Exact Two-Body Solutions and Quantum Defect Theory of Polar Molecular Gases with Van de Waals Potentials

In a recent experiment [Matsuda et al, Science 370, 1324 (2020)], a quasi two-dimensional (2D), long-lived and strongly interacting diatomic polar molecular gas was successfully prepared via controllable electric field technique. More interestingly, the effective positive and negative Van de Waals interactions would emerge when scanning the strength of the electric fields. Those results were also generalized to three-dimensional (3D) case in a later experiment [J. Li et al, Nature Physics 17, 1144 (2021)]. Motivated by these experiments, in this paper we provide the two-body exact solutions for the 2D and 3D Schr\"{o}dinger equation with isotropic Van de Waals interactions ($\pm1/r^{6}$). Furthermore, we build the analytical quantum defect theory (QDT) for quasi 2D and 3D based on these solutions and then apply QDT to study the scattering properties and bound states of two ultracold polar molecules confined in quasi-2D and 3D geometry. Interestingly, we find that for the attractive (repulsive) Van de Waals potential cases, the two-body short range potential can be approximated by an square barrier with infinity height (square potential with finite depth) which yields the wide (narrow) resonances of quantum defect parameter with dense (dilute) density. For the quasi-2D attractive case, the scattering resonance can happen simultaneously which is featured by the phase jumps when varying the scattering energy. The analytical expansions in the low energy agree well to all the results.