Excitations of one-dimensional supersolids with optical lattices

Based on mean-field Gross-Pitaevskii and Bogoliubov-de Gennes approaches, we investigate excitations of a one-dimensional soft-core interacting ultracold Bose gas under the effect of an optical lattice. It is found that no matter how deep the lattice is, at q→0 the lowest mode corresponds to a gapless phonon, ω12=v12q2, whereas the second lowest mode corresponds to a gapped optical phonon, ω22=Δ2±v22q2. Determination of the velocities v1,v2, the gap Δ, and the possible sign change in ω2 upon the change of lattice depth can give decisive measures to the transitions across various supersolid and solid states. The power law v1∼(fs)1/2 with fs the superfluid fraction is identified in the present system at the tight-binding regime.