  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 non-trivial, but
optimal strategies for sensing the basic and experimentally available trilinear
interactions using non-classical particle-like Fock states as a probe and
feasible measurement strategies. Remarkably, the optimal probing of nonlinear
coupling reaches estimation errors scaled down with $N^{-1/3}$ for overall $N$
of quanta in specific but available 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.
