Observation of mixed singlet-triplet ${\mathrm{Rb}}_2$ Rydberg molecules

We present high-resolution spectroscopy of ${\mathrm{Rb}}_{\text{2}}$ ultralong-range Rydberg molecules bound by mixed singlet-triplet electron-neutral atom scattering. The mixing of the scattering channels is a consequence of the hyperfine interaction in the ground-state atom, as predicted recently by Anderson et al. [Phys. Rev. A 90, 062518 (2014)]. Our experimental data enable the determination of the effective zero-energy singlet $s$-wave scattering length for Rb. We show that an external magnetic field can tune the contributions of the singlet and the triplet scattering channels and therefore the binding energies of the observed molecules. This mixing of molecular states via the magnetic field results in observed shifts of the molecular line which differ from the Zeeman shift of the asymptotic atomic states. Finally, we calculate molecular potentials using a full diagonalization approach including the $p$-wave contribution and all orders in the relative momentum $k$, and compare the obtained molecular binding energies to the experimental data.

Figure 1

(a) Schematic illustration of the matrix Hamiltonian [Eq. (1)], where for better readability the Rydberg hyperfine interaction for a single Rydberg $n{S}_{1/2}$ state is omitted. The respective terms are marked in different colors. Blank square indicates zero in the Hamiltonian. Basis states are listed according to the tensor operation ${\widehat{S}}_r\otimes {\widehat{S}}_g\otimes \widehat{I}$. (b) Excitation scheme to the Rydberg state including the fine structure. The small black arrows at the top indicate the Rydberg electron spin. ${\sigma }^{-}$ and $\pi$ denote the laser polarizations. The large detuning from the intermediate state allows for the usage of the $j$ basis for the Rydberg state.

Figure 2

Molecular potentials for the $40S\text{−}5S_{1/2}$ $(F=1$, 2) molecules as a function of the internuclear distance calculated with the full diagonalization method. The inclusion of the hyperfine interaction of the ground state atom causes splitting of the molecular potential into two: triplet (red) and mixed (green) adiabatic potential energy curves. The applied magnetic field $B=2.35$ G lifts the degeneracy of the triplet and mixed PECs and leads to further mixing of the states. Zero on the vertical axis corresponds to the asymptotes of the respective hyperfine states $F=1,2$ of the ground state in a zero magnetic field. The black dashed-dotted line indicates the position of the atomic spin-up Rydberg state, which is used as reference point henceforth in the paper.

Figure 3

The experiment addresses only the triplet and mixed molecular states with ground state hyperfine quantum number $F=2$. The solid red and green lines show the respective potentials obtained with the full matrix diagonalization method, including the $p$-wave contribution. In contrast, the red and green dotted lines show the calculated triplet and mixed potentials using the effective scattering length approach. It can be seen that the energy dependence of the scattering length and the $p$-wave scattering contribution are especially important for the shallow mixed potential energy curves. Zero on the vertical axis corresponds to the atomic spin-up Rydberg state (black dashed-dotted line). As an example a bound state wave function inside the accessible triplet potential is plotted in blue.

Figure 4

Molecular spectra for different principal quantum numbers $n$. The external applied magnetic field was $B=2.35$ G. The zero frequency position is set to the atomic transition from the $F=2$ ground state to the $nS$ Rydberg spin-up state. The large peak at around $-6.5$ MHz corresponds to the atomic transition to the spin-down Rydberg state. The red and green shaded areas show the calculated peak positions using the full diagonalization of the triplet and mixed molecular ground states, respectively. The shaded areas are used to reflect the sensitivity of the calculated binding energies to the chosen boundary conditions of the potential used in the Schrödinger equation. Likewise, the green and red straight lines are the calculated positions of the molecular states with the effective $s$-wave scattering length approach. For the triplet scattering length we take the fixed value of 15.7(1)${\mathrm{a}}_{\text{0}}$, while the singlet one ${\mathrm{a}}_s=-0.2a_0$ is fitted to the data. The green dashed line indicates the change of the singlet scattering length by $\pm 0.5$ ${\mathrm{a}}_0$. The gray lines are Lorentzian fits plotted for better visibility. Each spectrum is an average of 2000–6000 measurements with the standard error (of the mean) bars shown exemplary in the spectrum of $36S$.

Figure 5

Theoretical Zeeman map of the mixed and triplet molecular states corresponding to the asymptotic $40S$ Rydberg and $F=2$ ground state, calculated using the effective scattering length approach. Zero energy reference point is the atomic pair state $\left|m_{F_{\text{g}}}=2,m_{S_{\text{r}}}=1/2\right.〉$. The black dashed line shows the theoretical Zeeman shift of the atomic Rydberg spin down state, which we use for the calibration of the magnetic field in the experiment. The experimental data from our spectroscopy in the magnetic trap lie in the blue shaded region and are shown as dots in the inset: atomic Rydberg spin down state (black), triplet molecular ground state (red), and mixed molecular ground state (green). Straight lines show the calculated line positions using the effective scattering length approach; shaded areas indicate the results of the full diagonalization calculation (color code is the same as for experimental data). The atomic Rydberg spin down state is again shown by the black dashed line. Error bars are the 95% confidence intervals of the Lorentzian fit positions taken from more than 4000 measurements.