Efficient three-body calculations with a two-body mapped grid method

Abstract

We investigate the prospects of combining a standard momentum space approach for ultracold three-body scattering with efficient coordinate space schemes to solve the underlying two-body problem. In many of those schemes the two-body problem is numerically restricted up to a finite interparticle distance $r_\mathrm{b}$. We analyze effects of this two-body restriction on the two- and three-body level using pairwise square-well potentials that allow for analytic two-body solutions and more realistic Lennard-Jones van der Waals potentials to model atomic interactions. We find that the two-body $t$-operator converges exponentially in $r_\mathrm{b}$ for the square-well interaction. Setting $r_\mathrm{b}$ to 2000 times the range of the interaction, the three-body recombination rate can be determined accurately up to a few percent when the magnitude of the scattering length is small compared to $r_\mathrm{b}$, while the position of the lowest Efimov features is accurate up to the percent level. In addition we find that with the introduction of a momentum cut-off, it is possible to determine the three-body parameter in good approximation even for deep van der Waals potentials.