Spin squeezing in a many-body system is a witness for entanglement and can enable measurement sensitivities beyond those achievable by only classical correlations. Here, working with ultracold ${}^7\mathrm{Li}$ atoms in an optical lattice, we demonstrate spin squeezing using short-range contact interactions in both one and three dimensions. In 1D, spin squeezing is shown to be insensitive to density fluctuations caused by holes. In 3D, however, we observe that holes strongly modify the squeezing dynamics, hindering the recently predicted emergence of scalable squeezing in 3D $XXZ$ spin systems. By developing a new theoretical model to account for spin-density coupling, we obtain strong quantitative agreement with the observed squeezing dynamics, resulting in $\approx 2\mathrm{dB}$ at 7% hole fraction. Our observations highlight the importance of understanding spin-density coupling in the dynamics of many interacting spins and lay the groundwork for improved spin squeezing in systems with only short-range interactions.

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• Cold gases in optical lattices
• Entanglement in quantum gases
• Entanglement production
• Quantum metrology
• Quantum simulation
• Ultracold gases
• Optical lattices & traps