Quantum metrology with precision reaching beyond-$1/N$ scaling through $N$-probe entanglement generating interactions

Nonlinear interactions are recognized as potential resources for quantum metrology, facilitating parameter estimation precisions that scale as the exponential Heisenberg limit of $2^{-N}$. We explore such nonlinearity and propose an associated quantum measurement scenario based on the nonlinear interaction of $N$-probe entanglement generating form. This scenario provides an enhanced precision scaling of $D^{-N}/(N-1)!$ with $D > 2$ a tunable parameter. In addition, it can be readily implemented in a variety of experimental platforms and applied to measurements of a wide range of quantities, including local gravitational acceleration $g$, magnetic field, and its higher-order gradients.