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arXiv:1404.0324 · Physical Review A

Adiabatic hyperspherical representation for the three-body problem in two dimensions

José Paulo D'Incao, Brett D. Esry

Published Oct 16, 2014• 17 citations
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Abstract

We explore the three-body problem in two dimensions using the adiabatic hyperspherical representation. We develop the main equations in terms of democratic hyperangular coordinates and determine several symmetry properties and boundary conditions for both interacting and noninteracting solutions. From the analysis of the three-body effective potentials, we determine the threshold laws for low-energy three-body recombination, collision-induced dissociation, as well as inelastic atom-diatom collisions in two dimensions. Our results show that the hyperspherical representation can offer a simple and conceptually clear physical picture for three-body process in two dimensions which is also suitable for calculations using finite-range two-body interactions supporting a number of bound states.

DOI
10.1103/physreva.90.042707

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We study the three-body problem in one dimension for both zero- and finite-range interactions using the adiabatic hyperspherical approach. Particular emphasis is placed on the threshold laws for recombination, which are derived for all combinations of the parity and exchange symmetries. For bosons, we provide a numerical demonstration of several universal features that appear in the three-body system, and discuss how certain universal features in three dimensions are different in one dimension. We show that the probability for inelastic processes vanishes as the range of the pairwise interaction is taken to zero and demonstrate numerically that the recombination threshold law manifests itself for large scattering length.
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We address the few-body problem using the adiabatic hyperspherical representation. A general form for the hyperangular Green's function in ddd dimensions is derived. The resulting Lippmann-Schwinger equation is solved for the case of three particles with sss-wave zero-range interactions. Identical particle symmetry is incorporated in a general and intuitive way. Complete semianalytic expressions for the nonadiabatic channel couplings are derived. Finally, a model to describe the atom loss due to three-body recombination for a three-component Fermi gas of 6Li^{6}\mathrm{Li}6Li atoms is presented.
The 4He3^{4}\mathrm{He}_{3}4He3​ system is studied using the adiabatic hyperspherical representation. We adopt the current state-of-the-art helium interaction potential including retardation and the nonadditive three-body term to calculate all low-energy properties of the triatomic 4He^{4}\mathrm{He}4He system. The bound state energies of the 4He^{4}\mathrm{He}4He trimer are computed as well as the 4He+4He2^{4}\mathrm{He}+^{4}\mathrm{He}_{2}4He+4He2​ elastic scattering cross sections, the three-body recombination, and collision induced dissociation rates at finite temperatures. We also treat the system that consists of two 4He^{4}\mathrm{He}4He and one 3He^{3}\mathrm{He}3He atoms, and compute the spectrum of the isotopic trimer 4He23He^{4}\mathrm{He}_{2}^{3}\mathrm{He}4He23​He, the 3He+4He2^{3}\mathrm{He}+^{4}\mathrm{He}_{2}3He+4He2​ elastic scattering cross sections, the rates for three-body recombination, and the collision induced dissociation rate at finite temperatures. The effects of retardation and the nonadditive three-body term are investigated. Retardation is found to be significant in some cases, while the three-body term plays only a minor role for these systems.
The scattering length a associated with two-body interactions is the relevant parameter for near threshold processes in cold atom-atom collisions. For this reason zero-range potentials are traditionally used to model collective behaviour of dilute collections of bosons. The model is also used to compute three-body recombination rates, where it gives an a 4 law. In this paper we examine the applicability of the zero-range model to real physical systems. Hyperspherical adiabatic potentials obtained from the zero-range model are compared with published potentials based on realistic two-body interactions. From these comparisons it is possible to determine the regions where the model applies.