Optimal asymptotic precision bounds for nonlinear quantum metrology under collective dephasing

Interactions among sensors can provide, in addition to entanglement, an important resource for boosting the precision in quantum estimation protocols. Dephasing noise, however, remains a leading source of decoherence in state-of-the-art quantum sensing platforms. We analyze the impact of classical {\em collective dephasing with arbitrary temporal correlations} on the performance of generalized Ramsey interferometry protocols with quadratic encoding of a target frequency parameter. The optimal asymptotic precision bounds are derived for both product coherent spin states and for a class of experimentally relevant entangled spin-squeezed states of N qubit sensors. While, as in linear metrology, entanglement offers no advantage if the noise is Markovian, a precision scaling of N^-1 is reachable with classical input states in the quadratic setting, which is improved to N^-5/4 when temporal correlations are present and the Zeno regime is accessible. The use of nonclassical spin-squeezed states and a nonlinear readout further allows for an N^-3/2 precision scaling, which we prove is asymptotically optimal. We also show how to counter {\em noise-induced bias} by introducing a simple ratio estimator which relies on detecting two suitable system observables, and show that it remains asymptotically unbiased in the presence of dephasing, without detriment to the achievable precision.