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arXiv:2107.02676 · New Journal of Physics

Relativistic aspects of orbital and magnetic anisotropies in the chemical bonding and structure of lanthanide molecules

Eite Tiesinga, Jacek A. Kłos, Ming Li, Alexander N. Petrov, Svetlana Kotochigova

Published Aug 1, 2021• 11 citations

Abstract

Abstract The electronic structure of magnetic lanthanide atoms is fascinating from a fundamental perspective. They have electrons in a submerged open 4f shell lying beneath a filled 6s shell with strong relativistic correlations leading to a large magnetic moment and large electronic orbital angular momentum. This large angular momentum leads to strong anisotropies, i. e. orientation dependencies, in their mutual interactions. The long-ranged molecular anisotropies are crucial for proposals to use ultracold lanthanide atoms in spin-based quantum computers, the realization of exotic states in correlated matter, and the simulation of orbitronics found in magnetic technologies. Short-ranged interactions and bond formation among these atomic species have thus far not been well characterized. Efficient relativistic computations are required. Here, for the first time we theoretically determine the electronic and ro-vibrational states of heavy homonuclear lanthanide Er 2 and Tm 2 molecules by applying state-of-the-art relativistic methods. In spite of the complexity of their internal structure, we were able to obtain reliable spin–orbit and correlation-induced splittings between the 91 Er 2 and 36 Tm 2 electronic potentials dissociating to two ground-state atoms. A tensor analysis allows us to expand the potentials between the atoms in terms of a sum of seven spin–spin tensor operators simplifying future research. The strengths of the tensor operators as functions of atom separation are presented and relationships among the strengths, derived from the dispersive long-range interactions, are explained. Finally, low-lying spectroscopically relevant ro-vibrational energy levels are computed with coupled-channels calculations and analyzed.

DOI: 10.1088/1367-2630/ac1a9a

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Abstract The electronic structure of magnetic lanthanide atoms is fascinating from a fundamental perspective. They have electrons in a submerged open 4f shell lying beneath a filled 6s shell with strong relativistic correlations leading to a large magnetic moment and large electronic orbital angular momentum. This large angular momentum leads to strong anisotropies, i. e. orientation dependencies, in their mutual interactions. The long-ranged molecular anisotropies are crucial for proposals to use ultracold lanthanide atoms in spin-based quantum computers, the realization of exotic states in correlated matter, and the simulation of orbitronics found in magnetic technologies. Short-ranged interactions and bond formation among these atomic species have thus far not been well characterized. Efficient relativistic computations are required. Here, for the first time we theoretically determine the electronic and ro-vibrational states of heavy homonuclear lanthanide Er 2 and Tm 2 molecules by applying state-of-the-art relativistic methods. In spite of the complexity of their internal structure, we were able to obtain reliable spin–orbit and correlation-induced splittings between the 91 Er 2 and 36 Tm 2 electronic potentials dissociating to two ground-state atoms. A tensor analysis allows us to expand the potentials between the atoms in terms of a sum of seven spin–spin tensor operators simplifying future research. The strengths of the tensor operators as functions of atom separation are presented and relationships among the strengths, derived from the dispersive long-range interactions, are explained. Finally, low-lying spectroscopically relevant ro-vibrational energy levels are computed with coupled-channels calculations and analyzed.

Lanthanide atoms have an unusual electron configuration, with a partially filled shell of

f

orbitals. This leads to a set of characteristic properties that enable enhanced control over ultracold atoms and their interactions: large numbers of optical transitions with widely varying wavelengths and transition strengths, anisotropic interaction properties between atoms and with light, and a large magnetic moment and spin space present in the ground state. These features in turn enable applications ranging from narrow-line laser cooling and spin manipulation to evaporative cooling through universal dipolar scattering, to the observation of a rotonic dispersion relation, self-bound liquid-like droplets stabilized by quantum fluctuations, and supersolid states. In this short review, we describe how the unusual level structure of lanthanide atoms leads to these key features, and provide a brief and necessarily partial overview of experimental progress in this rapidly developing field.

