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.

Figure 1

Schematic representation of the system. Two movable dielectric membranes are placed inside a Fabry-Pérot cavity. Further details about the scheme are in the text.

Figure 2

Semilogarithmic plot of quantum and classical lower bounds of the variance $\text{Var}\left(\delta q_1\right)$ expressed in ${\text{m}}^2$ as a function of ${\mathrm{\Omega }}_l/{\omega }_m$ for (a) the CC model and (b) the TR model. The experimental values are taken from [22].

Figure 3

Semilogarithmic plot of distance $d$ in the trace norm as a function of ${\mathrm{\Omega }}_l/{\omega }_m$ for (a) the CC model and (b) the TR model. The experimental values are taken from [22].

Figure 4

(a) Semilogarithmic plot of distance $d$ in the trace norm as a function of the disorder $\delta q_1$ and (b) cavity frequency obtained by varying the prepared position of the first membrane with the disorder $\delta q_1$ and keeping $\delta q_2=0$, both for the TR model. In (b) the horizontal line refers to the cavity's natural frequency in the absence of membranes.

Figure 5

A log-log plot of the quantum lower bounds of the variances expressed in ${\text{m}}^2$ as a function of the temperature $T$ for (a) the CC model and (b) the TR model. Here $H_{12}^{-1}$ gives information on the correlation of the data used for estimating the two disorders $\delta q_1$ and $\delta q_2$.

Figure 6

(a) A log-log plot of the quantum lower bounds of the variances in ${\text{m}}^2$ as a function of the reflectivity $r$ and (b) a log-log plot of the mean photon number $\overline{n}$ in the cavity as a function of the reflectivity $r$, both for the TR model.