  Quantum parameter estimation, the ability to precisely obtain a classical
value in a quantum system, is very important to many key quantum technologies.
Many of these technologies rely on an optical probe, either coherent or
squeezed states to make a precise measurement of a parameter ultimately limited
by quantum mechanics. We use this technique to theoretically model, simulate
and validate by experiment the measurement and precise estimation of the
position of a cavity mirror. In non-resonant systems, the achieved estimation
enhancement from quantum smoothing over optimal filtering has not exceeded a
factor two, even when squeezed state probes were used. Using a coherent state
probe, we show that using quantum smoothing on a mechanically resonant
structure driven by a resonant forcing function can result significantly
greater improvement in parameter estimation than with non-resonant systems. In
this work, we show that it is possible to achieve a smoothing improvement by a
factor in excess of three times over optimal filtering. By using intra-cavity
light as the probe we obtain finer precision than has been achieved with the
equivalent quantum resources in free-space.
