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arXiv:1502.05636 · The Journal of Chemical Physics

Long-range interactions between polar bialkali ground-state molecules in arbitrary vibrational levels

Romain Véxiau, Maxence Lepers, M Aymar, Nadia Bouloufa-Maafa, Olivier Dulieu

Published Jun 2, 2015• 22 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/1.4921622

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CHAPTER 4. Long-range Interactions Between Ultracold Atoms and Molecules

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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.

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We address a growing interest in trapping and cooling of mixed-species alkali-metal atoms. Long-range coefficients that arise in the multipole expansion of molecular potentials for unlike alkali-metal dimers are calculated. The coefficients for the heteronuclear alkali-metal dimers corresponding to different molecular symmetries that separate to $\mathrm{nS}\ensuremath{-}{n}^{\ensuremath{'}}S,nS\ensuremath{-}{n}^{\ensuremath{'}}P,nS\ensuremath{-}{n}^{\ensuremath{'}}D,$ and $\mathrm{nP}\ensuremath{-}{n}^{\ensuremath{'}}P$ atomic levels are computed with high precision. We consider cases where in the infinite separation limit, one atom is in the ground state and the other is in one of the lowest S, P, and D excited states and both atoms are in their lowest excited P states. We find the long-range potentials for $\mathrm{Rb}(5S)\ensuremath{-}\mathrm{C}\mathrm{s}(6S),\mathrm{Rb}(6S)\ensuremath{-}\mathrm{C}\mathrm{s}(6S),\mathrm{K}(4S)\ensuremath{-}\mathrm{R}\mathrm{b}(5P){ }^{1,3}\ensuremath{\Sigma},\mathrm{K}(4P)\ensuremath{-}\mathrm{C}\mathrm{s}(6P){ }^{1,3}\ensuremath{\Sigma},$ and $\mathrm{Rb}(5S)\ensuremath{-}\mathrm{N}\mathrm{a}(3D){ }^{1,3}\ensuremath{\Sigma}$ molecules to be the most attractive. The $\mathrm{K}(4S)\ensuremath{-}\mathrm{R}\mathrm{b}(5P)$ dimer represents the best candidate molecule for ultracold photoassociative spectroscopy. We also find the $\mathrm{K}(4P)\ensuremath{-}\mathrm{C}\mathrm{s}(6P)$ and $\mathrm{Rb}(5S)\ensuremath{-}\mathrm{K}(4P)$ dimers to form, respectively, the most attractive and the most repulsive long-range potentials. The present calculation is in good agreement with experimentally determined value for the ${1/R}^{6}$ van der Waals coefficient for the interaction between $\mathrm{Cs}(6S)$ and $\mathrm{Li}(2P)$ atoms.

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Long-range interactions between ultracold atoms and molecules

Maxence Lepers, Olivier Dulieu

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 ($T\ll 1K$), 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.

arXiv:1703.02833•Mar 8, 2017•4 citations
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.
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.
The long-range part of the interatomic interactions plays a substantial role in the collisional dynamics of ultracold gases. Here, we report on the calculation of the isotropic and anisotropic C6{C}_{6}C6​ coefficients characterizing the van der Waals interaction between dysprosium or erbium atoms in the two lowest energy levels and the ground-state alkali-metal (Li, Na, K, Rb, Cs, Fr) or alkaline-earth-metal (Be, Mg, Ca, Sr, Ba) atoms. The calculations are done using the integral of dynamic dipole polarizabilities at so-called imaginary frequencies of the two interacting atoms. For all atom pairs, we find that the isotropic C6{C}_{6}C6​ coefficients are two or three orders of magnitude larger than the anisotropic ones. Those coefficients are essential for modeling collisional properties of heteronuclear quantum mixtures containing highly magnetic dysprosium or erbium atoms and alkali-metal or alkaline-earth-metal atoms.
We have studied the long-range interaction potentials of the heteronuclear alkali-metal-atom dimers in the case when one of the atoms is in the first excited state (nA{n}_{A}nA​P) and the other is in the ground state (nB{n}_{B}nB​S). The presentation of the general expressions for the interaction energies in the Hund's case (c) is followed by the calculation of the corresponding C6{C}_{6}C6​ constants for all heteronuclear alkali-metal-atom pairs. In the case of heavier perturbers (Rb and Cs) the C6{C}_{6}C6​ constants are found to be significantly influenced by the perturber's fine structure. The van der Waals interaction in the K-Rb system is exceptionally strong resulting in C6{C}_{6}C6​ constants considerably larger than in all other alkali-metal-atom heteronuclear systems. The intermediate-coupling region (ICR) is defined as the region of internuclear separation R where the interaction energy is of the order of the atomic spin-orbit splitting, while the exchange energy is still negligible. Analysis of the general form of the energy matrices in the ICR shows that the homonuclear alkali-metal-atom dimers can be treated as a special case. In the large-R limit, the ICR results reduce to those obtained in Hund's case (c). As an example, we have calculated the ICR interaction potentials of the K-Rb system, taking into account only the leading dipole-dipole interaction. Finally we have discussed the construction of the absorption coefficient in the wings of the resonance lines, both in the impact and in the quasistatic approximation.
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.
We address a growing interest in trapping and cooling of mixed-species alkali-metal atoms. Long-range coefficients that arise in the multipole expansion of molecular potentials for unlike alkali-metal dimers are calculated. The coefficients for the heteronuclear alkali-metal dimers corresponding to different molecular symmetries that separate to $\mathrm{nS}\ensuremath{-}{n}^{\ensuremath{'}}S,nS\ensuremath{-}{n}^{\ensuremath{'}}P,nS\ensuremath{-}{n}^{\ensuremath{'}}D,$ and $\mathrm{nP}\ensuremath{-}{n}^{\ensuremath{'}}P$ atomic levels are computed with high precision. We consider cases where in the infinite separation limit, one atom is in the ground state and the other is in one of the lowest S, P, and D excited states and both atoms are in their lowest excited P states. We find the long-range potentials for $\mathrm{Rb}(5S)\ensuremath{-}\mathrm{C}\mathrm{s}(6S),\mathrm{Rb}(6S)\ensuremath{-}\mathrm{C}\mathrm{s}(6S),\mathrm{K}(4S)\ensuremath{-}\mathrm{R}\mathrm{b}(5P){ }^{1,3}\ensuremath{\Sigma},\mathrm{K}(4P)\ensuremath{-}\mathrm{C}\mathrm{s}(6P){ }^{1,3}\ensuremath{\Sigma},$ and $\mathrm{Rb}(5S)\ensuremath{-}\mathrm{N}\mathrm{a}(3D){ }^{1,3}\ensuremath{\Sigma}$ molecules to be the most attractive. The $\mathrm{K}(4S)\ensuremath{-}\mathrm{R}\mathrm{b}(5P)$ dimer represents the best candidate molecule for ultracold photoassociative spectroscopy. We also find the $\mathrm{K}(4P)\ensuremath{-}\mathrm{C}\mathrm{s}(6P)$ and $\mathrm{Rb}(5S)\ensuremath{-}\mathrm{K}(4P)$ dimers to form, respectively, the most attractive and the most repulsive long-range potentials. The present calculation is in good agreement with experimentally determined value for the 1/R6{1/R}^{6}1/R6 van der Waals coefficient for the interaction between Cs(6S)\mathrm{Cs}(6S)Cs(6S) and Li(2P)\mathrm{Li}(2P)Li(2P) atoms.
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.