Enhanced multiparameter quantum estimation in cavity magnomechanics via a coherent feedback loop

Abstract

Multiparameter quantum metrology plays a fundamental role in uncovering and exploiting the distinctive features of quantum systems. In this work, we propose an effective and experimentally feasible scheme to significantly enhance the simultaneous quantum estimation of the photon magnon and magnon mechanical coupling strengths in a hybrid cavity magnon mechanical platform. Our approach relies on the assistance of a coherent feedback loop combined with the injection of a coherent driving field. We show that an appropriate tuning of the system and feedback parameters leads to a substantial reduction of the estimation errors associated with both coupling strengths. To quantify the metrological performance of the proposed scheme, we employ the quantum Cramer Rao bound (QCRB) as a fundamental benchmark for multiparameter estimation. We explicitly compute and compare the QCRBs derived from the symmetric logarithmic derivative (SLD) and the right logarithmic derivative (RLD) formalisms. Our results demonstrate that the RLD based QCRB is systematically lower than the SLD based bound, indicating superior estimation precision in the considered noncommutative estimation scenario. We further analyze the performance of heterodyne detection and show that, in suitable parameter regimes, the corresponding classical estimation precision closely approaches the ultimate quantum limit predicted by our scheme. Finally, we discuss the experimental feasibility of the proposed setup within currently available cavity magnon mechanical platforms. Owing to its general character, the framework developed here can be readily extended to the high precision estimation of other physical parameters in hybrid quantum systems.

Figures (9)

• Figure 1: Schematic of a hybrid cavity–magnonics system with coherent feedback. A microwave driving field (MWD) is injected into a controlled beam splitter (CBS), characterized by reflection and transmission coefficients $r$ and $\varepsilon$, respectively. The transmitted component propagates through two highly reflective mirrors (HRM1 and HRM2) before entering a single-port microwave cavity. Inside the cavity, a single-mode electromagnetic field coherently couples to a magnon mode, realized by a YIG sphere positioned near the maximum of the cavity magnetic field and subjected to a uniform bias magnetic field. The cavity output field is directed to a detection setup, from which the first-order moments (quadratures) and the covariance matrix are extracted, enabling a full reconstruction of the Gaussian state of the system. This reconstructed state serves as the prepared resource for metrological applications. Based on the measurement outcomes of the output field, a statistical estimator is constructed, allowing the inference of the parameters of interest and the assessment of the estimation precision.
• Figure 2: Ratio between the RLD-QCRB and SLD-QCRB as a function of the normalized cavity detuning $\Delta_a$. Panel (a) illustrates the dependence of this ratio on the environmental temperature $T$, while panel (b) presents its variation with the cavity dissipation rate $\kappa_a$. The pump power is fixed at $P=8.9 \;\text{mW}$, and the feedback parameters are set to $r=0.1$ and $\theta=\pi$. In panel (b), the temperature is kept constant at $T = 10 \,\text{mK}$.
• Figure 3: (a) Density plot of the $\mathrm{C}^{MI}$ as a function of the feedback parameters: the reflection coefficient $r$ and the phase shift $\theta$. (b) $\mathrm{C}^{MI}$ as a function of the magnon–cavity coupling strength $g_{ma}$ for different feedback configurations. (c) Density plot of the $\mathrm{C}^{MI}$ as a function of the detuning ratio $\Delta_a/\Delta_m$ and temperature $T$ (in K), with $r=0.5$ and $\theta=\pi$. In panels (a) and (b), the temperature is fixed at $T=10$ mK. All other system parameters are the same as in Fig. \ref{['rati']}.
• Figure 4: The most informative-QCRB $\mathrm{C}^{MI}$ as a function of the magnon–cavity coupling strength $g_{ma}$: (a) for different values of the cavity decay rate $\kappa_a$, and (b) for different input powers $P$, with $T = 10 \,\text{mK}$. In panel (b), we set $\kappa_a/2\pi=\kappa_m/2\pi= 0.4 \,\text{MHz}$, while all other parameters are the same as in Fig. \ref{['rati']}.
• Figure 5: Plot of the most informative-QCRB $\mathrm{C}^{MI}$ as a function of the normalized cavity detuning, $\Delta_a$. Panel (a) shows its variation for different values of the environmental temperature $T$. Panel (b) shows its variation for different values of the mechanical mode decay rate $\kappa_m$. The other system parameters are identical to those used in Fig. \ref{['rati']}.