Dynamic response of 1D bosons in a trap

TL;DR

This paper investigates the dynamic structure factor of a 1D interacting Bose gas in a trap, revealing how inhomogeneity and optical lattices influence its spectral features and response behavior.

Contribution

It provides a detailed analysis of the effects of trap inhomogeneity and optical lattices on the dynamic response of 1D Bose gases, including modifications to singularities and the emergence of Mott phases.

Findings

Softening of singular behavior at Lieb-1 mode due to inhomogeneity

Universal modification of power-law exponents in inhomogeneous systems

Identification of semi-circular peaks in S(q,ω) in the Mott regime

Abstract

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 of the gas density around the trap center leads to softening of the singular behavior of S(q,omega) at Lieb-1 mode compared to the behavior predicted for homogeneous 1D systems. Nevertheless, the density-averaged response remains a non-analytic function of q and omega at Lieb-1 mode in the limit of weak trap confinement. The exponent of the power-law non-analyticity 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 semi-circular 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 D. Clement et al., Phys. Rev. Lett. v. 102, p. 155301 (2009), based on an f-sum rule for the Bose-Hubbard model.