Heating rates in collisionally opaque alkali-metal atom traps: Role of secondary collisions

Grazing collisions with background gas are the major cause of trap loss and trap heating in atom traps. To first order, these effects do not depend on the trap density. In collisionally opaque trapped atom clouds, however, scattered atoms with an energy E larger than the effective trap depth ${\mathcal{E}}_{\mathrm{eff}},$ which are destined to escape from the atom cloud, will have a finite probability for a secondary collision. This results in a contribution to the heating rate that depends on the column density $〈\mathrm{nl}〉$ of the trapped atoms, i.e., the product of density and characteristic size of the trap. For alkali-metal atom traps, secondary collisions are quite important due to the strong long-range interaction with like atoms. We derive a simple analytical expression for the secondary heating rate, showing a dependency proportional to $〈\mathrm{nl}〉\hspace{0.5em}{\mathcal{E}}_{\mathrm{eff}}^{1/2}.$ When extrapolating to a vanishing column density, only primary collisions with the background gas will contribute to the heating rate. This contribution is rather small, due to the weak long-range interaction of the usual background gas species in an ultrahigh-vacuum system—He, Ne, or Ar—with the trapped alkali-metal atoms. We conclude that the transition between trap-loss collisions and heating collisions is determined by a cutoff energy $200\hspace{0.5em}\mu \mathrm{K}<~{\mathcal{E}}_{\mathrm{eff}}<~400\hspace{0.5em}\mu \mathrm{K},$ much smaller than the actual trap depth $\mathcal{E}$ in most magnetic traps. Atoms with an energy ${\mathcal{E}}_{\mathrm{eff}}<E<\mathrm{}\mathcal{E}$ escape into the Oort cloud: a mechanism of effective traploss in the microkelvin range of trap temperatures. We present results of secondary heating rates for the alkali-metal atoms Li through Cs as a function of the effective trap depth, the column density of the trap, and the species in the background gas. The predictions of our model are in good agreement with the experimental data of Myatt for heating rates in high-density ${}^{87}\mathrm{Rb}$-atom magnetic traps at JILA, including the effect of the rf shield and the composition of the background gas. It is shown that collisions with atoms from the Oort cloud also contribute to the heating rate. For ${}^{85}\mathrm{Rb}$ the calculated heating rate is below the experimentally observed value at JILA, supporting the idea that inelastic collisions in the trap are the major source of heating.