Estimation precision of parameter associated with Unruh-like effect

We study the dynamics of the quantum Fisher information (QFI) of acceleration, in the ultra-relativistic limit, for a two-level atom with the circular motion coupled to a massless scalar field without and with a reflecting boundary in the open quantum systems. The QFI with the larger acceleration reaches the stable value faster than that of the smaller acceleration. Without a boundary, there exists a peak value of QFI with a certain time. The QFI is the periodic function of the initial state parameter in a very short time, and it firstly takes the maximum in the ground state of the atom. The periodicity 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 QFI with the larger temperature arrives the maximum value faster than that of the smaller temperature. Without a boundary, the QFI of temperature is similar to that of acceleration. With a boundary, the QFI firstly takes the maximum in the ground state of the atom, which is different from the behavior of acceleration. The periodicity with a boundary lasts a longer time than unbounded case.