Estimation of disorders in the rest positions of two membranes in optomechanical systems

The formalism of quantum estimation theory is applied to estimate the disorders in the positions of two membranes positioned in a driven optical cavity. We consider the coupled-cavity and transmissive-regime models to obtain effective descriptions of this system for different reflectivity values of the membranes. Our models consist also of high-temperature Brownian motions of the membranes, losses of the cavity fields, the input-output formalism, and a balanced homodyne photodetection of the cavity output field. In this two-parameter estimation scenario, we compare the classical and quantum Fisher information matrices and evaluate the accuracies of the estimations. We show that models offer very different estimation strategies and the temperature does not have a detrimental effect on the estimation accuracies but makes it more difficult to attain the quantum optimal limit. Our analysis, based on recent experimental parameter values, also reveals that the best estimation strategies with unit efficient detectors are measurements of the quadratures of the output field.