Efimov physics in ${}^6\mathrm{Li}$ atoms

A new narrow three-atom loss resonance associated with an Efimov trimer crossing the three-atom threshold has recently been discovered in a many-body system of ultracold ${}^6\mathrm{Li}$ atoms in the three lowest hyperfine spin states at a magnetic field near 895 G. O’Hara and coworkers have used measurements of the three-body recombination rate in this region to determine the complex three-body parameter associated with Efimov physics. Using this parameter as the input, we calculate the universal predictions for the spectrum of Efimov states and for the three-body recombination rate in the universal region above 600 G where all three scattering lengths are large. We predict an atom-dimer loss resonance at $672\pm 2$ G associated with an Efimov trimer disappearing through an atom-dimer threshold. We also predict an interference minimum in the three-body recombination rate at $759\pm 1$ G where the three-spin mixture may be sufficiently stable to allow experimental study of the many-body system.

Figure 1

The scattering lengths in units of ${10}^3{a}_0$ for the three lowest hyperfine states of ${}^6\mathrm{Li}$ as functions of the magnetic field $B$ from 0 to 600 G [42]. The two vertical dotted lines mark the boundaries of the region in which the absolute values of all three scattering lengths are greater than $2{\ell }_{\mathrm{vdW}}$.

Figure 2

Three-body recombination rate constant $K_3$ as a function of the magnetic field $B$ from 0 to 600 G. The solid squares and dots are data points from Refs. [26] and [27], respectively. The curve is the absolutely normalized result for $K_3^{\mathrm{deep}}$ with ${\kappa }_{*}=76.8a_0^{-1}$ and ${\eta }_{*}=0.11$. The two vertical dotted lines mark the boundaries of the region in which the absolute values of all three scattering lengths are greater than $2{\ell }_{\mathrm{vdW}}$.

Figure 3

Energy of the Efimov trimer as a function of the magnetic field $B$ in the low-field region. The binding frequency $E_T^{(1)}/(2\pi \hslash )$ (dark solid line) and the frequency ${\Gamma }_T^{(1)}/(2\pi \hslash )$ associated with the width (difference between dark dashed lines) were obtained from the complex energy eigenvalue calculated using the parameters ${\kappa }_{*}=76.8a_0^{-1}$ and ${\eta }_{*}=0.11$. Also shown for comparison are the small-${\eta }_{*}$ approximation in Eqs. (24) and (25) for the binding frequency (light solid line) and for the frequency associated with the width (difference between light dashed lines). The two vertical dotted lines mark the boundaries of the region in which the absolute values of all three scattering lengths are greater than $2{\ell }_{\mathrm{vdW}}$.

Figure 4

The scattering lengths in units of ${10}^3{a}_0$ for the three lowest hyperfine states of ${}^6\mathrm{Li}$ as functions of the magnetic field $B$ from 600 G to 1200 G [42]. The three vertical lines mark the positions of the Feshbach resonances.

Figure 5

Three-body recombination rate constant $K_3$ as a function of the magnetic field $B$ from 600 G to 1200 G. The solid dots and triangles are data points from Ref. [27] at temperatures less than 180 nK and less than 30 nK, respectively. The curve between 834 G and 1200 G is a fit to that data. In the region from 600 G and 834 G, the curves are our predictions for the total three-body recombination rate (thick line) and the contribution from recombination into deep dimers (thin line) for ${\kappa }_{*}=80.7a_0^{-1}$ and ${\eta }_{*}=0.016$. The three vertical lines mark the positions of the Feshbach resonances.

Figure 6

Energies of the Efimov trimers as functions of the magnetic field $B$ in the high-field region. The solid curves are the predicted binding frequencies $E_T^{(n)}/(2\pi \hslash )$ for ${\kappa }_{*}=80.7a_0^{-1}$ and ${\eta }_{*}=0.016$. The dashed curves are the upper and lower error bars obtained by varying ${\kappa }_{*}$. The curves labeled 12, 23, and 13 are the atom-dimer thresholds. The dots indicate the points where the trimers disappear through the 1 + (23) atom-dimer threshold. The horizontal dotted line is the van der Waals frequency 154 MHz.

Figure 7

Widths of the Efimov trimers as functions of the magnetic field $B$ in the high-field region. The solid curves are the frequencies ${\Gamma }_T^{(n)}/(2\pi \hslash )$ associated with the widths for ${\kappa }_{*}=80.7a_0^{-1}$ and ${\eta }_{*}=0.016$. The dashed curves are the upper and lower error bars obtained by varying ${\eta }_{*}$.

Figure 8

Three-body recombination rate constant $K_3$ as a function of the magnetic field $B$ from 650 G to 850 G. The curves are our predictions for the total three-body recombination rate (thick solid line), the contribution from recombination into deep dimers (thin solid line), and the contributions from recombination into the (12) dimer (short-dashed line), the (23) dimer (dash-dotted line), and the (13) dimer (long-dashed line) for ${\kappa }_{*}=80.7a_0^{-1}$ and ${\eta }_{*}=0.016$.

Figure 9

Dimer relaxation rate constant ${\beta }_{1(23)}$ for (23) dimers and atoms of type 1 as a function of the magnetic field $B$. The magnetic field is measured relative to the position $B_{*}$ of the atom-dimer resonance. The curves are predictions using the approximation in Eq. (32) for three values of ${\eta }_{*}$: 0.006, 0.016, and 0.022.