Ultracold Akali Collisions

The nature of the interaction between ultracold atoms is sensitive to their internuclear separation distance r. When the collision partners are far apart, they are regarded as non-interacting and the state of the system is characterized by the internal degrees of the freedom of the individual atoms, namely their respective electronic and nuclear spins. However, as r begins to diminish, the atoms start to feel a weak attractive force represented by a sum of van der Waals terms with interaction energies on the order of 10−6 Hartree. This attractive force grows stronger as the particles continue to move towards each other, with associated interaction energies approaching ∼ 10−2 Hartree, until r reaches a point re, beyond which the atoms experience a strong repulsive force because the two cannot physically be on top of each other. Quantum defect theory (QDT) lends itself to calculations involving such systems because it exploits the natural separation of length and energy scales outlined above. In the simplest QDT approximation, two constants with respect to both field and energy, the singlet and triplet quantum defects, fully describe the short-range properties of the collision. These parameters are used to approximate the short-range reaction matrix Ksr with a frame transformation (FT) formula. In the long-range region, a collection of quantities that are smooth in energy and field characterize the physics. Moreover, at low collision energies, these long-range parameters behave as simple, analytic functions of energy to a good approximation. With the long-range parameters and Ksr in hand, the real properties of the atomic system, such as elastic cross sections, can be tabulated in a relatively few easy steps. We used this FT approximation to describe elastic s-wave collisions of 6Li, 7Li, 23Na, 39K and 87Rb atoms in which the particles enter and exit the lowest-lying interaction channel. For each, we calculated the elastic cross section as a function of magnetic field, finding that our FT method is able to reproduce the resonance features of a full coupled channels (FCC) calculation for systems where the hyperfine/Zeeman splitting of the collision channels is negligible in the area where the short- and long-range regions overlap