Universal range corrections to Efimov trimers for a class of paths to the unitary limit

Abstract

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, Γ 3 n , for each specific nlevel [Kievsky and Gattobigio, Phys. Rev. A 87, 052719 (2013)]. The value of Γ 3 n depends on the way the potentials is changed to tune the scattering length toward the unitary limit. In this work we analyze a particular path in which the length rB = a − aB, measuring the difference between the two-body scattering length a and the energy scattering length aB, results almost constant. Analyzing systems with very different scales, as atomic or nuclear systems, we observe that the finite-range parameter remains almost constant along the path with a numerical value of Γ 3 0 ≈ 0.87 for the ground state level. This observation suggests the possibility of constructing a single universal function that incorporate finite-range effects for this class of paths. The result is used to estimate the three-body parameter κ * in the case of real atomic systems brought to the unitary limit thought a broad Feshbach resonances. Furthermore, we show that the finite-range parameter can be put in relation with the two-body contact C2 at the unitary limit.

Universal Behavior of Few-Boson Systems Using Potential Models

“…First we have analysed the product κ n * a B from which we have defined the gaussian level function ∆ n (ξ ). In second place we have analysed the ratio a AD /a B and the gaussian values for this ratio have been used to determine a AD for the case of atom-dimer collision of three 4 He atoms at zero energy. From the results we observe that ξ dependence is well verified and, the gaussian model reproduces well the LM2M2 value for a AD .…”

“…As an application we calculate the value of the atom-dimer scattering length a AD for a system composed by three 4 He atoms. This system is well described by the realistic LM2M2 potential of Aziz [8] for which E 2 = 1.303 mK.…”

“…The potentials defined in Eqs. (4,5) can be used to describe the three-boson system close to the unitary limit. The Efimov law of Eq.…”

“…In fact the gaussian potentials can be used to determine the dependence on the angle ξ of the observables defined above. The first one, κ * a, is used to define the gaussian level function [4]. Moreover a gaussian potential verifying E 3 /E 2 = tan 2 ξ can be used to compute the ratio a AD /a at that angle.…”

“…Moreover a gaussian potential verifying E 3 /E 2 = tan 2 ξ can be used to compute the ratio a AD /a at that angle. After introducing range corrections we compute this ratio and, as an application, we determine a AD for the case of a system composed by three 4 He atoms.…”