Efimov physics and the three-body parameter within a two-channel framework

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.

Figure 1

The threshold scattering length $a^{(-)}$ at which the lowest universal Efimov trimer merges with the three-atom continuum plotted against the strength, $s_{\mathrm{res}}$, of the Feshbach resonance. The right-hand side corresponds to broad resonances. The different curves of the linetypes show results with different three-body parameters, ${\rho }_c$, in units of the atomic van der Waals length, $r_{\mathrm{vdW}}$. The values of ${\rho }_c$ decrease from top to bottom (precise values are given in the text). Experimental values are from ${}^{133}$Cs [7, 15], ${}^7$Li [16], ${}^{39}$K [17], and ${}^{85}$Rb [18].

Figure 2

Trimer binding energy $E_T$ vs scattering length $a$ for the one- and two-channel models with effective range $r_e=-5r_{\mathrm{vdW}}$. $a^{(-)}$ is the threshold scattering length for the existence of the lowest trimer. Finely dotted lines indicate atom-dimer threshold for positive $a$. Successive states have a scaling of 515 [1] and for visibility the axes are scaled by the power $1/8$.

Figure 3

The lowest adiabatic potentials multiplied by ${\rho }^2$ as functions of hyper-radius, $\rho$, for the zero-range one-channel model and three different effective ranges for the two-channel model.

Figure 4

Typical two-body neutral atom-atom potentials, including the hardcore van der Waals potential of Eq. (3), the Lennard-Jones potential, $V_{\text{LJ}}\left(r\right)=\frac{C_{10}}{r^{10}}-\frac{C_6}{r^6}$, and the Morse potential, $V_M\left(r\right)=4{\epsilon }_0(e^{-2\alpha (r-r_0)}-e^{-\alpha (r-r_0)})$. Here $C_6$, $C_{10}$, and $\alpha$ are parameters. They have all been fitted to have the same overall energy, ${\epsilon }_0$, at the radius of the potential minimum, $r_{\min }$.

Figure 5

Semianalytic results for $a^{(-)}$ plotted against number of bound states in the two-body van der Waals plus hard-core potential. The horizontal position of the experimental data is arbitrary. The grey band indicates the 15$\%$ margin around the mean value of $\sim$9.8.

Figure 6

Schematic of the (a) triangular and (b) linear configurations for an equal mass three-body system. ${\mathit{r}}_{12}$ and ${\mathit{r}}_{12,3}$ indicate the two Jacobi relative vectors.