  We address estimation of temperature for a micromechanical oscillator lying
arbitrarily close to its quantum ground state. Motivated by recent experiments,
we assume that the oscillator is coupled to a probe qubit via Jaynes-Cummings
interaction and that the estimation of its effective temperature is achieved
via quantum limited measurements on the qubit. We first consider the ideal
unitary evolution in a noiseless environment and then take into account the
noise due to non dissipative decoherence. We exploit local quantum estimation
theory to assess and optimize the precision of estimation procedures based on
the measurement of qubit population, and to compare their performances with the
ultimate limit posed by quantum mechanics. In particular, we evaluate the
Fisher information (FI) for population measurement, maximize its value over the
possible qubit preparations and interaction times, and compare its behavior
with that of the quantum Fisher information (QFI). We found that the FI for
population measurement is equal to the QFI, i.e., population measurement is
optimal, for a suitable initial preparation of the qubit and a predictable
interaction time. The same configuration also corresponds to the maximum of the
QFI itself. Our results indicate that the achievement of the ultimate bound to
precision allowed by quantum mechanics is in the capabilities of the current
technology.
