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arXiv:1508.06066 · Journal of Physics B Atomic Molecular and Optical Physics

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

Maxence Lepers, Goulven Quéméner, Eliane Luc-Koenig, Olivier Dulieu

Published Nov 24, 2015• 6 citations
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Abstract

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 term of the multipolar expansion, where R is the intermolecular distance, but also to additional terms scaling as 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 ... terms in the intermolecular potential energy. By calculating the reduced electric-quadrupole moment of erbium ground level 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.

DOI
10.1088/0953-4075/49/1/014004

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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 term of the multipolar expansion, where R is the intermolecular distance, but also to additional terms scaling as 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 ... terms in the intermolecular potential energy. By calculating the reduced electric-quadrupole moment of erbium ground level 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.
The term long-range interactions refers to electrostatic and magnetostatic potential energies between atoms and molecules with mutual distances ranging from a few tens to a few hundreds of Bohr radii. The energies involved are much smaller than the usual chemical bond energies. However, they are comparable with the typical kinetic energies of particles in an ultracold gas (T ≪ 1 K), so that the long-range interactions play a central role in its dynamics. The progress of research devoted to ultracold gases sheds a new light on the well-established topic of long-range interactions, because (i) the interacting atoms and molecules can be prepared in a well-defined quantum (electronic, vibrational, rotational, fine or hyperfine), ground or excited level; and (ii) long-range interactions can be tailored at will using external electromagnetic fields. In this chapter, we present the essential concepts and mathematical relations to calculate long-range potential energies. We start by deriving the multipolar expansion of the electrostatic interaction energy between classical charge distributions, both in Cartesian coordinates for pedagogical purpose, and in spherical coordinates for practical use. Then we combine multipolar expansion and quantum perturbation theory, to obtain the general first- and second-order energy corrections, including the well-known van der Waals energy. We consider two central examples in the current context of ultracold gases: (i) a pair of alkali metal atoms and (ii) a pair of alkali metal heteronuclear diatomic molecules in the presence of an electric field. We highlight the key role of the total angular momenta of the interacting particles and of the complex, irrespective of their electronic or nuclear, orbital or spin nature.
The term "long-range interactions" refers to electrostatic and magnetostatic potential energies between atoms and molecules with mutual distances ranging from a few tens to a few hundreds Bohr radii. The involved energies are much smaller than the usual chemical bond energies. However, they are comparable with the typical kinetic energies of particles in an ultracold gas (T≪1KT\ll 1KT≪1K), so that the long-range interactions play a central role in its dynamics. The progress of research devoted to ultracold gases shed a new light on the well-established topic of long-range interactions, because: (i) the interacting atoms and molecules can be prepared in a well-defined quantum (electronic, vibrational, rotational, fine or hyper-fine), ground or excited level; and (ii) long-range interactions can be tailored at will using external electromagnetic fields. In this chapter, we present the essential concepts and mathematical relations to calculate long-range potential energies. We start with deriving the multipolar expansion of the electrostatic interaction energy between classical charge distributions, both in Cartesian coordinates for pedagogical purpose, and in spherical coordinates for practical use. Then we combine multipolar expansion and quantum perturbation theory, to obtain the general first- and second-order energy corrections, including the well-known van der Waals energy. We consider two central examples in the current context of ultracold gases: (i) a pair of alkali-metal atoms and (ii) a pair of alkali-metal heteronuclear diatomic molecules submitted to an electric field. We highlight the key role of the total angular momenta of the interacting particles and of the complex, irrespective of their electronic or nuclear, orbital or spin nature.
The general formalism of the multipolar expansion of electrostatic interactions is applied to the calculation of the potential energy between an excited atom (without fine structure) and a ground-state diatomic molecule at large mutual separations. Both partners exhibit a permanent quadrupole moment so that their mutual long-range interaction is dominated by a quadrupole-quadrupole term, which is attractive enough to bind trimers. Numerical results are given for an excited Cs(62P6{}^{2}P62P) atom and a ground-state Cs2{\mathrm{Cs}}_{2}Cs2​ molecule. The prospects for achieving photoassociation of a cold-atom--dimer pair are thus discussed and found promising. The formalism can be generalized to the long-range interaction between molecules to investigate the formation of cold tetramers.
We have calculated the isotropic C6 coefficients characterizing the long-range van der Waals interaction between two identical heteronuclear alkali-metal diatomic molecules in the same arbitrary vibrational level of their ground electronic state X(1)Σ(+). We consider the ten species made up of (7)Li, (23)Na, (39)K, (87)Rb, and (133)Cs. Following our previous work [Lepers et al., Phys. Rev. A 88, 032709 (2013)], we use the sum-over-state formula inherent to the second-order perturbation theory, composed of the contributions from the transitions within the ground state levels, from the transition between ground-state and excited state levels, and from a crossed term. These calculations involve a combination of experimental and quantum-chemical data for potential energy curves and transition dipole moments. We also investigate the case where the two molecules are in different vibrational levels and we show that the Moelwyn-Hughes approximation is valid provided that it is applied for each of the three contributions to the sum-over-state formula. Our results are particularly relevant in the context of inelastic and reactive collisions between ultracold bialkali molecules in deeply bound or in Feshbach levels.
Doubly polar molecules, possessing an electric dipole moment and a magnetic dipole moment, can strongly couple to both an external electric field and a magnetic field, providing unique opportunities to exert full control of the system quantum state at ultracold temperatures. We propose a method for creating a purely long-range doubly polar homonuclear molecule from a pair of strongly magnetic lanthanide atoms, one atom being in its ground level and the other in a superposition of quasidegenerate opposite-parity excited levels [Phys. Rev. Lett. 121, 063201 (2018)]. The electric dipole moment is induced by coupling the excited levels with an external electric field. We derive the general expression of the long-range, Stark, and Zeeman interaction energies in the properly symmetrized and fully-coupled basis describing the diatomic complex. Taking the example of holmium, our calculations predict a few shallow long-range wells in the potential energy curves that support vibrational levels accessible by direct photoassociation from pairs of ground-level atoms.
We computed the long-range interactions between two identical polar bialkali molecules in their rovibronic ground level for all ten species involving Li, Na, K, Rb, and Cs, using accurate quantum chemistry results combined with available spectroscopic data. A huge van der Waals interaction is found for eight species in free space. The competition of the van der Waals interaction with the dipole-dipole interaction induced by an external electric field parallel or perpendicular to the intermolecular axis is investigated by varying the electric-field magnitude and the intermolecular distance. Our calculations predict a regime with the mutual orientation of the two molecules but with no preferential direction in the laboratory frame. A mechanism for the stimulated one-photon radiative association of a pair of ultracold polar molecules into ultracold tetramers is proposed, which would open the way towards the optical manipulation of ultracold polyatomic molecules.