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 $q\to 0$ the lowest mode corresponds to a gapless phonon, ${\omega }_1^2=v_1^2{q}^2$, whereas the second lowest mode corresponds to a gapped optical phonon, ${\omega }_2^2={\mathrm{\Delta }}^2\pm v_2^2{q}^2$. Determination of the velocities $v_1,v_2$, the gap $\mathrm{\Delta }$, and the possible sign change in ${\omega }_2$ upon the change of lattice depth can give decisive measures to the transitions across various supersolid and solid states. The power law $v_1\sim {\left(f_s\right)}^{1/2}$ with $f_s$ the superfluid fraction is identified in the present system at the tight-binding regime.