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Journal of Physics B Atomic Molecular and Optical Physics

Hyperspherical adiabatic eigenvalues for zero-range potentials

G. Gasaneo, J. H. Macek

Published May 9, 2002• 25 citations
Publisher

Abstract

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.

DOI
10.1088/0953-4075/35/10/304

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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.
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 threshold recombination coefficient for three bosons whose two-body scattering length aaa is positive and large compared with the nominal range of the potentials is shown to proceed via a crossing of hyperspherical adiabatic potential curves at $R\ensuremath{\approx}2.6a$. The hidden crossing theory gives a simple expression for the rate in terms of the scattering length and a phase $\ensuremath{\Delta}$. The phase depends upon the details of the interaction, and varies from species to species, however an upper limit to the rate which depends only upon the scattering length emerges from the hidden crossing theory.
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.
We use the hyperspherical adiabatic expansion method to formulate the angular Faddeev-like equation for N identical bosons including effects of two-body correlations. Eigenvalues and eigenfunctions determine the effective potentials in the coupled set of radial equations. The zero-range approximation is implemented on the free solutions as a boundary condition obtained by using the effective-range expansion of the two-body phase shift. This results in a new angular eigenvalue equation expressed as a transcendental equation in the index of the Jacobi functions corresponding to solutions for non-interacting particles. The collapse for attractive potentials is avoided by including scattering length, effective range and shape-parameter terms in the effective-range expansion of the phase shift. Correspondingly the hyperradial dependence in the eigenvalue equation is either a polynomial of first, second, or fourth order. Two-body correlations are included. The arithmetic eigenvalue equation allows derivation of analytic properties such as for example simple scaling relations for the particle number dependence. Excellent agreement with finite-range numerical results is obtained. The couplings between adiabatic channels are discussed. Including only the lowest adiabatic potential is shown to be sufficient for most applications on dilute systems.
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.
This work treats resonant collisions between five identical ultracold bosons in the framework of the adiabatic hyperspherical representation. The five-body recombination rate coefficient is quantified using a semiclassical description in conjunction with an analysis of the lowest five-body hyperspherical adiabatic potential curves in a scattering length regime with no universal weakly bound tetramers, trimers, or dimers. A comparison is made between these results and the only existing experimental measurement of five-body loss in an ultracold gas of bosonic cesium atoms and with the lone theoretical estimation of the loss rate. The recombination rate for the process B + B + B + B + B → B 4 + B is also computed in a different regime of scattering lengths where there is one universal bound tetramer by implementing a few-channel quantum scattering calculation based on five-body hyperspherical potential curves and nonadiabatic couplings. Our calculations predict regions where five-body recombination can cause decay of the atom cloud in an ultracold gas that is even faster than 3-body and 4-body recombination, which can ideally be tested by using the current generation of box traps having nearly uniform density.