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In the Bose-Hubbard model of cold atoms in an optical lattice, we consider only the short-range interaction or on-site interaction. Is it possible to realize long-range interaction similar to Coulomb interaction in the cold atom experiment?

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A way to get long-range interactions in cold atoms is to use polar molecules, which is a rapidly growing field in the cold atoms community.

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Check out the introduction of e.g. this paper for an example of long range Van der Waals forces, which are encountered in cold atom experiments. These forces occur between highly excited atoms called Rydberg atoms and behave as $n^{11}/r^6$, where $n$ is the principal quantum number, which is extraordinary high for these atoms.

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