Figure 2

(a) Samples of 100 random magnetic fields with the maximum magnitude β _m = 5 × 10⁶ χ (i.e., B _m = 4.83 mG). Color scales the optimal spin squeezing parameter during a driven-OAT process with a given magnetic field. (b) Typical mean spin direction and error bars averaged over 100 samples with β _m = 5 × 10⁶ χ. The purple diamonds denote the polar angle θ. (c) Averaged evolution of spin squeezing parameter in stray magnetic fields with β _m = 5 × 10⁶ χ (labeled by C, up triangles), 5 × 10² χ (M, down triangles), and 0.1χ (H, right triangles). The driven-OAT cycle time T _c = 1 × 10⁻⁴ χ ⁻¹. The solid grey curve represents the dynamics of TAT. (d) Optimal averaged spin squeezing parameter for different β _m . The results shown in panel (c) are marked correspondingly with the same type triangles. Three regions are identified as Heisenberg, crossover, and classical, denoting the quantum Heisenberg limit, the crossover, and the classical limit, respectively. The vertical dashed lines with β _H and β _C mark the boundaries between the three regions. The blue and red horizontal dashed lines are the classical (\({\xi }_{m}^{2}\approx 0.6\)) and Heisenberg limit (\({\xi }_{m}^{2}\approx \mathrm{1/}N\)), respectively. The green dashed line depicts the analytic results of Eq. (11) in the crossover region.