Approximate expression for the dynamic structure factor in the Lieb-Liniger model

that the dynamic structure factor (DSF) of the 1D Bose gas demonstrates power-law behaviour along the limiting dispersion curve of the collective modes and calculated the corresponding exponents exactly. Combining these recent results with a previously obtained strong-coupling expansion we present an interpolation formula for the DSF of the 1D Bose gas. The obtained expression is further consistent with exact low energy exponents from Luttinger liquid theory and shows nice agreement with recent numerical results. Cigar-shaped traps with cold alkali atoms have recently been used to obtain a quasi-1D quantum degenerate Bose gas, where atomic motion in the transverse dimensions is confined to zero-point quantum oscillations, in weak and strong interaction regimes [1, 2]. Theoretically, we may describe the system as a one-dimensional rarefied gas where interactions of bosonic atoms can be described well by effective δ-function interactions [3]. Thus the Lieb-Liniger model [4, 5] is applicable. Being exactly solvable in the uniform case, the model, however, does not admit complete analytic solutions for the correlation functions. Up to now, this has been an outstanding problem in 1D physics [6, 7]. Here, we propose an approximate formula for the DSF of the Lieb-Liniger gas that is consistent with known results in accessible limits and power laws. Dynamical density-density correlations, which can be measured by the two-photon Bragg scattering [8, 9], are described by the dynamic structure factor (DSF) [10] S(q, ω) = L ∫ dtd