Quantum phases of a Feshbach-resonant atomic Bose gas in one dimension
  We study an atomic Bose gas with an s-wave Feshbach resonance in a one-dimensional optical lattice, with the densities of atoms and molecules incommensurate with the lattice. At zero temperature, most of the parameter region is occupied by a phase in which the superfluid fluctuations of atoms and molecules are the predominant ones, due to the phase fluctuations of atoms and molecules being locked by a Josephson coupling between them. When the density difference between atoms and molecules is commensurate with the lattice, two additional phases may exist: the two component Luttinger liquid where both the atomic and molecular sectors are gapless, and the inter-channel charge density wave where the relative density fluctuations between atoms and molecules are frozen at low energy. 
Yu-Wen Lee and Yu-Li Lee