Scaling of the interaction in BECs at large scattering lengths

We have studied the scaling of the interaction in Bose-Einstein condensates of ultracold alkali-metal gases for large scattering lengths and momenta where corrections to the mean field approximation become important. We find that the effective interaction in the metastable, open channel, gaseous phase scales well with the scattering length in the range analyzed. Based on this we show that for increasing scattering lengths, or equivalently increasing densities, the system becomes less correlated, and that at large scattering lengths Bragg scattering experiments can directly measure the effective two-body potential in momentum space. This work is motivated by the recent Bragg-scattering measurements in ⁸⁵Rb by Papp et al. [Phys. Rev. Lett. 101, 135301 (2008)], where the results in the line shifts show clear deviations from the simple contact interaction. We show that those results are well described by a soft spheres potential with parameters chosen to scale in scattering length units. So far the resolution in the experiments does not reveal details on the frequency dependence in the dynamic structure function S(k,ω) and we show that the Feynman spectrum determines the measured line shifts. We also construct the effective atom-atom interaction from two coupled channels, open and closed, assuming that the Feshbach resonance dominates the closed channel. The resonance energy and the scattering length a of the system are tunable by magnetic fields. We derive the T-matrix of such a system and use renormalization to calculate the bound state energy as a function of the magnetic field and make comparison with available experiments. The s-wave phase shifts determine the local, effective open-channel interaction, but if no scaling is used in the cut-off parameters of the renormalization the phase shift resembles more and more the ones obtained from the contact interaction with increasing scattering length. This leads to clear deviations from the measured line shift experiments.