  Fluctuations of the atomic positions are at the core of a large class of
unusual material properties ranging from quantum para-electricity to high
temperature superconductivity. Their measurement in solids is the subject of an
intense scientific debate focused on seeking a methodology capable of
establishing a direct link between the variance of the atomic displacements and
experimentally measurable observables. Here we address this issue by means of
non-equilibrium optical experiments performed in shot-noise limited regime. The
variance of the time dependent atomic positions and momenta is directly mapped
into the quantum fluctuations of the photon number of the scattered probing
light. A fully quantum description of the non-linear interaction between
photonic and phononic fields is benchmarked by unveiling the squeezing of
thermal phonons in $\alpha$-quartz.