The properties of the electronic ground state of the polar and paramagnetic chromium--closed-shell-atom molecules have been investigated. State-of-the-art ab initio techniques have been applied to compute the potential energy curves for the chromium--alkaline-earth-metal-atom, Cr

X

(

X=\mathrm{Be}

, Mg, Ca, Sr, Ba), and chromium--ytterbium, CrYb, molecules in the Born-Oppenheimer approximation for the high-spin ${X}^{7}{\ensuremath{\Sigma}}^{+}$ electronic ground state. The spin restricted open-shell coupled cluster method restricted to single, double, and noniterative triple excitations, RCCSD(T), was employed and the scalar relativistic effects within the Douglas-Kroll-Hess Hamiltonian or energy-consistent pseudopotentials were included. The permanent electric dipole moments and static electric dipole polarizabilities were computed. The leading long-range coefficients describing the dispersion interaction between the atoms at large interatomic distances

{C}_{6}

are also reported. The molecules under investigation are examples of species possessing both large magnetic and electric dipole moments making them potentially interesting candidates for ultracold many-body physics studies.

The interaction between a Rydberg electron and a neutral atom situated inside its extended orbit is described via contact interactions for each atom-electron scattering channel. In ultracold environments, these interactions lead to long-range molecules with binding energies typically ranging from 10 to $1\ifmmode\times\else\texttimes\fi{}{10}^{4}$ MHz. These energies are comparable to the relativistic and hyperfine structure of the separate atomic components. Studies of molecular formation aiming to reproduce observations with spectroscopic accuracy must therefore include the hyperfine splitting of the neutral atom and the spin-orbit splittings of both the Rydberg atom and the electron-atom interaction. Adiabatic potential energy curves and permanent electric dipole moments are presented for

X=\mathrm{Be}

and

{\mathrm{Cs}}_{2}

. The influence of spin degrees of freedom on the potential energy curves and multipole moments probed in recent experimental work is elucidated, and the observed dipole moments of butterfly molecules are explained by the generalized

^{3}P_{J}

pseudopotential derived here.

We unravel some peculiar properties of ultra-long-range Rydberg molecules formed by an

s

-state

^{87}\mathrm{Rb}

Rydberg atom and a corresponding ground-state atom whose electronic orbitals are spherically symmetric and therefore should not be influenced by the presence of weak magnetic fields. However, the electron-atom interaction, which establishes the molecular bond, is under certain conditions subject to a sizable spin-orbit coupling and, hence, sensitive to the magnetic field. This mechanism can be harnessed to counterintuitively align the

s

-state molecules with respect to the field axis. We demonstrate this by analyzing the angular-dependent Born-Oppenheimer potential energy surfaces and the supported vibrational molecular states. Our predictions open interesting possibilities for accessing the physics of relativistic electron-atom scattering experimentally.

This paper reviews current experimental and theoretical progress in the study of dipolar quantum gases of ground and meta-stable atoms with a large magnetic moment. We emphasize the anisotropic nature of Feshbach resonances due to coupling to fast-rotating resonant molecular states in ultracold s-wave collisions between magnetic atoms in external magnetic fields. The dramatic differences in the distribution of resonances of magnetic (7)S3 chromium and magnetic lanthanide atoms with a submerged 4f shell and non-zero electron angular momentum is analyzed. We focus on dysprosium and erbium as important experimental advances have been recently made to cool and create quantum-degenerate gases for these atoms. Finally, we describe progress in locating resonances in collisions of meta-stable magnetic atoms in electronic P-states with ground-state atoms, where an interplay between collisional anisotropies and spin-orbit coupling exists.

Ultralong-range Rydberg molecules, composed of an excited Rydberg atom and a ground-state atom, are characterized by large bond lengths, dipole moments, sensitivity to external fields, and an unusual binding mechanism based on low-energy elastic electron scattering. Although Rydberg molecules formed between alkali atoms have received the most attention, the additional complexity found in atoms with more than a single valence electron poses new theoretical challenges as well as new possibilities for control and design of the molecular structure. In this paper, we extend the theory of Rydberg molecules to include the additional spin coupling of the Rydberg states of a multivalent atom. We employ this theory to describe the properties of Rydberg molecules composed of mercury atoms. We calculate the potential energy curves of both heteronuclear (

{\mathrm{Hg}}^{*}\mathrm{Rb}

) and homonuclear (

{\mathrm{Hg}}^{*}\mathrm{Hg}

) molecules. In the former case, we propose the realization of long-range spin entanglement and remote spin flip. In the latter, we show how long-lived metastable molecular states of

{\mathrm{Hg}}^{*}\mathrm{Hg}

exist as resonances above the dissociation threshold.