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.