We propose and analyse a scheme to cool atoms in an optical lattice to ultra-low temperatures within a Bloch band, and away from commensurate filling. The protocol is inspired by ideas from dark state laser cooling, but replaces electronic states with motional levels, and spontaneous emission of photons by emission of phonons into a Bose-Einstein condensate, in which the lattice is immersed. In our model, achievable temperatures correspond to a small fraction of the Bloch band width, and are much lower than the reservoir temperature.
Griessner, A., Daley, A. J., Clark, S. R., Jaksch, D., & Zoller, P. (2006). Dark state cooling of atoms by superfluid immersion. Physical Review Letters, 97(22), . https://doi.org/10.1103/PhysRevLett.97.220403