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arXiv:1209.4553 · Physical Review A

Universal three-body recombination via resonant d -wave interactions

Jia Wang, José Paulo D'Incao, Yujun Wang, Chris H. Greene

Published Dec 20, 2012• 37 citations
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Abstract

For a system of three identical bosons interacting via short-range forces, when two of the atoms are about to form a two-body $s$-wave dimer, the Efimov effect takes place leading to the formation of an infinite number of three-body (Efimov) states. The lowest Efimov state crosses the three-body break-up threshold when the $s$-wave two-body scattering length is $a\ensuremath{\approx}\ensuremath{-}9.73{r}_{\mathrm{vdW}}$, ${r}_{\mathrm{vdW}}$ being the van der Waals length. This article focuses on a generalized version of this Efimov scenario, where two of the atoms are about to form a two-body $d$-wave dimer, resulting in strong $d$-wave interactions. In his paper [Phys. Rev. A 62, 050702(R) (2000)], Bo Gao predicted that for broad resonances the $d$-wave dimer is always formed when $a\ensuremath{\approx}0.956{r}_{\mathrm{vdW}}$. Here we find that a single universal three-body state associated with the $d$-wave dimer is also formed near the three-body breakup threshold at $a\ensuremath{\approx}1.09{r}_{\mathrm{vdW}}$, or alternatively ${a}_{2}=0.902{r}_{\mathrm{vdW}}$, where ${a}_{2}$ is the two-body $d$-wave scattering length. The signature of such a universal three-body state is signaled experimentally by an enhancement of the three-body recombination rate. The three-body effective potential curves that are crucial for understanding the recombination dynamics are also calculated and analyzed. An improved method to calculate the couplings, effective potential curves, and recombination rate coefficients is presented.

