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Physical Review A

Scaling limit of weakly bound triatomic states

T. Frederico, Lauro Tomio, A. Delfino, Antonio Eduardo Assis Amorim

Published Jul 1, 1999• 71 citations
Publisher

Abstract

The usefulness of a scale-independent approach to identify Efimov states in three-body systems is shown by comparing such an approach with a realistic calculation in the case of three helium atoms. We show that the scaling limit is realized in practice in this case, and suggest its application to study other similar systems, including the case where two kinds of atoms are mixed. We also consider the observed large scattering length of $\mathrm{the}{ }^{87}\mathrm{Rb}$ dimer to estimate the critical value of the ground-state energy of the corresponding trimer $(>~1.5 \mathrm{mK})$, in order to allow for one Efimov state above the ground state.

DOI
10.1103/physreva.60.r9

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The usefulness of a scale-independent approach to identify Efimov states in three-body systems is shown by comparing such an approach with a realistic calculation in the case of three helium atoms. We show that the scaling limit is realized in practice in this case, and suggest its application to study other similar systems, including the case where two kinds of atoms are mixed. We also consider the observed large scattering length of the87Rb\mathrm{the}{ }^{87}\mathrm{Rb}the87Rb dimer to estimate the critical value of the ground-state energy of the corresponding trimer (> 1.5mK)(>~1.5 \mathrm{mK})(> 1.5mK), in order to allow for one Efimov state above the ground state.
The mean-square radii of the molecules 4He3,{}^{4}{\mathrm{He}}_{3},4He3​, ${}^{4}{\mathrm{He}}_{2}{\ensuremath{-}}^{6}\mathrm{Li},$ ${}^{4}{\mathrm{He}}_{2}{\ensuremath{-}}^{7}\mathrm{Li},$ and ${}^{4}{\mathrm{He}}_{2}{\ensuremath{-}}^{23}\mathrm{Na}$ are calculated using a three-body model with contact interactions. They are obtained from a universal scaling function calculated within a renormalized scheme for three particles interacting through pairwise Dirac-$\ensuremath{\delta}$ interaction. The root-mean-square distance between two atoms of mass mA{m}_{A}mA​ in a triatomic molecule are estimated to be of the order of $\mathcal{C}\sqrt{{\ensuremath{\Elzxh}}^{2}/[{m}_{A}{(E}_{3}\ensuremath{-}{E}_{2})]},$ where E2{E}_{2}E2​ is the dimer and E3{E}_{3}E3​ is the trimer binding energies, and C\mathcal{C}C is a constant (varying from $\ensuremath{\sim}0.6$ to $\ensuremath{\sim}1),$ which depends on the ratio between E2{E}_{2}E2​ and E3.{E}_{3}.E3​. Considering previous estimates for the trimer energies, we also predict the sizes of rubidium and sodium trimers in atomic traps.
The discrete Efimov scaling behavior, well known in the low-energy spectrum of three-body bound systems for large scattering lengths (unitary limit), is identified in the energy dependence of an atom-molecule elastic cross section in mass-imbalanced systems. That happens in the collision of a heavy atom with mass mH{m}_{H}mH​ with a weakly bound dimer formed by the heavy atom and a lighter one with mass ${m}_{L}\ensuremath{\ll}{m}_{H}$. Approaching the heavy-light unitary limit, the sss-wave elastic cross section $\ensuremath{\sigma}$ will present a sequence of zeros or minima at collision energies following closely the Efimov geometrical law. Our results, obtained with Faddeev calculations and supplemented by a Born-Oppenheimer analysis, open a perspective to detecting the discrete scaling behavior from low-energy scattering data, which is timely in view of the ongoing experiments with ultracold binary mixtures having strong mass asymmetries, such as lithium and cesium or lithium and ytterbium.
We consider a system of three 4{}^{4}4He atoms, which is so far the simplest realistic three-body system exhibiting the Efimov effect, in order to analyze deviations from the universal Efimov three-body spectrum. We first calculate the bound states using a realistic two-body potential, and then analyze how they can be reproduced by simple effective models beyond Efimov's universal theory. We find that the nonuniversal variations of the ground and first excited states can be well reproduced by models parametrized with only three quantities: the scattering length and effective range of the original potential, and a small three-body force. Furthermore, the three-body parameter, which fixes the origin of the infinite set of three-body levels, is found to be consistent with recent experimental observations in other atomic species.
We report the observation of a three-body recombination resonance in an ultracold gas of cesium atoms at a very large negative value of the s-wave scattering length. The resonance is identified as the second triatomic Efimov resonance, which corresponds to the situation where the first excited Efimov state appears at the threshold of three free atoms. This observation, together with a finite-temperature analysis and the known first resonance, allows the most accurate demonstration to date of the discrete scaling behavior at the heart of Efimov physics. For the system of three identical bosons, we obtain a scaling factor of 21.0(1.3), close to the ideal value of 22.7.
We calculated, using seven realistic $^{4}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}$ potentials in the literature, the Efimov spectra of the 4He^{4}\mathrm{He}4He trimer and tetramer and analyzed the universality of the systems. The three-(four-)body Schr\"odinger equations were solved fully nonadiabatically with the high-precision calculation method employed in our previous work on the 4He^{4}\mathrm{He}4He trimer and tetramer [Phys. Rev. A 85, 022502 (2012); Phys. Rev. A 85, 062505 (2012)]. We found the following universality in the four-boson system: (i) The critical scattering lengths at which the tetramer ground and excited states couple to the four-body threshold are independent of the choice of the two-body realistic potentials in spite of the difference in the short-range details and do not contradict the corresponding values observed in the experiments in ultracold alkali-metal atoms when scaled with the van der Waals length rvdW{r}_{\mathrm{vdW}}rvdW​, and (ii) the four-body hyperradial potential has a repulsive barrier at the four-body hyperradius ${R}_{4}\ensuremath{\approx}3\phantom{\rule{0.16em}{0ex}}{r}_{\mathrm{vdW}}$, which prevents the four particles from getting close together to explore nonuniversal features of the interactions at short distances. This result is an extension of the universality in Efimov trimers that the appearance of the repulsive barrier at the three-body hyperradius ${R}_{3}\ensuremath{\approx}2\phantom{\rule{0.16em}{0ex}}{r}_{\mathrm{vdW}}$ makes the critical scattering lengths independent of the short-range details of the interactions as reported in the literature and also in the present work for the 4He^{4}\mathrm{He}4He trimer with the realistic potentials.