Dynamic response of one-dimensional bosons in a trap

We calculate the dynamic structure factor S(q,omega) of a one-dimensional (1D) interacting Bose gas confined in a harmonic trap. The effective interaction depends on the strength of the confinement enforcing the (1D) motion of atoms; interaction may be further enhanced by superimposing an optical lattice on the trap potential. In the compressible state, we find that the smooth variation in the gas density around the trap center leads to softening of the singular behavior of S(q,omega) at the first Lieb excitation mode compared to the behavior predicted for homogeneous 1D systems. Nevertheless, the density-averaged response S(q,omega) remains a nonanalytic function of q and omega at the first Lieb excitation mode in the limit of weak trap confinement. The exponent of the power-law nonanalyticity is modified due to the inhomogeneity in a universal way and thus bears unambiguously the information about the (homogeneous) Lieb-Liniger model. A strong optical lattice causes formation of Mott phases. Deep in the Mott regime, we predict a semicircular peak in S(q,omega) centered at the on-site repulsion energy, omega=U. Similar peaks of smaller amplitudes exist at multiples of U as well. We explain the suppression of the dynamic response with entering into the Mott regime, observed recently by Cleacutement [Phys. Rev. Lett. 102, 155301 (2009)], based on an f-sum rule for the Bose-Hubbard model.