A dark state of Chern bands: Designing flat bands with higher Chern number

We introduce a scheme by which flat bands with higher Chern number \vert C\vert>1 | C | > 1 can be designed in ultracold gases through a coherent manipulation of Bloch bands. Inspired by quantum-optics methods, our approach consists in creating a ``dark Bloch band" by coupling a set of source bands through resonant processes. Considering a \Lambda Λ system of three bands, the Chern number of the dark band is found to follow a simple sum rule in terms of the Chern numbers of the source bands: C_D\!=\!C_1+C_2-C_3 C D = C 1 + C 2 − C 3 . Altogether, our dark-state scheme realizes a nearly flat Bloch band with predictable and tunable Chern number C_D C D . We illustrate our method based on a \Lambda Λ system, formed of the bands of the Harper-Hofstadter model, which leads to a nearly flat Chern band with C_D\!=\!2 C D = 2 . We explore a realistic sequence to load atoms into the dark Chern band, as well as a probing scheme based on Hall drift measurements. Dark Chern bands offer a practical platform where exotic fractional quantum Hall states could be realized in ultracold gases.