Three-body problem for ultracold atoms in quasi-one-dimensional traps

C. Mora; R. Egger; A. O. Gogolin

doi:10.1103/PhysRevA.71.052705

paper

Three-body problem for ultracold atoms in quasi-one-dimensional traps

arXiv:cond-mat/0412225 · doi:10.1103/PhysRevA.71.052705

Abstract

We study the three-body problem for both fermionic and bosonic cold atom gases in a parabolic transverse trap of lengthscale $a_{⊥}$ . For this quasi-one-dimensional (1D) problem, there is a two-body bound state (dimer) for any sign of the 3D scattering length $a$ , and a confinement-induced scattering resonance. The fermionic three-body problem is universal and characterized by two atom-dimer scattering lengths, $a_{a d}$ and $b_{a d}$ . In the tightly bound `dimer limit', $a_{⊥} / a \to \infty$ , we find $b_{a d} = 0$ , and $a_{a d}$ is linked to the 3D atom-dimer scattering length. In the weakly bound `BCS limit', $a_{⊥} / a \to - \infty$ , a connection to the Bethe Ansatz is established, which allows for exact results. The full crossover is obtained numerically. The bosonic three-body problem, however, is non-universal: $a_{a d}$ and $b_{a d}$ depend both on $a_{⊥} / a$ and on a parameter $R^{*}$ related to the sharpness of the resonance. Scattering solutions are qualitatively similar to fermionic ones. We predict the existence of a single confinement-induced three-body bound state (trimer) for bosons.

arXiv comment20 pages, 6 figures, accepted for publication in PRA, appendix on the derivation of an integral formula for the Hurvitz zeta function

Three-body problem for ultracold atoms in quasi-one-dimensional traps

Abstract

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