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 Cramér-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.

We calculate a measure of superposition macroscopicity M for a superposition of screening current states in a superconducting flux qubit (SFQ), by relating M to the action of an instanton trajectory connecting the potential wells of the flux qubit. When a magnetized Bose-Einstein condensed (BEC) gas containing NB ∼ O(10 6 ) atoms is brought into a O(1) µm proximity of the flux qubit in an experimentally realistic geometry, we demonstrate the appearance of a two-to five-fold amplification of M over the bare value without the BEC, by calculating the instantion trajectory action from the microscopically derived effective flux Lagrangian of a hybrid quantum system composed of the flux qubit and a spin-F atomic Bose gas. Exploiting the connection between M and the maximal metrological usefulness of a multimode superposition state, we show that amplification of M in the ground state of the hybrid system is equivalent to a decrease in the quantum Cramér-Rao bound for estimation of an externally applied flux. Our result therefore demonstrates the increased usefulness of the BEC-SFQ hybrid system as a sensor of ultraweak magnetic fields below the standard quantum limit.