Towards the production of ultracold ground-state RbCs molecules: Feshbach resonances, weakly bound states, and the coupled-channel model

We have studied interspecies scattering in an ultracold mixture of ${}^{87}$Rb and ${}^{133}$Cs atoms, both in their lowest-energy spin states. The three-body loss signatures of 30 incoming $s$- and $p$-wave magnetic Feshbach resonances over the range 0 to 667 G have been cataloged. Magnetic field modulation spectroscopy was used to observe molecular states bound by up to 2.5 $\text{MHz}\times h$. We have created RbCs Feshbach molecules using two of the resonances. Magnetic moment spectroscopy along the magnetoassociation pathway from 197 to 182 G gives results consistent with the observed and calculated dependence of the binding energy on magnetic field strength. We have set up a coupled-channel model of the interaction and have used direct least-squares fitting to refine its parameters to fit the experimental results from the Feshbach molecules, in addition to the Feshbach resonance positions and the spectroscopic results for deeply bound levels. The final model gives a good description of all the experimental results and predicts a large resonance near 790 G, which may be useful for tuning the interspecies scattering properties. Quantum numbers and vibrational wave functions from the model can also be used to choose optimal initial states of Feshbach molecules for their transfer to the rovibronic ground state using stimulated Raman adiabatic passage.

Figure 1

Molecular potential energy curves $V_0\left(R\right)$ and $V_1\left(R\right)$ for the singlet and triplet states of RbCs correlating with two separated ${}^2{S}_{1/2}$ ground-state atoms. The inset shows an expanded view of the long-range potentials separating to the four different hyperfine thresholds at zero field, labeled by $(f_{{}^{87}\mathrm{Rb}},f_{{}^{133}\mathrm{Cs}})$, but no longer by singlet or triplet.

Figure 2

Thresholds for ${}^{87}$Rb${}^{133}$Cs and the bins (vertical boxes) below each threshold within which each vibrational state must lie for any value of the scattering length. The horizontal boxes, 1.5 GHz deep, show the energy range within which bound levels can cause resonances at the $|1,1\rangle +|3,3\rangle$ and $|2,-1\rangle +|3,3\rangle$ thresholds at fields under 500 G. The vibrational bin for each hyperfine state that contains bound states that can cause resonances at the lowest threshold is colored. Selected levels are shown (short horizontal lines) as a function of $L$ ($s$, $p$, $d$, $f$, $g$ for $L=0$, 1, 2, 3, 4, respectively) for the specific choice of scattering length that gives a least-bound state for $L=0$ with 110(20) kHz below threshold. For fields below 500 G, only levels $|n(f_{\mathrm{Rb}},f_{\mathrm{Cs}})\rangle =|-2(1,3)\rangle$ and $\left|-6\right(2,4)\rangle$ can cause resonances at the $|1,1\rangle +|3,3\rangle$ threshold and only levels $\left|-4\right(1,4)\rangle$ and $\left|-2\right(2,3)\rangle$ can cause resonances at the $|2,-1\rangle +|3,3\rangle$ threshold.

Figure 3

Example of a Feshbach resonance scan showing simultaneous loss for Rb and Cs near 197.06(5) G for a hold time of 200 ms. The maximum for the number of Rb atoms to the right of the resonance is attributed to the zero crossing for the interspecies scattering length. The difference in the positions of the minima and the maximum as indicated gives an estimate for the resonance width $\Delta$.

Figure 4

Measurement of the binding energy of the Feshbach molecules. This is an example of a field modulation free-bound resonance scan showing simultaneous loss for Rb and Cs for a hold time of 900 ms. The modulation frequency is held fixed at $f_{\mathrm{m}}=330$ kHz.

