Alignment of $D$-State Rydberg Molecules

We report on the formation of ultralong-range Rydberg $D$-state molecules via photoassociation in an ultracold cloud of rubidium atoms. By applying a magnetic offset field on the order of 10 G and high resolution spectroscopy, we are able to resolve individual rovibrational molecular states. A full theory, using a Fermi pseudopotential approach including $s$- and $p$-wave scattering terms, reproduces the measured binding energies. The calculated molecular wave functions show that in the experiment we can selectively excite stationary molecular states with an extraordinary degree of alignment or antialignment with respect to the magnetic field axis.

Figure 1

Spectra of the $44D$, $J=5/2$, $m_J=1/2$ (a) and $42D$, $J=5/2$, $m_J=5/2$ (b) states, where the ion detector signal is plotted against the relative frequency of the excitation laser. The molecular lines appear energetically below the dissociation limit of the molecules which defines the energy offset. The energy difference is the binding energy of the molecule necessary to form the bond between the Rydberg and a ground state atom. Apart from the photoassociation spectra, two additional insets are shown: the calculated APES as a function of the spherical coordinates ($R$, $\mathrm{\Theta }$) and the angular distribution ${|Y_{l=2}^{m=(0,2)}|}^2$ (a,b) of the atomic electron orbitals relevant for the topological structure of the APES. In (a) the potential provides bound states in potential wells localized at $\mathrm{\Theta }=0$, $\pi$ (axial states) and at $\mathrm{\Theta }=\pi /2$ (toroidal states). In the spectrum of (a) we depict two individual spectra taken with different laser intensities which are separated with a black line at $-3.2\mathrm{MHz}$: the left one (blue) was taken at a high intensity to resolve the axial molecules and the right one (black) at a low intensity to decrease the power broadening of the atomic line such that the toroidal molecules can be resolved. For better visibility, a moving average (red line) is included and the data are scaled by a factor of 3. The dashed lines (gray) mark the experimental peak positions of the molecules whereas colored diamonds indicate the calculated binding energies of the axial (red) and toroidal (green) molecular states. In (b) the APES only provides bound states for the toroidal ($\mathrm{\Theta }=\pi /2$) configuration. For both spectra the error bars (standard deviation) are determined from 20 independent measurements.

Figure 2

Molecular binding energies for the $m_J=5/2$ states plotted against the rovibrational excitation numbers $\nu$ for principal quantum numbers $n$ ranging from 41 to 49. For increasing $n$ the states are colored brighter. The calculated binding energies (diamonds) are plotted with a horizontal offset to the experimental ones (circles) to improve readability. The insets depict the scaled probability densities of certain rovibrational states for $\rho$ and $z$ ranging from $2000a_0$ to $3300a_0$ and from $-1500a_0$ to $1500a_0$, respectively.

Figure 3

Molecular binding energies for the $m_J=1/2$ states plotted against the axial rovibrational excitation numbers (a) and the toroidal rovibrational excitation numbers (b), respectively, for principal quantum numbers $n$ ranging from 42 to 46. For increasing $n$ the states are colored brighter. The calculated binding energies (diamonds) are plotted with a horizontal offset to the experimental ones (circles) to improve readability. The insets depict the scaled probability densities in the $z$ and $\rho$ direction of selected rovibrational states. For the axial rovibrational excitation numbers (a) $\rho$ ranges from 0 to $3500a_0$ and $z$ from 0 to $3500a_0$, whereas the toroidal rovibrational excitation numbers (b) range from $\rho =2700a_0$ to $3500a_0$ and $z=-2250a_0$ to $2250a_0$.