Hyperfine, rotational, and vibrational structure of the $a^3{\Sigma }_u^{+}$ state of ${}^{87}\mathrm{Rb}$${}_2$

We have performed high-resolution two-photon dark-state spectroscopy of an ultracold gas of ${}^{87}\mathrm{Rb}$${}_2$ molecules in the $a^3{\Sigma }_u^{+}$ state at a magnetic field of about 1000 G. The vibrational ladder as well as the hyperfine and low-lying rotational structure are mapped out. Energy shifts in the spectrum are observed due to singlet-triplet mixing at binding energies as deep as a few hundred $\text{GHz}\times h$. This information, together with data from other sources, is used to optimize the potentials of the $a^3{\Sigma }_u^{+}$ and $X^1{\Sigma }_g^{+}$ states in a coupled-channel model. We find that the hyperfine structure depends weakly on the vibrational level. This provides a possible explanation for inaccuracies in recent Feshbach resonance calculations.

Figure 1

Dark-state spectroscopy scheme for the ${}^{87}\mathrm{Rb}$${}_2$ $a^3{\Sigma }_u^{+}$ potential. Lasers 1 and 2 couple the molecular levels $|i\rangle$ and $|v\rangle$ to the excited level $|e\rangle$ with Rabi frequencies ${\Omega }_{1,2}$, respectively. Laser 1 is kept on resonance while laser 2 can be tuned to any level of the $a^3{\Sigma }_u^{+}$ potential. (Inset) Bound-state level of Feshbach molecules as a function of magnetic field $B_z$. The dashed line gives the position of the Feshbach resonance. The dot marks the Feshbach molecule state used in the experiments.

Figure 2

(a) Binding energies $E_B(v)$ for the state $a^3{\Sigma }_u^{+}$, where $v$ is the vibrational quantum number. The line is the result of a coupled-channel model calculation after optimization of the Born-Oppenheimer potential. Five levels were not measured. (b) Residues, that is, the difference between experimental data and theory. Large error bars belong to early measurements without simultaneous wave-meter readings of both lasers.

Figure 3

Scans of two vibrational levels: (a) $v=6$, (b) $v=0$. Plotted is the remaining molecular fraction as a function of the binding energy $E_B$, which basically corresponds to the laser difference frequency. The scans were recorded using an excited state $|e\rangle$ with $1_g$ character. The labels s and d indicate rotational states $N=0$ and 2, respectively. The numbers after the labels s or d indicate the position in the spectrum.

Figure 4

Scan of the hyperfine, rotational, and Zeeman structure in the $a^3{\Sigma }_u^{+}(v=6)$ manifold using an excited level $|e\rangle$ with $0_g^{-}$ character. The $x$ axis shows the binding energy $E_B$. The quantum number $f$ is shown below each of the three groups of lines. Thick black lines above the spectrum indicate states with $N=0$; gray lines correspond to states with $N=2$. The upper row of numbers indicates the expectation values of the magnetic quantum number $m_f$ at the magnetic field of 1005.8 G.

Figure 5

Energy splitting $\Delta E_B$ between the s1 level ($N=0$) and the d1 level ($N=2$) for a given vibrational quantum number $v$. Large error bars correspond to early measurements without simultaneous wave-meter readings (see text). The continuous line is a calculation based on the coupled-channel model.

Figure 6

(a) Progression of the substructure of the vibrational manifold at $B=1005.8$ G. Shown are calculated $a^3{\Sigma }_u^{+}$ levels with quantum numbers $I\approx 3$, $M=2$, $N=0,2$, and $v=0\dots 34$. Thick black lines correspond to $N=0$, gray ones to $N=2$. The s2 level serves as energy reference ($\Delta E_B=0$) in each vibrational substructure. In addition to the triplet levels, nearby singlet levels of the $X^1{\Sigma }_g^{+}$ potential are also shown (thick black crosses, $N=0$; gray crosses, $N=2$). (b) Symbols as in panel (a). To discuss the singlet-triplet mixing from a theoretical point of view, we now also include lines with $I\approx 1$ but only $N=0$ for clarity. On the bottom, the approximate quantum numbers $(I,f)$ are given. See text for details.

Figure 7

“Discrete second derivative” of $E_B(v)$, that is, ${\delta }^2{E}_B(v)=E_B(v+1)-2E_B(v)+E_B(v-1)$. The solid curve connects the points from the s1 levels, while the diamonds correspond to the s2 state.

Figure 8

Observation of strongly perturbed singlet levels. We compare sections of line spectra for the vibrational levels $v=28$, 31, and 33. As in Fig. 6 the s2 level is chosen as energy reference $\Delta E_B=0$. The lines (and crosses) above the experimentally observed spectrum are from coupled-channel calculations. Black lines (crosses) represent $N=0$ levels, while thin gray lines (crosses) correspond to $N=2$ levels for $S=1$ ($S=0$, respectively). The lines that originate from the singlet states are labeled with “${}^1\Sigma$ ”.

Figure 9

Normalized transition matrix element ${\Omega }_2/(2\pi \sqrt{I_2})$ between the excited level $|e\rangle$ ($v^{\prime }=13,1_g$ character) in the $1^3{\Sigma }_g^{+}$ potential and the s1 level with vibrational quantum number $v$ in the lowest triplet potential $a^3{\Sigma }_u^{+}$.