Anisotropic optical trapping as a manifestation of the complex electronic structure of ultracold lanthanide atoms: The example of holmium

The efficiency of optical trapping is determined by the atomic dynamic dipole polarizability, whose real and imaginary parts are associated with the potential energy and photon-scattering rate, respectively. In this article we develop a formalism to calculate analytically the real and imaginary parts of the scalar, vector, and tensor polarizabilities of lanthanide atoms. We assume that the sum-over-state formula comprises only transitions involving electrons in the valence orbitals like 6s,0.16em0ex5d,0.16em0ex6p, and $7s$, while transitions involving $4f$ core electrons are neglected. Applying this formalism to the ground level of configuration 4fq6s², we restrict the sum to transitions implying the 4fq6s6p configuration, which yields polarizabilities depending on two parameters: an effective transition energy and an effective transition dipole moment. Then, by introducing configuration-interaction mixing between 4fq6s6p and other configurations, we demonstrate that the imaginary part of the scalar, vector, and tensor polarizabilities is very sensitive to configuration-interaction coefficients, whereas the real part is not. The magnitude and anisotropy of the photon-scattering rate are thus strongly related to the details of the atomic electronic structure. Those analytical results agree with our detailed electronic-structure calculations of the energy levels, Land\'e g factors, transition probabilities, polarizabilities, and van der Waals C₆ coefficients, previously performed on erbium and dysprosium and presently performed on holmium. Our results show that, although the density of states decreases with increasing q, the configuration interaction between 4fq6s6p,0.16em0ex4f^q-15d6s², and 4f^q-15d²6s is surprisingly stronger in erbium ($q=12$) than in holmium ($q=11$), itself stronger than in dysprosium ($q=10$).