Parameter estimation in $n$-dimensional massless scalar field

Quantum Fisher information(QFI) associated with local metrology has been used to parameter estimation in open quantum systems. In this work, we calculated the QFI for a moving Unruh-DeWitt detector coupled with massless scalar fields in $n$-dimensional spacetime, and analyzed the behavior of QFI with various parameters, such as the dimension of spacetime, evolution time, and Unruh temperature. We discovered that the QFI of state parameter decreases monotonically from $1$ to $0$ over time. Additionally, we noted that the QFI for small evolution times is several orders of magnitude higher than the QFI for long evolution times. We also found that the value of QFI decreases at first and then stabilizes as the Unruh temperature increases. It was observed that the QFI depends on initial state parameter $\theta$, and $F_{\theta}$ is the maximum for $\theta=0$ or $\theta=\pi$, $F_{\phi}$ is the maximum for $\theta=\pi/2$. We also obtain that the maximum value of QFI for state parameters varies for different spacetime dimensions with the same evolution time.