Four-body long-range interactions between ultracold weakly-bound diatomic molecules

TL;DR

This paper characterizes the complex long-range interactions between ultracold diatomic molecules, revealing additional anisotropic terms beyond the usual dipole-dipole interactions, with implications for understanding molecular behavior at ultracold temperatures.

Contribution

The paper introduces a detailed multipolar expansion approach for long-range interactions between weakly-bound diatomic molecules, including higher-order terms and anisotropic effects, exemplified with Er$_2$ Feshbach molecules.

Findings

Magnetic dipole interactions produce R^{-3} terms and additional R^{-5}, R^{-7} terms.

Van der Waals interactions contribute R^{-6} and higher order terms.

Electric-quadrupole interactions are negligible compared to magnetic and van der Waals energies.

Abstract

Using the multipolar expansion of electrostatic and magnetostatic potential energies, we characterize the long-range interactions between two weakly-bound diatomic molecules, taking as an example the paramagnetic Er$_2$ Feshbach molecules which were produced recently. Since inside each molecule, individual atoms conserve their identity, the intermolecular potential energy can be expanded as the sum of pairwise atomic potential energies. In the case of Er$_2$ Feshbach molecules, we show that the interaction between atomic magnetic dipoles gives rise to the usual $R^{-3}$ term of the multipolar expansion, with $R$ the intermolecular distance, but also to additional terms scaling as $R^{-5}$, $R^{-7}$, and so on. Those terms are due to the interaction between effective molecular multipole moments, and are strongly anisotropic with respect to the orientation of the molecules. Similarly the atomic pairwise van der Waals interaction results in $R^{-6}$, $R^{-8}$, ... terms in the intermolecular potential energy. By calculating the reduced electric-quadrupole moment of erbium ground level $\langle J=6||\hat{Q}_2||J=6\rangle = -1.305$ a.u., we also demonstrate that the electric-quadrupole interaction energy is negligible with respect to the magnetic-dipole and van der Waals interaction energies. The general formalism presented in this article can be applied to calculate the long-range potential energy between arbitrary charge distributions composed of almost free subsystems.