Ideal Optical Flux Lattices

The realization of fractional quantum Hall (FQH) states in cold atomic gases is a long-standing goal in quantum simulation. Established approaches, including rapidly rotating gases and tight-binding lattices, are often hampered by low interaction energies and small many-body energy gaps. While optical flux lattices (OFLs) can achieve higher effective magnetic flux densities, standard two-state configurations generate highly non-uniform fields, and extensions to multi-state systems introduce significant experimental complexity. Here, we present a new paradigm for engineering robust FQH phases in OFLs using only two internal atomic states. We show that the introduction of an additional scalar potential provides a generic mechanism for creating Chern bands that are simultaneously essentially flat and "ideal." Drawing on concepts from moir\'e materials, these desirable properties arise by tuning lattice parameters to certain $N$-flat manifolds $(N=1,2,\dots)$, where the $1$-flat manifold shares its origin with certain "magic-angle" conditions. We provide a variety of examples of how to design optical flux lattices, including dark-state OFLs, to achieve these goals. This method allows for precise tuning of band flatness and stabilizes both Abelian and non-Abelian FQH phases. Our scheme is compatible with existing experimental capabilities using vector polarizability, opening practical routes to exploring strongly correlated topological physics with cold atoms.