  Tripartite interactions play a fundamental role in the quantum information
processing and quantum technology. However, it is generally difficult to
realize strong tripartite coupling. We investigate the estimation of a
tripartite coupling strength in a hybrid setup composed of a single
nitrogen-vacancy (NV) center and a micromagnet. A time-independent parametric
drive can be utilized to increase the estimation precision of the tripartite
coupling strength. By calculating the quantum Fisher information (QFI), we can
obtain the optimal estimation precision by measuring the eigenstate of the
tripartite system. At the critical position, the QFI is divergent due to that
the preparation time of the eigenstate is divergent. When the system is
subjected to a dissipation, the QFI near the critical point of the
driven-dissipation phase transition is analytically obtained. The direct
intensity measurement is the optimal measurement near the dissipation phase
transition point. In addition, we quantify the robustness of an imperfect
measurement operator by the measurement noise susceptibility based on the error
propagation formula. We find that the direct intensity measurement is enough
robust against small measurement disturbance from a coherent drive. But it can
be disturbed by the nonlinear anti-harmonic measurement noise, especially near
the critical point.
