  We derive families of optimal and near-optimal probe states for quantum
estimation of the coupling constants of a general two-mode number-conserving
bosonic Hamiltonian describing one-body and two-body dynamics. We find that the
optimal states for estimating the dephasing of the modes, the self-interaction
strength, and the contact interaction strength are related to the NOON states,
whereas the optimal states for estimation of the intermode single particle
tunneling amplitude are superpositions of antipodal SU(2) coherent states. For
estimation of the amplitude of pair tunneling and the amplitude of
density-dependent single particle tunneling processes, respectively, we
introduce classes of variational superposition probe states that provide near
perfect saturation of the corresponding quantum Cram\'{e}r-Rao bounds. We show
that the ground state of the pair tunneling term in the Hamiltonian has a high
fidelity with the optimal states for estimation of a single particle tunneling
amplitude, suggesting that high-performance probes for tunneling amplitude
estimation may be produced by tuning the two-mode system through a quantum
phase transition.
