Feshbach resonances in ultracold ${}^{85}\mathrm{Rb}\text{−}{}^{87}\mathrm{Rb}$ and ${}^6\mathrm{Li}\text{−}{}^{87}\mathrm{Rb}$ mixtures

We present an analysis of experimentally accessible magnetic Feshbach resonances in ultracold heteronuclear ${}^{85}\mathrm{Rb}\text{−}{}^{87}\mathrm{Rb}$ and ${}^6\mathrm{Li}\text{−}{}^{87}\mathrm{Rb}$ mixtures. Using recent experimental measurements of the triplet scattering lengths for ${}^6\mathrm{Li}\text{−}{}^{87}\mathrm{Rb}$ and ${}^7\mathrm{Li}\text{−}{}^{87}\mathrm{Rb}$ mixtures and Feshbach resonances for one combination of atomic states, we create model potential curves and fine-tune them to reproduce the measured resonances and to predict the location of several experimentally relevant resonances in Li-Rb collisions. To model ${}^{85}\mathrm{Rb}\text{−}{}^{87}\mathrm{Rb}$ collisions, we use accurate ${\mathrm{Rb}}_2$ potentials obtained previously from the analysis of experiments on ${}^{87}\mathrm{Rb}\text{−}{}^{87}\mathrm{Rb}$ collisions. We find resonances that occur at very low magnetic fields, below $10\mathrm{G}$, which may be useful for entanglement generation in optical lattices or atom chip magnetic traps.

Figure 1

Elastic scattering cross sections for $s$-wave collisions of ${}^{85}\mathrm{Rb}$ and ${}^{87}\mathrm{Rb}$ atoms in the lowest weak magnetic field seeking hyperfine state ${\mid 2,-2⟩}_{85}\otimes {\mid 1,-1⟩}_{87}$ versus the applied magnetic field (upper panel). The collision energy is $144\mathrm{nK}$ (see text). The lower panels show more details of the resonances.

Figure 2

(Color online) Magnetic field dependence of $s$-wave elastic scattering cross sections for ${}^{85}\mathrm{Rb}\text{−}{}^{87}\mathrm{Rb}$ in different hyperfine states. Full line, ${\mid 2,0⟩}_{85}\otimes {\mid 1,-1⟩}_{87}$; long dashed line, ${\mid 2,-1⟩}_{85}\otimes {\mid 1,-1⟩}_{87}$; short dashed line, ${\mid 2,1⟩}_{85}\otimes {\mid 1,-1⟩}_{87}$. The collision energy is $144\mathrm{nK}$.

Figure 3

(Color online) Molecular bound state energies versus magnetic field computed with the asymptotic bound state model. The threshold for the ${\mid \frac{1}{2},\frac{1}{2}⟩}_{{}^6\mathrm{Li}}\otimes {\mid 1,1⟩}_{{}^{87}\mathrm{Rb}}$ collision channel is shown by the solid line while the dashed (dotted) lines indicate the $s$-wave ($p$-wave) states. These molecular state energies were computed given the least bound states $E_S^l$ of the optimal singlet and triplet potentials of $E_0^{l=0}=-0.106{\mathrm{cm}}^{-1}$, $E_0^{l=1}=-0.0870{\mathrm{cm}}^{-1}$, and $E_1^{l=0}=-0.137{\mathrm{cm}}^{-1}$, $E_1^{l=1}=-0.116{\mathrm{cm}}^{-1}$. The predicted resonance locations are close to the actual locations determined by the full coupled-channel calculation and are indicated by the solid dots. Near $890\mathrm{G}$ $\left(A\right)$ the threshold crosses a $p$-wave molecular state and the corresponding $p$-wave elastic scattering cross section shown in Fig. 5 is observed to rapidly diverge and then return to the background level. Likewise, near $1070\mathrm{G}$ $\left(B\right)$ the threshold crosses both a $p$-wave and an $s$-wave molecular state and both the $s$-wave and $p$-wave elastic scattering cross sections are affected. Finally, near $1300\mathrm{G}$ $\left(C\right)$ a second $s$-wave-induced Feshbach resonance occurs.

Figure 4

(Color online) Locus of points in the $(E_{\text{singlet}},E_{\text{triplet}})$ parameter space where an $s$-wave resonance occurs at one of the two experimentally determined locations 882.02 [gray (green)] or $1066.92\mathrm{G}$ [dark (red)] for atoms in the ${\mid \frac{1}{2},\frac{1}{2}⟩}_{{}^6\mathrm{Li}}\otimes {\mid 1,1⟩}_{{}^{87}\mathrm{Rb}}$ state. The dotted lines indicate the approximate values for $E_{\text{singlet}}$ and $E_{\text{triplet}}$ beyond which a new bound state enters the potential at zero energy. There are four regions (I–IV) indicated on the plot where an $s$-wave resonance occurs simultaneously at 882.02 and at $1066.92\mathrm{G}$. Region V indicates a range of values for which an $s$-wave resonance occurs at 1066.92 while a $p$-wave resonance (not represented in this plot) occurs at $882\mathrm{G}$. For each of these five candidate regions, the character of the predicted elastic cross sections as a function of magnetic field was studied, and the results of this analysis are discussed in the text.

Figure 5

(Color online) Magnetic field dependence of the $s$-wave (upper panel) and $p$-wave (lower panel) elastic scattering cross sections for atoms in the ${\mid \frac{1}{2},\frac{1}{2}⟩}_{{}^6\mathrm{Li}}\otimes {\mid 1,1⟩}_{{}^{87}\mathrm{Rb}}$ state. These results are from the coupled-channel calculations for a collision energy of $144\mathrm{nk}$ and using the optimal singlet and triplet potentials. Only the $m_l=0$ contribution to the $p$-wave elastic scattering cross section is shown. Two $s$-wave resonances occur at 1065 and $1278\mathrm{G}$, while two $p$-wave resonances occur at 882 and $1066\mathrm{G}$.