  Quantum oscillators in Gaussian states are often characterized by average
occupation numbers that refer to a basis of eigenstates of the non-interacting
oscillator Hamiltonian. We argue that quantum state purity is a more
appropriate characteristic of such states, which can be applied to oscillators
of any dimensionality. For a one-dimensional oscillator, the state purity is
directly related to a thermal occupation number defined with respect to the
number state basis in which the oscillator's quantum state is thermal. Thus, it
naturally introduces a more versatile definition of an average occupation
number. We study optomechanical sideband cooling of one- and two-dimensional
mechanical oscillators in particular, and derive exact analytical expressions
for the maximal mechanical state purity achievable in the quantum backaction
limit. In the case of a one-dimensional oscillator, we show that the thermal
occupation number related to purity can be well approximated by the average
phonon number in the weak-coupling regime, but that the two differ in the
regime of ultrastrong optomechanical coupling or in cases where the
oscillator's resonance frequency is strongly renormalized.
