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.

Figure 1

Scaling laws for the quantum gain. (a) The quantum metrological gain ${\xi }^{-2}$ optimized over the preparation time $\chi t$ for linear ($L$), nonlinear ($NL$), quadratic ($Q$), and MAI measurement strategies as a function of $N$ compared to the standard quantum limit and the Heisenberg limit (HL). Solid lines represent the analytical scaling laws in the limit of large $N$ including finite size corrections. The leading term is indicated in the plot. (b) Modification of the scaling laws for the MAI method in the presence of ballistic noise with fluctuations that scale with $\gamma$ for $\epsilon =0.05$ (see text).

Figure 2

Discontinuous scaling law in the presence of ballistic dephasing. Quantum metrological gain of the MAI method ${\xi }_{\mathrm{MAI},\mathrm{bal}}^{-2}$ with $\tau =-t$, $\epsilon =0.05$, $\gamma =0.65$ as a function of $\alpha$: Larger values of $1-\alpha$ correspond to longer preparation times as we set $\chi t=\sigma N^{-\alpha }$. The normalized quantum gain ${\xi }_{\mathrm{MAI},\mathrm{bal}}^{-2}/(N^{1-\gamma }/4\epsilon )$ (top) and ${\xi }_{\mathrm{MAI},\mathrm{bal}}^{-2}/N^{2-2\alpha }$ (bottom) illustrates how the scaling in the thermodynamic limit, Eqs. (7) and (8) for short and longer times, respectively, is approached as $N$ increases. At ${\alpha }_c$ (6) the transition becomes discontinuous in the thermodynamic limit.