Bose Einstein condensation: Its role in the excitations of liquid helium and in trapped Bose gases

The role of Bose-Einstein condensation (BEC) in determining the properties of Bose systems at ultracold temperatures is investigated. First, we present a model of the dynamic structure factor S(Q,o) of liquid 4He as observed in inelastic neutron scattering measurements beyond the roton (Q ≳ 2.0 A-1). We separate the dynamic susceptibility into chi = chiS + chi'R. chi S involves states in the condensate and chi' R states above the condensate only. We find that the weight of chiS scales with the condensate fraction n0(T) and vanishes at Tlambda. chi' R is broad and largely temperature independent and a low energy intensity broadening arises from the thermal broadening of the phonon-roton (p-r) modes. Secondly, we investigate ultracold Bose gases with repulsive and attractive interactions confined in a spherical harmonic trap over a broad range of densities using model potentials and variational Monte Carlo (VMC) at T = 0 K. In the case of repulsive interactions, the Bosons are represented by hard spheres (HS)s interacting by a HS potential. We change the densities of the Bosons by increasing the s-wave scattering length a. We find that the VMC total and VMC condensate density distributions are similar in shape, they are flat nearly at the higher densities. Further the Thomas-Fermi approximation becomes invalid and the condensate is substantially depleted at the higher densities. In the case of attractive interactions, we model the interactions by a hard core square well (HCSW). We change the densities of these systems by keeping the hard core diameter, a c, fixed and increasing the potential depth V 0 or by increasing both of them simultaneously while keeping a fixed. We find that a Bose gas with attractive interactions undergoes a first order phase transition from the gas to the liquid state at a value of N|a| ≈ 0.574 in agreement with the value predicted by Gross-Pitaevskii (GP) theory. The condensate depletion is mainly driven by the HC diameter in the presence of an attractive well. The scattering length is a useful representation of the HCSW at the low densities, but at the higher densities the HCSW is not represented well anymore by the scattering length.