Analysis of threshold effects in ultracold atomic collisions

At the ultracold temperatures which occur in cold atom traps and Bose-Einstein condensates, only a few partial waves contribute to the scattering of ground-state alkali-metal atoms, and cross sections are extremely sensitive to threshold effects. We present an analysis of these threshold effects, using a generalized multichannel quantum defect theory (GMQDT) to construct a close-coupled scattering wave function which is analytic in energy across thresholds. We illustrate the theory using the hyperfine transitions in Na+Na collisions, and show that it gives results completely equivalent to the usual close-coupled cross sections. The virtue of the GMQDT is that it treats both open and closed channels on an equal footing, and all interchannel dynamics is summarized in a single real symmetric matrix Y(E) which is essentially constant across thresholds (and often over excursions of energy which exceed the hyperfine splittings). The multichannel threshold energy behavior can then be related to calculable properties of the individual channels that are being closed. Many of the smaller spin depolarization cross sections are determined by very long-range α²/R³ spin-spin interactions which are not well treated by GMQDT, and we correct these specific elements with a perturbative distorted-wave approximation which yields the observed threshold dependences and brings the GMQDT into perfect agreement with the exact close-coupled results.