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arXiv:1005.2911 · Physical Review A

Green’s functions and the adiabatic hyperspherical method

Seth T. Rittenhouse, Nirav P. Mehta, Chris H. Greene

Published Aug 12, 2010• 40 citations
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Abstract

We address the few-body problem using the adiabatic hyperspherical representation. A general form for the hyperangular Green's function in $d$ dimensions is derived. The resulting Lippmann-Schwinger equation is solved for the case of three particles with $s$-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 $^{6}\mathrm{Li}$ atoms is presented.

DOI
10.1103/physreva.82.022706

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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.
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.
Three‐body recombination in cold atomic collisions has been successfully modeled using the hyperspherical close‐coupling theory. This representation maps many‐body fragmentation channels onto simple radial equations similar to those for two‐body processes. Computation of hyperspherical adiabatic energy eigenvalues is a major task using this theory. We have developed asymptotic representations of the adiabatic eigenvalues based upon energy dependent zero‐range potentials. Our asymptotic expressions are compared with ab initio calculations as well as hyperspherical adiabatic calculations using zero range potentials with constant M that have appeared in the literature.
The adiabatic hyperspherical technique is applied to investigate the asymptotic behavior of three-body systems. Here our attention is focused on deriving the analytical expressions for the asymptotic effective potential curves and the corresponding channel wave functions of the system, which, so far, cannot be obtained otherwise by conventional numerical methods to the required accuracy. Firstly, we consider a system of three charged particles of which two have opposite charges. We study such asymptotic regions in which one particle is far away from the other two, leaving them in either the ground state or, more importantly, excited ones. Asymptotic adiabatic hyperspherical potential curves and channel wave functions, valid through 1/R$\sp4$ and 1/R$\sp2$ respectively, are obtained. The wave functions and potentials include the polarization potential characteristic of the polarized orbital approximation. Terms specific to the separation in hyperspherical coordinates are identified. This investigation is then extended to three-body systems such as Helium and other iso-electronic ions in which two of the particles are identical. A symmetrization treatment of such systems leads us to a reasonable approximation to their asymptotic potential curves. Secondly, we study three neutral bosons with two-body short range interactions characterized by the so-called effective range r$\sb{\rm o}$ and scattering length c$\sb{\rm o}$. Analytical expressions for the effective hyper-radial potential are also obtained without explicit use of the two-body potential. The analytical results are then used to compare the general asymptotic behaviors of three-body systems interacting via short range potentials with those interacting via long range Coulomb ones. Possible loosely bound states of the system have also been analyzed quantitatively and the result confirms the prediction by Efimov. At the same time, some new characteristics of the three-body system are discussed. This investigation, once again, demonstrates the adiabatic hyperspherical method as a comprehensive framework for quantitative treatment of three-body systems.
Hyperspherical coordinates provide a systematic way of describing three-body systems. Solving three-body Schrödinger equations in an adiabatic hyperspherical representation is the focus of this thesis. An essentially exact solution can be found numerically by including nonadiabatic couplings using either a slow variable discretization or a traditional adiabatic method. Two different types of three-body systems are investigated: (1) rovibrational states of the triatomic hydrogen ion H+3 and (2) ultracold collisions of three identical bosons.
We study three-body recombination in two dimensions for systems interacting via short-range two-body interactions in the regime of large scattering lengths. Using the adiabatic hyperspherical representation, we derive semianalytical formulas for three-body recombination in both weakly and deeply bound diatom states. Our results demonstrate the importance of long-range corrections to the three-body potentials by showing how they alter the low-energy and scattering-length dependence of the recombination rate for both bosonic and fermionic systems, which exhibit suppressed recombination if compared to the three-dimensional case. We verify these results through numerical calculations of recombination for systems with finite-range interactions and supporting a few two-body bound states. We also study finite-range effects for the energies of the universal three-identical-bosons states and find a slow approach to universal predictions as a function of the scattering length.
We present numerical solutions of the Schr\"odinger equation for three identical bosons from very large to essentially infinite two-body sss-wave scattering lengths aaa. We have considered scattering lengths up to $2\ifmmode\times\else\texttimes\fi{}{10}^{5}\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}$ and solved the hyperangular part of the Schr\"odinger equation for distances up to 106a.u.{10}^{6}\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}106a.u. From these, we obtained the three-body effective potentials, hyperspherical channel functions and the asymptotic behavior of the nonadiabatic couplings in order to characterize the main aspects of the Efimov states. These results allow us to test and quantify the assumptions related to the Efimov effect.