Understanding Quantum Defect Theory for Cold Atoms and Molecules

Interactions between ultracold diatomic molecules are characterized by a deep potential energy well at short range, with a shallow long-range tail that prevails to large molecular separation. Quantum defect theory (QDT) exploits this separation of length and energy scales so that in the simplest single-channel approximation, one parameter which is weakly dependent on energy, the \textit{quantum defect}, can be used to describe the short-range properties of the collision. The quantum defect can either be calculated numerically from an \textit{ab initio }method or found from the solution of an effective short-range potential designed such that it reproduces the essential short-range physics. The long-range solutions are efficiently found using the Milne phase amplitude method. We present simple single-channel QDT models that make use of this separation of length and energy scales to treat real properties of ultracold atomic and molecular collisions.