Squeezing Towards the Heisenberg Limit with Locally Interacting Spins

We propose a robust approach to spin squeezing with local interactions that approaches the Heisenberg limit of phase sensitivity. To generate the requisite entanglement, we generalize the paradigmatic two-axis countertwisting Hamiltonian -- akin to squeezing by parametric amplification -- to systems with power-law interactions, incorporating a Heisenberg coupling that aids in spreading correlations and protects the collective spin coherence. The resulting time to approach the Heisenberg limit scales sublinearly with particle number in 2D dipolar and 3D van der Waals interacting systems. Our protocol is robust to disorder and density fluctuations, and can be implemented in near-term experiments with molecules, Rydberg atoms, and solid-state spins.