Quantum spin ice represents a paradigmatic example on how the physics of
frustrated magnets is related to gauge theories. In the present work we address
the problem of approximately realizing quantum spin ice in two dimensions with
cold atoms in optical lattices. The relevant interactions are obtained by
weakly admixing van der Waals interactions between laser admixed Rydberg states
to the atomic ground state atoms, exploiting the strong angular dependence of
interactions between Rydberg p-states together with the possibility of
designing step-like potentials. This allows us to implement Abelian gauge
theories in a series of geometries, which could be demonstrated within state of
the art atomic Rydberg experiments. We numerically analyze the family of
resulting microscopic Hamiltonians and find that they exhibit both classical
and quantum order by disorder, the latter yielding a quantum plaquette valence
bond solid. We also present strategies to implement Abelian gauge theories
using both s- and p-Rydberg states in exotic geometries, e.g. on a 4-8 lattice.Comment: 26 pages, 16 figure