Quantum estimation of tripartite coupling in Spin-Magnon-Mechanical Hybrid Systems

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.