Optimal control for multi-parameter quantum estimation with time-dependent Hamiltonians

We investigate simultaneous estimation of multi-parameter with time-dependent Hamiltonians. We analytically obtain the maximum quantum Fisher information matrix for two-parameter encoded in time-dependent three-level systems. The optimal coherent control scheme is proposed to increase the simultaneous multi-parameter estimation precisions. In an example of a spin-1 particle in a uniformly rotating magnetic field, the optimal coherent Hamiltonians for different parameters can be chosen to be completely same. However, in general, the optimal coherent Hamiltonians for different parameters are incompatible. In this situation, we suggest a variance method to obtain the optimal coherent Hamiltonian for simultaneously estimating multiple parameters, and obtain the optimal simultaneous estimation precision of two-parameter in a three-level Landau-Zener Hamiltonian.