Interaction effects on atomic laboratory trapped Bose-Einstein condensates

We discuss the effect of inter-atoms interactions on the condensation temperature Tc of an atomic laboratory trapped Bose-Einstein condensate. We show that, in the mean-field Hartree-Fock and semiclassical approximations, interactions produce a shift ΔTc/Tc0 ≈ b1(a/λTc) + b2(a/λTc)2 + ψ[a / λTc] with a the s-wave scattering length, λT the thermal wavelength and ψ[a / λTc] a non-analytic function such that ψ[0] = ψ′[0] = ψ′′[0] = 0 and |ψ′′′[0]| = ∞. Therefore, with no more assumptions than Hartree-Fock and semiclassical approximations, interaction effecs are perturbative to second order in a / λTc and the expected non-perturbativity of physical quantities at critical temperature appears only to third order. We compare this finding with different results by other authors, which are based on more than the Hartree-Fock and semiclassical approximations. Moreover, we obtain an analytical estimation for b2 ≃ 18.8 which improves a previous numerical result. We also discuss how the discrepancy between b2 and the empirical value of b2 = 46 ± 5 may be explained with no need to resort to beyond-mean field effects