Prediction of Feshbach resonances from three input parameters

We have developed a model of Feshbach resonances in gases of ultracold alkali-metal atoms using the ideas of multichannel quantum-defect theory. Our model requires just three parameters describing the interactions—the singlet and triplet scattering lengths, and the long-range van der Waals coefficient—in addition to known atomic properties. Without using any further details of the interactions, our approach can accurately predict the locations of resonances. It can also be used to find the singlet and triplet scattering lengths from measured resonance data. We apply our technique to ${}^6\text{L}\text{i-}{}^{40}\text{K}$ and ${}^{40}\text{K}\text{−}{}^{87}\text{R}\text{b}$ scattering, obtaining good agreement with experimental results, and with the more computationally intensive coupled channels technique.

Figure 1

(Color online) Bound-state energies (lower panel) and scattering phase shift ${\text{sin}}^2{\delta }_0$ (upper panel) for the $ac$ channel of ${}^6\text{L}\text{i-}{}^{40}\text{K}$ as a function of magnetic field, calculated from our three parameter MQDT model. The inset shows a close-up of the 15.81 mT resonance, illustrating the narrow region within which the Feshbach molecular state has an appreciable entrance channel component [1].

Figure 2

(Color online) Comparison of ${}^{40}\text{K}\text{−}{}^{87}\text{R}\text{b}$ $aa$ resonance locations predicted by different approaches. The upper panel shows scattering length vs magnetic field. The MQDT results are given by the approach presented in this Rapid Communication. Other results are given by coupled channels calculations with potentials of the form $-C_6/r^6+C_{12}/r^{12}$ (6–12), $-C_6/r^6-C_8/r^8+C_{12}/r^{12}$ (6–8–12), and the potentials of Ref. [26] (full). The full potentials were also used to calculate the bound-state spectrum (“○”), which is compared to an MQDT calculation (“$+$”) in the lower panel. Bound states are numbered according to the vibrational quantum number of the singlet and triplet states which give rise to them. Note that the two plots have different ranges of magnetic field.