  We provide a full quantum mechanical analysis of a weak energy measurement of
a driven mechanical resonator. We demonstrate that measurements too weak to
resolve individual mechanical Fock states can nonetheless be used to
unambiguously detect the non-classical energy fluctuations of the driven
mechanical resonator, i.e. "phonon shot noise". We also show that the third
moment of the oscillator's energy fluctuations provides a far more sensitive
probe of quantum effects than the second moment, and that measuring the third
moment via the phase shift of light in an optomechanical setup directly yields
the type of operator ordering postulated in the theory of full-counting
statistics.
