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Physical Review A

Universalities for the Efimov-favored system with finite-range interactions

M.-S. Wu, Hui-Li Han, Ting-Yun Shi

Published Jun 9, 2016• 5 citations
Publisher

Abstract

Universal properties of Efimov states for some triatomic systems composed by one light and two heavy atoms are investigated using the finite-range model potentials. We have successfully obtained accurate values of binding energies and sizes for three successive Efimov states. Compared with the predicted scaling constants of the zero-range theory, the ground Efimov states are found to have the largest finite-range corrections. There exists a universal size-binding momentum relation for Efimov states which can be described by the formula ${\mathcal{R}}_{{n}^{*}}{k}_{{n}^{*}}=\sqrt{(1+{s}_{0}^{2})/3}$ derived with zero-range approximation. It is found that for the Efimov states that have large scaled sizes, their size-binding momentum relations follow the universal formula well. Our calculations demonstrate that size is an important characteristic in determining the universalities of Efimov states.

DOI
10.1103/physreva.93.062507

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Universal properties of Efimov states for some triatomic systems composed by one light and two heavy atoms are investigated using the finite-range model potentials. We have successfully obtained accurate values of binding energies and sizes for three successive Efimov states. Compared with the predicted scaling constants of the zero-range theory, the ground Efimov states are found to have the largest finite-range corrections. There exists a universal size-binding momentum relation for Efimov states which can be described by the formula Rn∗kn∗=(1+s02)/3{\mathcal{R}}_{{n}^{*}}{k}_{{n}^{*}}=\sqrt{(1+{s}_{0}^{2})/3}Rn∗​kn∗​=(1+s02​)/3​ derived with zero-range approximation. It is found that for the Efimov states that have large scaled sizes, their size-binding momentum relations follow the universal formula well. Our calculations demonstrate that size is an important characteristic in determining the universalities of Efimov states.
Universal behavior has been found inside the window of Efimov physics for systems with N=4,5,6N=4,5,6N=4,5,6 particles. Efimov physics refers to the emergence of a number of three-body states in systems of identical bosons interacting via a short-range interaction becoming infinite at the verge of binding two particles. These Efimov states display a discrete scale invariance symmetry, with the scaling factor independent of the microscopic interaction. Their energies in the limit of zero-range interaction can be parametrized, as a function of the scattering length, by a universal function. We have found, using the form of finite-range scaling introduced by A. Kievsky and M. Gattobigio [Phys. Rev A 87, 052719 (2013)], that the same universal function can be used to parametrize the ground and excited energy of $N\ensuremath{\le}6$ systems inside the Efimov-physics window. Moreover, we show that the same finite-scale analysis reconciles experimental measurements of three-body binding energies with the universal theory.
Some aspects of universal properties are discussed in few-atom as well as in few- nucleon systems. When the two-body scattering length a is large the three-body system presents some peculiar characteristics such as the Efimov effect. In the present contribution, the zerorange theory deduced by V. Efimov is compared to results obtained using potential models. Accordingly range corrections are introduced. These corrections appear as a shift in the variable K * a with K * the three-body parameter. The three-boson spectrum, atom-dimer scattering and the recombination rate at threshold are analyzed using the finite-range theory proposed. In the case of nuclear systems, n – d scattering is discussed using the universal formula derived by Efimov. Furthermore, the spectrum up to N = 6 is discussed using the central Volkov potential.
Using potential models, we analyze range corrections to the universal law dictated by the Efimov theory of three bosons. In the case of finite-range interactions, we have observed that at first order, it is necessary to supplement the theory with one finite-range parameter ${\mathrm{\ensuremath{\Gamma}}}_{n}^{3}$ for each specific nnn level [A. Kievsky and M. Gattobigio, Phys. Rev. A 87, 052719 (2013)]. The value of ${\mathrm{\ensuremath{\Gamma}}}_{n}^{3}$ depends on the way the potentials are changed to tune the scattering length toward the unitary limit. In this work, we analyze a particular path in which the length ${r}_{B}=a\ensuremath{-}{a}_{B}$, measuring the difference between the two-body scattering length aaa and the energy-scattering length aB{a}_{B}aB​, is almost constant. Analyzing systems with very different scales, such as atomic or nuclear systems, we observe that the finite-range parameter remains almost constant along the path with a numerical value of ${\mathrm{\ensuremath{\Gamma}}}_{0}^{3}\ensuremath{\approx}0.87$ for the ground-state level. This observation suggests the possibility of constructing a single universal function that incorporates finite-range effects for this class of paths. The result is used to estimate the three-body parameter ${\ensuremath{\kappa}}_{*}$ in the case of real atomic systems brought to the unitary limit through broad Feshbach resonances. Furthermore, we show that the finite-range parameter can be put in relation to the two-body contact C2{C}_{2}C2​ at the unitary limit.
We calculate shallow three-body bound states in the universal regime, defined by Efimov, with inclusion of both scattering length and effective range parameters. We find corrections to the universal scaling laws for large binding energies. For narrow resonances we find a distinct nonmonotonic behavior of the threshold at which the lowest Efimov trimer merges with the three-body continuum. The origin of the three-body parameter is related to the two-body atom-atom interactions in a physically clear model. Our results demonstrate that experimental information from narrow Feshbach resonances and/or mixed systems are of vital importance to pin down the relation of two- and three-body physics in atomic systems.
The bound states of a system consisting of two heavy identical atoms and one light electron interacting through finite-range pairwise potentials are explored, focusing on their dependence on the electron-atom scattering length. In the case of an exact resonance in the electron-atom interaction, the binding energy of an electron yields an effective 1/r21/{r}^{2}1/r2 potential for the relative motion of the atoms. One major finding is a universal potential that depends on the polarization length, which goes beyond the Efimov region. An analytic expression for that potential is extracted from numerical calculations. The spectrum of the e+Rb+Rbe+\mathrm{Rb}+\mathrm{Rb}e+Rb+Rb system produced by the electron-atom polarization interaction exhibits three main sections: a nonuniversal transition region, a quasi-Efimov region, and a densely packed Efimov region.
We calculate the low-lying energy spectrum for three trapped identical bosons interacting via a finite-range potential. The spectrum is compared to results from the zero-range model. The thresholds for trimer binding and atom-dimer binding are extracted and related to effective range corrections. Effective range corrections to Efimov physics and Borromean binding are two aspects of the same effect, and we connect these two regions qualitatively. The Borromean window becomes slightly narrower for substantial effective ranges. The structure at the atom-dimer threshold is an atom far away from the dimer and the major energy correction is due to the change of dimer energy with effective range. This structure becomes less pronounced when the effective range increases.