Interaction effects on atomic laboratory trapped Bose-Einstein condensates

We discuss the effect of inter-atoms interactions on the condensation temperature T c of an atomic laboratory trapped Bose-Einstein condensate. We show that, in the mean-field Hartree-Fock and semiclassical approximations, interactions produce a shift Δ T c /T c 0 ≈ b 1 (a/λ T c ) + b 2 (a/λ T c ) 2 + ψ[a / λ T c ] with a the s-wave scattering length, λ T the thermal wavelength and ψ[a / λ T c ] 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 / λ T c 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 b 2 ≃ 18.8 which improves a previous numerical result. We also discuss how the discrepancy between b 2 and the empirical value of b 2 =46 ± 5 may be explained with no need to resort to beyond-mean field effects. Copyright EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2013