DOI
10.1103/physreva.86.062511

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For a system of three identical bosons interacting via short-range forces, when two of the atoms are about to form a two-body sss-wave dimer, the Efimov effect takes place leading to the formation of an infinite number of three-body (Efimov) states. The lowest Efimov state crosses the three-body break-up threshold when the sss-wave two-body scattering length is a\ensuremath≈\ensuremath−9.73rvdWa\ensuremath{\approx}\ensuremath{-}9.73{r}_{\mathrm{vdW}}a\ensuremath≈\ensuremath−9.73rvdW​, rvdW{r}_{\mathrm{vdW}}rvdW​ being the van der Waals length. This article focuses on a generalized version of this Efimov scenario, where two of the atoms are about to form a two-body ddd-wave dimer, resulting in strong ddd-wave interactions. In his paper [Phys. Rev. A 62, 050702(R) (2000)], Bo Gao predicted that for broad resonances the ddd-wave dimer is always formed when a\ensuremath≈0.956rvdWa\ensuremath{\approx}0.956{r}_{\mathrm{vdW}}a\ensuremath≈0.956rvdW​. Here we find that a single universal three-body state associated with the ddd-wave dimer is also formed near the three-body breakup threshold at a\ensuremath≈1.09rvdWa\ensuremath{\approx}1.09{r}_{\mathrm{vdW}}a\ensuremath≈1.09rvdW​, or alternatively a2=0.902rvdW{a}_{2}=0.902{r}_{\mathrm{vdW}}a2​=0.902rvdW​, where a2{a}_{2}a2​ is the two-body ddd-wave scattering length. The signature of such a universal three-body state is signaled experimentally by an enhancement of the three-body recombination rate. The three-body effective potential curves that are crucial for understanding the recombination dynamics are also calculated and analyzed. An improved method to calculate the couplings, effective potential curves, and recombination rate coefficients is presented.
Few-body effects play an important role for the understanding of ultracold quantum gases. We make use of an effective field theory approach to investigate various aspects of universal few-body physics close to a Feshbach resonance. That is the regime where the scattering length is large compared to all other length scales of the system and thus determines the observables. It is also the regime where the Efimov effect whose main characteristic is the occurrence of a sequence of three-body bound states becomes important. At unitarity, i.e., for diverging scattering length, the ratio of binding energies of neighboring states approaches a constant. The existence of those trimers can be captured by a single three-body parameter and influences observables such as recombination rates. Starting from adequate Lagrangians, we derive atom-dimer scattering amplitudes that contain all information of interest about the three-particle systems. With this method, we first investigate the scattering of atoms and dimers at finite temperature in the presence of the Efimov effect. We calculate the dimer relaxation rate and obtain good agreement with available experimental data. Furthermore, heteronuclear mixtures exhibiting large interspecies scattering length are studied in detail. If bosons are the majority species, the Efimov effect occurs in an S-wave channel and we are able to compute three-body recombination and dimer relaxation rates. The results are compared to the outcome of two existing experiments. For mainly fermionic systems, the Efimov effect is only present in an overall P-wave and three-body recombination at threshold vanishes. However, in mixtures of atoms and dimers, scattering cross sections and dimer relaxation show the typical Efimov behavior of log-periodicity. We also consider two-dimensional Bose gases, where the Efimov effect does not occur. For those systems, we derive an equation including the first non-universal corrections and deduce three-body observables such as binding energies and atom-dimer scattering parameters.
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.
We construct a simple model to calculate the trimer bound state energy ET(n){E}_{T}^{(n)}ET(n)​ and three-body recombination rate K30{K}_{3}^{0}K30​ of three identical bosons with van der Waals interaction without using any two- or three-body fitting parameter. Using this simple model, we investigate the influence of the van der Waals finite-range effect on ET(n){E}_{T}^{(n)}ET(n)​ and K30{K}_{3}^{0}K30​. Our calculation shows that the finite-range effect leads to the ground trimer state energy ET(0){E}_{T}^{(0)}ET(0)​ not crossing the atom-dimer threshold, and the scaled three-body recombination rate K30/a4{K}_{3}^{0}/{a}^{4}K30​/a4 deviating from the universal three-body theory. The results of our simple model agree within a few percent with other theoretical works with van der Waals interaction and also the experimental data.
Equal mass fermionic trimers with two different spin components near the unitary limit are shown to possess a universal van der Waals bound or resonance state near sss-wave unitarity, when ppp-wave interactions are included between the particles with equal spin. Our treatment uses a single-channel Lennard-Jones interaction with long-range two-body van der Waals potentials. While it is well known that there is no true Efimov effect that would produce an infinite number of bound states in the unitary limit, we demonstrate that another type of universality emerges for the symmetry ${L}^{\mathrm{\ensuremath{\Pi}}}={1}^{\ensuremath{-}}$. The universality is a remnant of Efimov physics that exists in this system at ppp-wave unitarity, and it leads to modified threshold and scaling laws in that limit. Application of our model to the system of three lithium atoms studied experimentally by Du et al. [Phys. Rev. Lett. 102, 250402 (2009)] provides an interpretation of their measured three-body recombination loss rates.
A new narrow three-atom loss resonance associated with an Efimov trimer crossing the three-atom threshold has recently been discovered in a many-body system of ultracold 6Li^{6}\mathrm{Li}6Li atoms in the three lowest hyperfine spin states at a magnetic field near 895 G. O'Hara and coworkers have used measurements of the three-body recombination rate in this region to determine the complex three-body parameter associated with Efimov physics. Using this parameter as the input, we calculate the universal predictions for the spectrum of Efimov states and for the three-body recombination rate in the universal region above 600 G where all three scattering lengths are large. We predict an atom-dimer loss resonance at $672\ifmmode\pm\else\textpm\fi{}2$ G associated with an Efimov trimer disappearing through an atom-dimer threshold. We also predict an interference minimum in the three-body recombination rate at $759\ifmmode\pm\else\textpm\fi{}1$ G where the three-spin mixture may be sufficiently stable to allow experimental study of the many-body system.
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