Scaling laws for the sensitivity enhancement of non-Gaussian spin states

We identify the large-$N$ scaling of the metrological quantum gain offered by over-squeezed spin states that are accessible by one-axis-twisting, as a function of the preparation time. We further determine how the scaling is modified by relevant decoherence processes and predict a discontinuous change of the quantum gain at a critical preparation time that depends on the noise. Our analytical results provide recipes for optimal and feasible implementations of quantum enhancements with non-Gaussian spin states in existing experiments, well beyond the reach of spin squeezing.