Enhancement of the formation of ultracold ${}^{85}\mathrm{Rb}_2$ molecules due to resonant coupling

We have studied the effect of resonant electronic-state coupling on the formation of ultracold ground-state ${}^{85}\mathrm{Rb}_2$. Ultracold ${\mathrm{Rb}}_2$ molecules are formed by photoassociation (PA) to a coupled pair of $0_u^{+}$ states, $0_u^{+}\left(P_{1∕2}\right)$ and $0_u^{+}\left(P_{3∕2}\right)$, in the region below the $5S+5P_{1∕2}$ limit. Subsequent radiative decay produces high vibrational levels of the ground state, $X{}^1\Sigma _g^{+}$. The population distribution of these $X$-state vibrational levels is monitored by resonance-enhanced two-photon ionization through the $2{}^1\Sigma _u^{+}$ state. We find that the populations of vibrational levels $v^{″}=112–116$ are far larger than can be accounted for by the Franck-Condon factors for $0_u^{+}\left(P_{1∕2}\right)\to X{}^1\Sigma _g^{+}$ transitions with the $0_u^{+}\left(P_{1∕2}\right)$ state treated as a single channel. Further, the ground-state molecule population exhibits oscillatory behavior as the PA laser is tuned through a succession of $0_u^{+}$ state vibrational levels. Both of these effects are explained by a calculation of transition amplitudes that includes the resonant character of the spin-orbit coupling of the two $0_u^{+}$ states. The resulting enhancement of more deeply bound ground-state molecule formation will be useful for future experiments on ultracold molecules.

Figure 1

(Color online) Selected potential curves for ${\mathrm{Rb}}_2$ from Ref. 14, showing our scheme for forming ultracold ${}^{85}\mathrm{Rb}_2$ molecules in the $X{}^1\Sigma _g^{+}$ state. Photoassociation (PA) is used to excite high-$v$ levels of the coupled $0_u^{+}$ states, $0_u^{+}\left(P_{1∕2}\right)$ and $0_u^{+}\left(P_{3∕2}\right)$, converging to the $5S+5P_{1∕2}$ and $5S+5P_{3∕2}$ atomic limits. These states correlate at short range to the $A{}^1\Sigma _u^{+}$ and $b{}^3\Pi _u$ states. Spontaneous emission (SE) from these levels populates the ground $X$ state. Pulsed laser light at $600\mathrm{nm}$ is used for ionization detection.

Figure 2

(Color online) The squares of the vibrational eigenfunctions of the $0_u^{+}$ coupled states for binding energies of $9.39{\mathrm{cm}}^{-1}$ and $11.27{\mathrm{cm}}^{-1}$ (red solid line, triplet component; blue dotted line, singlet component). The vertical position of the square of each wave function corresponds to its binding energy. The potentials (thick solid lines) are drawn in order to illustrate how the structure of the coupled wave functions can be understood in terms of classical turning points.

Figure 3

(Color online) Franck-Condon factors between ground-state vibrational levels $v^{″}$ and the $0_u^{+}$ vibrational levels with binding energies of $9.39{\mathrm{cm}}^{-1}$ (top) and $11.27{\mathrm{cm}}^{-1}$ (bottom). Enhancement of Franck-Condon factors by an order of magnitude due to the resonant coupling is observed for the $X{}^1\Sigma _g^{+}$ levels $v^{″}=113$ $(E_{\mathrm{bind}}=6.9{\mathrm{cm}}^{-1})$ and $v^{″}=114$ $(E_{\mathrm{bind}}=5.0{\mathrm{cm}}^{-1})$.

Figure 4

(Color online) Solid curve with solid (red) diamonds: sums of calculated Franck-Condon factors from each of 25 vibrational levels of the coupled $0_u^{+}$ states to ground-state levels with $v^{″}=112–116$, revealing strong oscillations in ground-state molecule formation. The $0_u^{+}$ level energies are shown as a function of PA laser detuning below the $5S+5P_{1∕2}$ limit for comparison with experimental data. Crosses: experimental data set 1. Circles: experimental data set 2, taken on a different day under improved conditions as explained in Sec. 4. The typical systematic error bar shown is based on upper and lower bounds for systematic corrections as explained in the text.

Figure 5

(Color online) Comparison of experimental detection laser scans for PA detunings corresponding to the minimum and maximum effects of resonant coupling as predicted from the calculated curve in Fig. 4. Ionization signals for binding energies of (a) $6.02{\mathrm{cm}}^{-1}$ (Min), (b) $9.39{\mathrm{cm}}^{-1}$ (Max), (c) $11.27{\mathrm{cm}}^{-1}$ (Min), and (d) $16.27{\mathrm{cm}}^{-1}$ (Max) are shown. For easy comparison, the vibrational levels $v^{″}$ are marked by vertical lines.