Efimov–van der Waals universality for ultracold atoms with positive scattering lengths

We study the universality of the three-body parameters for systems relevant for ultracold quantum gases with positive $s$-wave two-body scattering lengths. Our results account for finite-range effects and their universality is tested by changing the number of deeply bound diatomic states supported by our interaction model. We find that the physics controlling the values of the three-body parameters associated with the ground and excited Efimov states is constrained by a variational principle and can be strongly affected by $d$-wave interactions that prevent both trimer states from merging into the atom-dimer continuum. Our results enable comparisons to current experimental data and they suggest tests of universality for atomic systems with positive scattering lengths.

Figure 1

Energy of Efimov states calculated near the first three poles of $a, \lambda ={\lambda }_1^{*}, {\lambda }_2^{*}$, and ${\lambda }_3^{*}$, in our model potential in Eq. (2). Near ${\lambda }_1^{*}$, Efimov states are true bound states, while near ${\lambda }_2^{*}$ and ${\lambda }_3^{*}$ , Efimov states are resonant states with the corresponding widths indicated as the error bars. Approximated values for $a_{-}, a_{*}$, and $a_{+}$ are also indicated.

Figure 2

(a) Binding energies, $E_b=E_{2\mathrm{b}}-E_{3\mathrm{b}}$, of Efimov states near ${\lambda }_1^{*}$ and ${\lambda }_2^{*}$, showing that both ground and first excited Efimov states fail to merge into the atom-dimer threshold (see text). In (b) and (c) we show a blow up of (a) near the second and first excited Efimov states, respectively.

Figure 3

(a) Atom-dimer scattering length $a_{\mathrm{ad}}$ and (b) corresponding loss rate $\beta$ displaying resonant behavior due to Efimov resonances associated with the first and second excited Efimov states. The values of the three-body parameters $a_{*,1}$ and $a_{*,2}$ are also indicated. The dashed curve gives the analytical zero range results from Ref. [23].

Figure 4

Three-body recombination $K_3$ displaying interference minima associated with the ground and first excited Efimov states. Values of the three-body parameters $a_{+,0}$ and $a_{+,1}$ are indicated The dashed curve gives the analytical zero range result in the absence of deeply bound dimers [2, 38]. Inset: Thermally averaged recombination rate $\langle K_3\rangle$ illustrating the temperature dependence of $\langle a_{+,1}\rangle$.