Synthesizing complex spin networks with spin-motion coupled neutral atoms in photonic crystals

We develop a toolbox for realizing ``fully programmable'' d-dimensional pairwise interacting lattice spin systems with spin-motion coupled neutral atoms in the vicinity of 1D photonic crystal waveguides. The enabling platform thereby allows to synthesize a wide range of strongly interacting quantum materials by way of vacuum-engineered interatomic kinetic interactions. We demonstrate the versatility of our assembly language approach towards arbitrary SU(2)-lattice spin models with explicit constructions of familiar Hamiltonians for perfect state transfer in 1D spin chains, lattice gauge theories, and topologicallyquantum spin liquids. We further construct Dzyaloshinski-Moriya interaction for the realization ofspin liquids and long-range random quantum magnets with spin-glass phase.