  The formalism of quantum estimation theory, focusing on the quantum and
classical Fisher information, is applied to the estimation of the coupling
strength in an optomechanical system. In order to estimate the optomechanical
coupling, we have considered a cavity optomechanical model with non-Markovian
Brownian motion of the mirror and employed input-output formalism to obtain the
cavity output field. Our estimation scenario is based on balanced homodyne
photodetection of the cavity output field. We have explored the difference
between the associated measurement-dependent classical Fisher information and
the quantum Fisher information, thus addressing the question of whether it is
possible to reach the lower bound of the mean squared error of an unbiased
estimator by means of balanced homodyne detection. We have found that the phase
of the local oscillator in the homodyne detection is crucial; certain
quadrature measurements allow very accurate estimation.
