Subwavelength vacuum lattices and atom-atom interactions in two-dimensional photonic crystals

Quantum simulation with cold atoms in optical lattices is an attractive avenue for explorations of quantum many-body physics. A principal challenge in the field is to increase the energy and length scales in current set-ups, thereby reducing temperature and coherence-time requirements. Here, we present a new paradigm for high-density, two-dimensional optical lattices in photonic crystal waveguides. Specially engineered two-dimensional photonic crystals provide a practical platform to trap atoms and engineer their interactions in ways that surpass the limitations of current technologies and enable investigations of novel quantum many-body matter. Our schemes remove the constraint on the lattice constant set by the free-space optical wavelength in favour of deeply sub-wavelength atomic arrays. We further describe possibilities for atom-atom interactions mediated by photons in two-dimensional photonic crystal waveguides with energy scales several orders of magnitude larger than for exchange interactions in free-space lattices and with the capability to engineer strongly long-range interactions.