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arXiv:1301.5857 · Physical Review Letters

Quantum Theory of Reactive Collisions for 1/rn Potentials

Krzysztof Jachymski, Michał Krych, Paul S. Julienne, Zbigniew Idziaszek

Published May 23, 2013• 55 citations
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Abstract

We develop a general quantum theory for reactive collisions involving power-law potentials (-1/r(n)) valid from the ultracold up to the high-temperature limit. Our quantum defect framework extends the conventional capture models to include the nonuniversal case when the short-range reaction probability P(re)<1. We present explicit analytical formulas as well as numerical studies for the van der Waals (n=6) and polarization (n=4) potentials. Our model agrees well with recent merged beam experiments on Penning ionization, spanning collision energies from 10 mK to 30 K [Henson et al., Science 338, 234 (2012)].

DOI
10.1103/physrevlett.110.213202

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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.
A simple quantum-defect model gives analytic expressions for the complex scattering length and threshold collision rates of ultracold molecules. If the probability of reaction in the short-range part of the collision is high, the model gives universal rate constants for s- and p-wave collisions that are independent of short-range dynamics. This model explains the magnitudes of the recently measured rate constants for collisions of two ultracold 40K87Rb molecules, or an ultracold 40K atom with the 40K87Rb molecule [S. Ospelkaus et al., Science 327, 853 (2010).
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.
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{}^{40}40K87{}^{87}87Rb +++ 40{}^{40}40K87{}^{87}87Rb are calculated.