  In an open quantum system, we study the evolution of a two-level atom as a
detector which interacts with given environments. For a uniformly accelerated
two-level atom coupled to a massless scalar field in the Minkowski vacuum, when
it evolves for a certain time, we find that there exists a peak value for the
quantum Fisher information (QFI) of acceleration, which indicates that the
optimal precision of estimation is achieved when choosing an appropriate
acceleration $a$. QFI has different behaviors for different initial state
parameters $\theta$ of the atom, displaying periodicity. However, the
periodicity fades away with the evolution of time, which means that the initial
state cannot affect the later stable quantum state. Furthermore, adding a
boundary, we observe that the peak value of QFI increases when the atom is
close to the boundary, which shows that QFI is protected by the boundary. Here,
QFI fluctuates, and there may exist two peak values with a certain moment,
which expands the detection range of the acceleration. Therefore, we can
enhance the estimation precision of acceleration by choosing an appropriate
position and acceleration $a$. The periodicity of QFI with respect to the
initial state parameter $\theta$ lasts a longer time than the previous
unbounded case, which indicates that the initial state is protected by the
boundary. Finally, for a thermal bath with a boundary, QFI of temperature has
no more than one peak value with a certain moment, which is different from QFI
of acceleration with a boundary. The periodicity also lasts a longer time than
unbounded case, which shows the initial quantum state of the atom is protected
by the boundary for two cases.
