Optimal quantum sensing of the nonlinear bosonic interactions using Fock states

Nonlinear processes with individual quanta beyond bilinear interactions are essential for quantum technology with bosonic systems. Diverse coherent splitting and merging of quanta in them already manifest in the estimation of their nonlinear coupling from observed statistics. We derive nontrivial but optimal strategies for sensing the experimentally feasible trilinear interactions, served here as basic gates and examples of multimode nonlinear interactions, using nonclassical particlelike Fock states as a probe and feasible measurement strategies. Remarkably, the optimal probing of nonlinear coupling reaches estimation errors scaled down with N-3/2 for overall N of quanta in specific and high-quality Fock states in all interacting modes. It can reveal unexplored aspects of nonlinear dynamics relevant to using such nonlinear processes in bosonic experiments with trapped ions and superconducting circuits and opens further developments of quantum technology with them.