Level structure of deeply bound levels of the $c{}^3{\mathrm{\Sigma }}_g{}^{+}$ state of ${}^{87}\mathrm{Rb}_2$

We spectroscopically investigate the hyperfine, rotational, and Zeeman structure of the vibrational levels $v^{\prime }=0,7,13$ within the electronically excited $c{}^3{\mathrm{\Sigma }}_g{}^{+}$ state of ${}^{87}\mathrm{Rb}_2$ for magnetic fields of up to $1000\text{G}$. As spectroscopic methods we use short-range photoassociation of ultracold Rb atoms as well as photoexcitation of ultracold molecules which have been previously prepared in several well-defined quantum states of the $a{}^3\mathrm{\Sigma }{}_u^{+}$ potential. As a by-product, we present optical two-photon transfer of weakly bound Feshbach molecules into $a{}^3\mathrm{\Sigma }{}_u^{+}$, $v=0$ levels featuring different nuclear spin quantum numbers. A simple model reproduces well the molecular level structures of the $c{}^3{\mathrm{\Sigma }}_g{}^{+}$ vibrational states and provides a consistent assignment of the measured resonance lines. Furthermore, the model can be used to predict the relative transition strengths of the lines. From fits to the data we extract for each vibrational level the rotational constant, the effective spin-spin interaction constant, as well as the Fermi contact parameter and the anisotropic hyperfine constant. In an alternative approach, we perform coupled-channel calculations where we fit the relevant potential energy curves, spin-orbit interactions, and hyperfine functions. The calculations reproduce the measured hyperfine level term frequencies with an average uncertainty of $\pm 9\mathrm{MHz}$, similar as for the simple model. From these fits we obtain a section of the potential energy curve for the $c{}^3{\mathrm{\Sigma }}_g{}^{+}$ state which can be used for predicting the level structure for the vibrational manifold $v^{\prime }=0$ to 13 of this electronic state.

Figure 1

Molecular potentials $X{}^1{\mathrm{\Sigma }}_g{}^{+}$, $a{}^3\mathrm{\Sigma }{}_u^{+}$, and $c{}^3{\mathrm{\Sigma }}_g{}^{+}$ and the relevant vibrational levels. The inset shows the level structure in the vicinity of the Feshbach resonance which is indicated by the arrow. The circle represents the frequency location of the Feshbach state at $B=999.9\text{G}$.

Figure 2

Spectra of the hyperfine levels of the $c{}^3{\mathrm{\Sigma }}_g{}^{+}$, $0_g^{-}$, $v^{\prime }=13$, $J^{\prime }=2$ manifold. The data are taken for various initial levels of the $a{}^3\mathrm{\Sigma }{}_u^{+}$ state: (a) photoassociation of colliding atoms, (b) $|a,R=0\rangle$, (c) $|a,R=2\rangle$, and (d) Feshbach molecules (for the notation of the molecular states, see Table 1). The measurements (a)–(c) are carried out at a $B$ field of about $5\mathrm{G}$ while (d) is obtained at a $B$ field of $999.9\mathrm{G}$. In (b)–(d) blue (dark) data points correspond to $\pi$ transitions and orange (light-gray) data points correspond to $\sigma$ transitions. For the spectrum of (a) a mixture of both polarizations was used. The bars give the standard mean error. For each resonance dip in (b), (c), and (d) the laser power was adjusted individually to optimize the signal. The solid lines are model curve fits to our data. ${\nu }_0=294000\text{GHz}$.

Figure 3

Level structure of the $a{}^3\mathrm{\Sigma }{}_u^{+}$, $v=0$, $m_F=2$ manifold. Lines are calculations based on a coupled-channel model [27]. Blue (dark) lines are states without rotational excitation ($R=0$); orange (light-gray) lines are $R=2$ states. Dashed (solid) lines correspond to total nuclear spin $I=1$ (3), respectively. The total angular momenta $f$ and $F$ are given for each level for $B=0\mathrm{G}$. The circles at $B=999.9\mathrm{G}$ represent dark-state spectroscopy measurements (see Table $\mathrm{S}\mathrm{I}$ of the Supplemental Material for the precise frequency values [28]), while the squares show level positions which are used as initial states for spectroscopy of the $c{}^3{\mathrm{\Sigma }}_g{}^{+}$ manifold.

