Estimation precision of parameter associated with Unruh-like effect

We study the quantum Fisher information (QFI) of acceleration, in the open quantum systems, for a two-level atom with the circular motion coupled to a massless scalar field in the Minkowski vacuum without and with a reflecting boundary in the ultra-relativistic limit. As we amplify $a$, the saturation time decreases for $\theta\neq\pi$, but first increases and then decreases for $\theta=\pi$. Without a boundary, there exists a peak value of QFI with a certain time. The QFI varies with the initial state parameter $\theta$, and firstly takes peak value in the ground state of the atom. The variation of QFI with respect to $\theta$ gradually fades away with the evolution of time. With a boundary, the detection range of acceleration has been expanded. The QFI firstly takes the maximum in the excited state of the atom. In addition, we study the QFI of temperature for a static atom immersed in a thermal bath without and with a boundary. The relation between the saturation time and $T$ is similar to $a$. Without a boundary, the QFI of temperature is similar to that of acceleration. With a boundary, the QFI firstly takes peak value in the ground state of the atom, which is different from the behavior of acceleration. The results provide references for the detection of Unruh-like effect.