Spin-spin interaction and spin squeezing in an optical lattice

We show that by displacing two optical lattices with respect to each other, we may produce interactions similar to the ones describing ferromagnetism and antiferromagnetism in condensed matter physics. We also show that particularly simple choices of the interaction lead to spin squeezing, which may be used to improve the sensitivity of atomic clocks. Spin squeezing is generated even with partially, and randomly, filled lattices, and our proposal may be implemented with current technology.