Figure 4

Level spectrum for $c{}^3{\mathrm{\Sigma }}_g{}^{+}$, $1_g$, $v^{\prime }=13$, $m_F^{\prime }=2$ at $B=999.9\text{G}$. The measurements (blue long bars) are taken with $\pi$-polarized light. The initial states for the spectroscopy are indicated on the left. The vertical solid lines show calculations of the levels with $I^{\prime }=3$, and dashed ones with $I^{\prime }=1$. Above the figure the ranges of the rotational states $J^{\prime }$ are indicated. ${\nu }_0=294000\text{GHz}$.

Figure 5

Spectra for $c{}^3{\mathrm{\Sigma }}_g{}^{+}$, $v^{\prime }=13$, $1_g$ (a) and $0_g^{-}$ (b) at $B\approx 5\text{G}$. The boxes represent measurements while calculations are represented by lines. The width of the boxes is always $40\text{MHz}$ and helps to identify the different frequency scales in the figure. Blue (dark) boxes correspond to measurements with $\pi$-polarized light, and orange (light-gray) boxes indicate measurements with $\sigma$-polarized light. White boxes illustrate a mixture of both. The initial states are given on the left. Solid lines belong to $I^{\prime }=3$, and dashed ones to $I^{\prime }=1$. Above the figure the ranges of the rotational states $J^{\prime }$ are indicated. ${\nu }_0=294000\text{GHz}$.

Figure 6

Zeeman shifts of $c{}^3{\mathrm{\Sigma }}_g{}^{+}$, $v^{\prime }=0$ energy levels. Symbols are measurements while lines are calculations. The legend indicates for each plot symbol the initial state and the polarization corresponding to the observed transition. The nuclear spin $I^{\prime }$ and $m_F^{\prime }$ quantum numbers are given via the line color and line style, respectively (see legend). On the right, groups of levels are assigned with the corresponding $F^{\prime }$ ($J^{\prime }$) quantum numbers for a magnetic field of $\approx 0\mathrm{G}$ ($\approx 1000\mathrm{G}$). Dashed vertical lines mark the magnetic field values of 5 and $999.9\mathrm{G}$. ${\nu }_0=281000\text{GHz}$.

Figure 7

Influence of various interaction terms on the level spectrum. Shown are calculations for $v^{\prime }=13$, $I^{\prime }=3$, $m_F^{\prime }=2$ at a magnetic field of $B\approx 5\mathrm{G}$. From left to right the effective spin-spin coupling, the rotation, the diagonal, and the off-diagonal contributions of the hyperfine interaction are subsequently turned on to values which correspond to those of the parameters given in Table 2. The circles on the right represent experimental results.

Figure 8

Relative transition strengths of lines. Shown are experimental loss spectra (dots) by photoassociation toward the states $c{}^3{\mathrm{\Sigma }}_g{}^{+}$, $v^{\prime }=13$, $1_g$ (a) and $0_g{}^{-}$ (b) for fixed laser power and exposure time at $B\approx 5\text{G}$. The solid lines are calculations. The vertical dashed lines mark the center frequencies of the transitions obtained from our simple model. ${\nu }_0=294000\text{GHz}$.

Figure 9

Potential scheme (solid lines) of molecular states with $g$ symmetry at the $s+p$ atom pair asymptote and their constructed spin-orbit coupling $C_{\text{mol},b}$ as a function of the internuclear distance $r$ (dashed lines). ${\nu }_{\text{ref}}/c=12737.6{\mathrm{cm}}^{-1}$ is the term energy of the $5s+5p$ atomic pair asymptote.