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arXiv:2007.12460 · Physical Review A

Quantum reactive scattering in the long-range ion-dipole potential

Tomasz Wasak, Zbigniew Idziaszek

Published Feb 23, 2021• 1 citation
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Abstract

An ion and a polar molecule interact by an anisotropic ion-dipole potential scaling as $\ensuremath{-}\ensuremath{\alpha}\phantom{\rule{0.16em}{0ex}}cos(\ensuremath{\theta})/{r}^{2}$ at large distances. Due to its long-range character, it modifies the properties of angular wave functions, which are no longer given by spherical harmonics. In addition, an effective centrifugal potential in the radial equation can become attractive for low angular momenta. In this paper, we develop a general framework for an ion-dipole reactive scattering, focusing on the regime of large $\ensuremath{\alpha}$. We introduce modified spherical harmonics as solutions of the angular part of the Schr\"odinger equation and derive several useful approximations in the limit of large $\ensuremath{\alpha}$. We present a formula for the scattering amplitude expressed in terms of the modified spherical harmonics and we derive expressions for the elastic and reactive collision rates. The solutions of the radial equation are given by Bessel functions, and we analyze their behavior in two distinct regimes corresponding, basically, to attractive and repulsive long-range centrifugal potentials. Finally, we study reactive collisions in the universal regime, where the short-range probability of loss or reaction is equal to unity.

DOI
10.1103/physreva.103.023324

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•Jan 1, 2010•0 citations
An ion and a polar molecule interact by an anisotropic ion-dipole potential scaling as \ensuremath−\ensuremathαcos(\ensuremathθ)/r2\ensuremath{-}\ensuremath{\alpha}\phantom{\rule{0.16em}{0ex}}cos(\ensuremath{\theta})/{r}^{2}\ensuremath−\ensuremathαcos(\ensuremathθ)/r2 at large distances. Due to its long-range character, it modifies the properties of angular wave functions, which are no longer given by spherical harmonics. In addition, an effective centrifugal potential in the radial equation can become attractive for low angular momenta. In this paper, we develop a general framework for an ion-dipole reactive scattering, focusing on the regime of large \ensuremathα\ensuremath{\alpha}\ensuremathα. We introduce modified spherical harmonics as solutions of the angular part of the Schr"odinger equation and derive several useful approximations in the limit of large \ensuremathα\ensuremath{\alpha}\ensuremathα. We present a formula for the scattering amplitude expressed in terms of the modified spherical harmonics and we derive expressions for the elastic and reactive collision rates. The solutions of the radial equation are given by Bessel functions, and we analyze their behavior in two distinct regimes corresponding, basically, to attractive and repulsive long-range centrifugal potentials. Finally, we study reactive collisions in the universal regime, where the short-range probability of loss or reaction is equal to unity.
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).
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.
We describe a method for evaluating analytical long-range contributions to scattering lengths for some potentials used in atomic physics. We assume that an interaction potential between colliding particles consists of two parts. The form of a short-range component, vanishing beyond some distance from the origin (a core radius), need not be given. Instead, we assume that a set of short-range scattering lengths due to that part of the interaction is known. A long-range tail of the potential is chosen to be an inverse power potential, a superposition of two inverse power potentials with suitably chosen exponents or the Lent potential. For these three classes of long-range interactions a radial Schrodinger equation at zero energy may be solved analytically with solutions expressed in terms of the Bessel, Whittaker and Legendre functions, respectively. We utilize this fact and derive exact analytical formulae for the scattering lengths. The expressions depend on the short-range scattering lengths, the core radius and parameters characterizing the long-range part of the interaction. Cases when the long-range potential (or its part) may be treated as a perturbation are also discussed and formulae for scattering lengths linear in strengths of the perturbing potentials are given. It is shown that for some combination of the orbital angular momentum quantum number and an exponent of the leading term of the potential the derived formulae, exact or approximate, take very simple forms and contain only polynomial and trigonometric functions. The expressions obtained in this paper are applicable to scattering of charged particles by neutral targets and to collisions between neutrals. The results are illustrated by accelerating convergence of scattering lengths computed for e - -Xe and Cs-Cs systems.
We consider two ultracold particles confined in a spherically symmetric harmonic trap and interacting via isotropic potential with absorbing boundary conditions at short range that models reactive scattering. First, we apply the contact pseudopotential with complex scattering length and investigate the properties of eigenenergies and eigenfunctions for different trap states as a function of the real and imaginary parts of the scattering length. In the analyzed case, the eigenenergies take complex values and their imaginary part can be interpreted in terms of decay rates. Then we introduce the model of a square well with absorbing boundary conditions that is later used to investigate the effects of the finite range of interaction on the properties of a reactive scattering in the trap. Finally, we analyze the decay rates for some reactive alkali dimer molecules assuming short-range probability of reaction equal to unity. In this case, the complex scattering length is given by the universal value related to the mean scattering length of the van der Waals interaction between molecules.
We study polar alkali-metal dimer scattering in a quasi-one-dimensional geometry for both reactive and nonreactive species. Elastic and reactive rates are computed as a function of the amplitude of a static electric field within a purely long-range model with suitable boundary conditions at shorter range. We describe the diatomic molecules as rigid rotors and results are compared to the fixed-dipole approximation. We show in particular that for molecules with a sufficiently strong induced dipole moment oriented perpendicular to the trap axis, the long-range repulsive interaction leads to the suppression of short-range processes. Such shielding effect occurs for both reactive and nonreactive molecules, preventing two-body reactions as well as losses due to complex-mediated processes [M. Mayle et al., Phys. Rev. A 85, 062712 (2012)] from occurring. The present results demonstrate the possibility to suppress loss rates in current ultracold molecule experiments using one-dimensional confinement.