Rigorous calculation of heating in alkali-metal traps by background gas collisions

The finite depth $\mathcal{E}$ of an atom trap results in an upper bound for the energy transfer in collisions with the background gas that will result in heating but not in loss of an atom. The energy transfer rate is accurately predicted as function of the well depth by applying a versatile semiempirical model function for the small-angle differential cross section, covering the full range from pure diffractive scattering to classical scattering. Simple scaling laws for the energy transfer rate are presented that can be readily applied. For the diffraction dominated regime we find an energy transfer rate proportional to $(\mathcal{E}/{\mathcal{E}}_{\mathrm{ref}}{)}^2$ with ${\mathcal{E}}_{\mathrm{ref}}{=(k}_B{T}_b\hspace{0.5em}{\theta }_0^2/4),$ a system-dependent energy determined by the ambient temperature $T_b$ and the diffraction angle ${\theta }_0.$ In the classical regime we find the usual result of an energy transfer rate proportional to $(\mathcal{E}/{\mathcal{E}}_{\mathrm{ref}}{)}^{5/6}.$