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arXiv:atom-ph/9605009 · Physical Review Letters

Three-Body Recombination of Ultracold Atoms to a Weakly Bound s Level

Peter O. Fedichev, M. W. Reynolds, Georgy V. Shlyapnikov

Published Sep 30, 1996• 286 citations
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Abstract

We discuss three-body recombination of ultracold atoms to a weakly bound $s$ level. In this case, characterized by large and positive scattering length $a$ for pair interaction, we find a repulsive effective potential for three-body collisions, which strongly reduces the recombination probability. In the zero temperature limit we obtain a universal relation, independent of the detailed shape of the interaction potential, for the (event) rate constant of three-body recombination: ${\ensuremath{\alpha}}_{\mathrm{rec}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3.9\ensuremath{\Elzxh}{a}^{4}/m$, where $m$ is the atom mass.

DOI
10.1103/physrevlett.77.2921

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We discuss three-body recombination of ultracold atoms to a weakly bound sss level. In this case, characterized by large and positive scattering length aaa for pair interaction, we find a repulsive effective potential for three-body collisions, which strongly reduces the recombination probability. In the zero temperature limit we obtain a universal relation, independent of the detailed shape of the interaction potential, for the (event) rate constant of three-body recombination: \ensuremathαrec=3.9\ensuremath\Elzxha4/m{\ensuremath{\alpha}}_{\mathrm{rec}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3.9\ensuremath{\Elzxh}{a}^{4}/m\ensuremathαrec​=3.9\ensuremath\Elzxha4/m, where mmm is the atom mass.
We use universality in two-body dipolar physics to study three-body recombination. We present results for the universal structure of weakly bound two-dipole states that depend only on the s-wave scattering length (a). We study threshold three-body recombination rates into weakly bound dimer states as a function of the scattering length. A Fermi golden rule analysis is used to estimate rates for different events mediated by the dipole-dipole interaction and a phenomenological contact interaction. The three-body recombination rate in the limit where a≫D contains terms which scale as a{4}, a{2}D{2}, and D4, where D is the dipolar length. When a≪D, the three-body recombination rate scales as D4.
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.
Few-body systems with large scattering length aaa have universal properties that do not depend on the details of their interactions at short distances. The rate constant for three-body recombination of bosonic atoms of mass mmm into a shallow dimer scales as $\ensuremath{\hbar}{a}^{4}∕m$ times a log-periodic function of the scattering length. We calculate the leading and subleading corrections to the rate constant, which are due to the effective range of the atoms, and study the correlation between the rate constant and the atom-dimer scattering length. Our results are applied to 4He^{4}\mathrm{He}4He atoms as a test case.
Generally, the three-body recombination (TBR) of the ultracold helium atoms in the zero-collision-energy limit could be described by the original zero-range (Efimov) universal function [E. Braaten, D. Kang, and L. Platter, Phys. Rev. A 75, 052714 (2007)] using the scattering length aaa of the He-He interaction and a three-body parameter a∗{a}_{*}a∗​ as input, where a∗{a}_{*}a∗​ is associated with the positive scattering length at which an interference minimum in the TBR rate occurs at the threshold. This universal property has been reinvestigated by considering various post-Born-Oppenheimer (post-BO) effects on the relevant scattering process. It is found that a linear correction to 1/a∗1/{a}_{*}1/a∗​ versus 1/a1/a1/a should be introduced to give a better description of these considered TBR cases because of the finite-range effects. Particularly, such a correction may be ascribed not only to the general scattering properties associated with the Efimov channel at the short hyperradial distance, but also to the tunneling probability in the lowest incident channel. In addition, a successful fit of the modified universal function that includes correctly the finite-range effects [E. Garrido, M. Gattobigio, and A. Kievsky, Phys. Rev. A 88, 032701 (2013)] or the zero-range universal function with a linearly variational 1/a∗1/{a}_{*}1/a∗​ versus 1/a1/a1/a to numerically accurate data also indicates a clear manifestation of the van der Waals universality. Finally, the post-BO effects on the TBR process for the partial wave with the total angular momentum J>0J>0J>0 are also discussed.
For atoms with large scattering length, the dependence of the three-body recombination rate on the collision energy is determined by the scattering length and the Efimov three-body parameters and can be expressed in terms of universal functions of a single scaling variable. We use published results on the three-body recombination rate for 4He^{4}\mathrm{He}4He atoms to constrain the universal functions. We then use those universal functions to calculate the three-body recombination rate for other atoms with large scattering length at nonzero temperature. The constraints from the 4He^{4}\mathrm{He}4He results are strong if the scattering length is near the minimum of the three-body recombination rate at threshold. We apply our results to 133Cs^{133}\mathrm{Cs}133Cs atoms with a large positive scattering length, and compare them with experimental results from the Innsbruck group.