Anisotropic optical trapping of ultracold erbium atoms

We calculate the complex dynamic dipole polarizability of ground-state erbium, a rare-earth atom that was recently Bose-condensed [1]. This quantity determines the trapping conditions of cold atoms in an optical trap. The polarizability is calculated with the sum-over-state formula inherent to second-order perturbation theory. The summation is performed on transition energies and transition dipole moments from ground-state erbium, which are computed using the Racah-Slater least-square fitting procedure provided by the Cowan codes [2]. This allows us to predict several yet unobserved energy levels in the range 25000-31000 cm$^{-1}$ above the ground state. Regarding the trapping potential, we find that ground-state erbium essentially behaves like a spherically-symmetric atom, in spite of its large electronic angular momentum. We find a mostly isotropic van der Waals interaction between two ground-state erbium atoms, with a coefficient C$_{6}^{\mathrm{iso}}=$1760 a.u.. On the contrary, the photon-scattering rate is strongly anisotropic with respect to the polarization of the trapping light [3]. \\[4pt] [1] K. Aikaw et al,. \textit{Phys. Rev. Lett.} \textbf{108}, 210401 (2012).\\[0pt] [2] R.D. Cowan. \textit{The theory of atomic structure and spectra}. University of California Press (1981).\\[0pt] [3] M. Lepers, J.-F. Wyart, and O. Dulieu. \textit{Phys. Rev. A}, in press (2014).