From Independent to Joint: Enhancing Quantum Phase and Correlation Factor Estimation by Squeezed Reservoir Engineering

High-precision quantum parameter estimation is fundamental to the advancement of quantum metrology. Although reservoir engineering provides a powerful approach to improve estimation by tailoring system-environment interactions, the role of the squeezing phase and correlations arising from the sequential utilization of the same squeezed reservoir remains inadequately explored. In this work, we employ a correlated squeezed-thermal reservoir to enhance the precision of estimating the phase parameter $\phi$ and the correlation factor $\mu$, both individually and simultaneously. We show that the squeezing phase $\Phi$ is crucial for achieving quantum-enhanced precision, with optimal phase-matching conditions that depend strongly on $\mu$. Specifically, we derive the near-optimal phase-matching relations aimed at maximizing the quantum Fisher information (QFI) for both $\phi$ and $\mu$, as well as minimizing the total variance $\Delta_{\rm sim}$ in joint estimation. Furthermore, we show that the joint estimation variance is dominated by $F_{\phi}$, which motivates our search for the phase-matching conditions that minimize $\Delta_{\text{sim}}$. Through the ratio $R$ of variances, we demonstrate that joint estimation conserves quantum resources and maintains high precision when the squeezing phase is optimized for $F_{\phi}$, despite the inherent incompatibility of the parameters. These findings provide practical insights into reservoir engineering strategies for high-precision quantum sensing and information processing.