselves just on
purely quantum objects. On the base of this consideration, we propose
experiments aiming to observe possible quantum gravity effects on macroscopic
mechanical oscillators that are preliminary prepared in a high purity state,
and we report on the status of their realization.
