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Physical Review A

Quantum defect theory for the van der Waals plus dipole-dipole interaction

Gao‐Ren Wang, Ting Xie, Yin Hui Huang, Wei Zhang, Shu‐Lin Cong

Published Dec 14, 2012• 15 citations
Publisher

Abstract

We investigate the scattering dynamics governed by the long-range van der Waals plus dipole-dipole interaction potential, $\ensuremath{-}{C}_{6}/{R}^{6}\ensuremath{-}{C}_{3}/{R}^{3}$, which describes the long-range interaction between two polar molecules in an electric field. In the spirit of quantum defect theory, a set of parameters which are nearly constants in the threshold regime is defined to characterize the scattering process. Using appropriate boundary conditions for the scattering wave functions and relevant parameters, we explore the quantum reflection by and quantum tunneling through the long-range potential. As a sample application, the reactive collision rates of ${}^{40}$K${}^{87}$Rb $+$ ${}^{40}$K${}^{87}$Rb are calculated.

DOI
10.1103/physreva.86.062704

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We investigate the scattering dynamics governed by the long-range van der Waals plus dipole-dipole interaction potential, \ensuremath−C6/R6\ensuremath−C3/R3\ensuremath{-}{C}_{6}/{R}^{6}\ensuremath{-}{C}_{3}/{R}^{3}\ensuremath−C6​/R6\ensuremath−C3​/R3, which describes the long-range interaction between two polar molecules in an electric field. In the spirit of quantum defect theory, a set of parameters which are nearly constants in the threshold regime is defined to characterize the scattering process. Using appropriate boundary conditions for the scattering wave functions and relevant parameters, we explore the quantum reflection by and quantum tunneling through the long-range potential. As a sample application, the reactive collision rates of 40{}^{40}40K87{}^{87}87Rb +++ 40{}^{40}40K87{}^{87}87Rb are calculated.
Progress in ultracold experiments with polar molecules requires a clear understanding of their interactions and reactivity at ultralow collisional energies. Two important theoretical steps in this process are the characterization of interaction potentials between molecules and the modeling of the reactive scattering mechanism. Here, we report on the ab initio calculation of isotropic and anisotropic van der Waals interaction potentials for polar KRb and RbCs colliding with each other or with ultracold atoms. Based on these potentials and two short-range scattering parameters, we then develop a single-channel scattering model with flexible boundary conditions. Our calculations show that, at low temperatures (and in the absence of an external electric field), the reaction rates between molecules or between molecules and atoms have a resonant character as a function of the short-range parameters. We also find that both the isotropic and anisotropic van der Waals coefficients have significant contributions from dipole coupling to excited electronic states. Their values can differ dramatically from those solely obtained from the permanent dipole moment. A comparison with recently obtained reaction rates of fermionic 40 K 87 Rb shows that the experimental data cannot be explained by a model where the short-range scattering parameters are independent of the relative orbital angular momentum or partial wave.
Abstract In this paper, we have provided exact two-body solutions to the 2D and 3D Schrödinger equations with isotropic van der Waals potentials of the form ± 1 / r 6 . Based on these solutions, we developed an analytical quantum defect theory (QDT) applicable to both quasi-2D and 3D geometries, and applied it to study the scattering properties and bound-state spectra of ultracold polar molecules confined in these geometries. Interestingly, we find that in the attractive (repulsive) van der Waals potential case, the short-range interaction can be effectively modeled by an infinite square barrier (finite square well), which leads to narrow and dense (broad and sparse) resonance structures in the quantum defect parameter. In the quasi-2D attractive case, shape resonances can appear in an ordered fashion across different partial waves, characterized by sharp phase jumps as the scattering energy is varied. Furthermore, the low-energy analytical expansions derived from QDT show excellent agreement with the exact numerical results, validating the accuracy and usefulness of our analytical approach in describing two-body physics governed by long-range van der Waals interactions.
We investigate collisions of polar molecules in quasi-2D traps in the presence of an external electric field perpendicular to the collision plane. We use the quantum-defect model characterized by two dimensionless parameters: yyy and sss. The first of them is related to the probability of the reaction at short distances, whereas the latter one defines the wave function phase at short distances. For yyy close to unity we obtain universal collision rates determined by the quantum reflection process from the long-range part of the interaction potential that depends only on the van der Waals coefficient, dipole-dipole interaction and the trap frequency. For small short-range reaction probabilities collision rates are highly nonuniversal and trap induced shape resonances are visible. For high dipole moments we observe the damping of reactive collisions, which can stabilize the ultracold gas of polar molecules. The calculations were performed with help of multichannel wave funcion propagation by imposing short-range boundary condition derived from the quantum-defect model.
By generalizing Gao's approach [] for solving the Schrödinger equation for an isotropic van der Waals (vdW) potential to the systems with a multiscale anisotropic long-range interaction, we derive the solutions for the Schrödinger equation for an anisotropic dipole-dipole interaction plus an isotropic attractive vdW potential, i.e., C d ( 1 − 3 cos 2 θ ) / r 3 − C 6 / r 6 , which is projected to the subspace with angular momentum l ≤ l cut , with l cut being an arbitrary angular-momentum cutoff. Here θ is the polar angle of the coordinate r and r = | r | . The asymptotic behaviors of these solutions for r → 0 and r → ∞ are obtained. The generalization of our approach for the case of an anisotropic vdW potential is also discussed. These results can be used in the research of collisions and chemical reactions between ultracold polar molecules in a static electric field. Our approach to derive the solutions can be applied to the systems with a general long-range potential ∑ λ = 2 λ max V λ ( θ , φ ) / r λ , with φ being the azimuthal angle of r , and thus can be used in collision problems between various kinds of molecules, ions, or atoms (e.g., the collisions between ultracold dipolar atoms).
We consider low energy threshold reactive collisions of particles interacting via a van der Waals potential at long range in the presence of external confinement and give analytic formulas for the confinement modified scattering in such circumstances. The reaction process is described in terms of the short range reaction probability. Quantum defect theory is used to express elastic and inelastic or reaction collision rates analytically in terms of two dimensionless parameters representing phase and reactivity. We discuss the modifications to Wigner threshold laws for quasi-one-dimensional and quasi-two-dimensional geometries. Confinement-induced resonances are suppressed due to reactions and are completely absent in the universal limit where the short-range loss probability approaches unity.