Figure 5

Weakly bound states of RbCs obtained by free-bound (red, dark-gray) and bound-free (green, gray) magnetic-field modulation spectroscopy, together with levels calculated for the final, fitted potentials (blue solid lines). All levels are shown relative to dissociation limit Rb $|1,1\rangle$ $+$ Cs $|3,3\rangle$ at the given magnetic field value $B$. Avoided crossings between the least-bound state and the ramping $n=-2$ levels are shown as arrows. The smaller panels labeled (a), (b), and (c) refer to the areas on the larger panel marked by rectangles and the same labels. The quantity $2\beta$ is the minimum separation in energy between the two states.

Figure 6

Calculated (solid line) and measured (red points) molecular magnetic moment along the magnetoassociation route starting at the 197.06 G Feshbach resonance. The two horizontal error bars mark the estimated field values for two weak avoided crossings, which we detect by variations in the cloud displacement at fixed gradient field; see inset. The crossings are visible here as spikes in the calculated curve and are indicated more explicitly in Figs. 5b and 5c. (Inset) Cloud displacement as a function of magnetic field $B$ near the weak avoided crossing at $B=189.50$ G. The strong variation in the displacement data indicates the presence of the crossing. Similar data are obtained for the weak avoided crossing near $B=185.17$ G.

Figure 7

Study of the second-order spin-orbit interaction energy $V_{1,1}\left(R\right)-V_{1,0^{-}}\left(R\right)$ as a function of internuclear separation $R$. The solid circles are the result of the ab initio RCI VB electronic structure calculation. The green dash-dotted line is a fit to the ab initio values using the functional form of Eq. (12). The blue line corresponds to the second-order spin-orbit interaction energy optimized to reproduce the location of the observed magnetic Feshbach resonances for $A_{2\mathrm{SO}}^{\mathrm{long}}=-0.03310$. For comparison, the absolute value of the corresponding splitting due to the magnetic dipole-dipole interaction is shown by a red dashed line.

Figure 8

Weakly bound states of RbCs for $M=4$ at fields up to 550 G, calculated using the final fitted potentials, together with the scattering length at the $|1,1\rangle +|3,3\rangle$ threshold, calculated at $E=160$ nK. Bound states are plotted in a color corresponding to their value of $M_F$, as shown on the figure. States arising from $L=2$ ($d$ states) are shown as solid lines, and from $L=0$ ($s$ states) are shown as dashed lines. The resonance positions are marked on the scattering length plot as vertical lines with the same color as the bound state that they arise from. The slanted text on the left-hand axis and below the bottom axis indicates which vibrational and hyperfine manifold the $L=2$ bound states arise from. The least-bound state $\left|-1\right(1,3\left)s\right(1,3)\rangle$ is within the thickness of the zero line on this scale. The observed positions of incoming $s$-wave resonances are shown as arrows above the plot.

Figure 9

The $p$-wave scattering lengths at the $|1,1\rangle +|3,3\rangle$ threshold for the three values of $M$ allowed for $p$-wave scattering of $\mathrm{Rb}+\mathrm{Cs}$, calculated at $E=7\mu$K using the final fitted potentials. Results for $M_F=3$, 4, and 5 are shown as blue (dark gray), magenta (gray), and green (light gray), respectively. The observed positions of $p$-wave resonances are shown as arrows at the top of the graph.

Figure 10

The real and imaginary parts of the complex scattering length at the Rb $|2,-1\rangle$ $+$ Cs $|3,3\rangle$ threshold, calculated using the final fitted potentials: $s$ wave (red, dark gray, left-hand axis) and $p$-wave (green, light gray, right-hand axis). The observed resonance positions are shown as arrows at the top of the graph. The three components of the $p$-wave scattering length are indistinguishable on this scale.

Figure 11

RbCs scattering length at the $|1,1\rangle +|3,3\rangle$ threshold at fields above 560 G, calculated at $E=160$ nK using the final fitted potential. Resonance positions are marked by vertical lines, with the value of $M_F$ of the corresponding bound state indicated using the same color scheme as in Fig. 8.