Achieving optimal multiparameter quantum estimation in optomechanics

Accurate simultaneous estimation of temperature and effective detuning is crucial for understanding, controlling, and optimizing optomechanical systems. Temperature characterizes the thermal state and mechanical properties, while detuning governs the light-motion interaction, influencing stability and enabling diverse quantum applications. Precise estimates of these parameters are therefore essential to exploit the full potential of such systems in high-precision metrology and the exploration of macroscopic quantum phenomena. In this work we investigate the simultaneous estimation of the effective detuning <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi mathvariant="normal">Δ</a:mi> </a:math> and the mirror temperature <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>T</c:mi> </c:math> in a Fabry-Pérot cavity system, with a particular focus on the performance of heterodyne detection. Using analytical expressions for the quantum Fisher information matrices (QFIM) derived from the symmetric logarithmic derivative (SLD) and the right logarithmic derivative (RLD), we demonstrate that the quantum Cramér-Rao bound associated with the RLD is consistently lower than that based on the SLD, highlighting superior estimation precision. This gap between the SLD- and RLD-based bounds becomes especially pronounced in certain physical regimes, emphasizing the relevance of choosing the appropriate formalism in noncommutative estimation scenarios. From a physical perspective, the improved estimation performance stems from the fact that temperature directly influences thermal fluctuations in the mechanical mode, while the effective detuning modulates the strength of the optomechanical interaction, shaping the information exchange between optical and mechanical subsystems. We showed that using higher optical power, greater cavity decay rate, increased photon and phonon numbers, and reduced coupling strength improves the precision of both parameter estimates. These physical mechanisms affect the available quantum resources in the system—such as squeezing, entanglement, and state purity—which in turn determine the ultimate sensitivity achievable. Furthermore, our analysis shows that heterodyne detection can, under certain conditions, approach the ultimate precision set by the QFIM. Thus, a measurement strategy based on heterodyne detection could enable the efficient estimation of temperature, achieving performance close to that of ideal quantum-limited strategies.