  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.
