Interaction effects on atomic laboratory trapped Bose-Einstein condensates

We discuss the effect of inter-atoms interactions on the condensation temperature T <Subscript> c </Subscript> 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 <Subscript> c </Subscript>/T <Subscript> c </Subscript> <Superscript>0</Superscript> ≈ b <Subscript>1</Subscript>(a/λ<Subscript> T c </Subscript>) + b <Subscript>2</Subscript>(a/λ<Subscript> T c </Subscript>)<Superscript>2</Superscript> + ψ[a / λ <Subscript> T c </Subscript>] with a the s-wave scattering length, λ <Subscript> T </Subscript> the thermal wavelength and ψ[a / λ <Subscript> T c </Subscript>] 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 <Subscript> T c </Subscript> 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 <Subscript>2</Subscript> ≃ 18.8 which improves a previous numerical result. We also discuss how the discrepancy between b <Subscript>2</Subscript> and the empirical value of b <Subscript>2</Subscript>=46 ± 5 may be explained with no need to resort to beyond-mean field effects. Copyright EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2013