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Physical Review A

Long-range potentials for two-species alkali-metal atoms

M. Marinescu, H. R. Sadeghpour

Published Jan 1, 1999• 145 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.59.390

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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 nS\ensuremath−n\ensuremath′S,nS\ensuremath−n\ensuremath′P,nS\ensuremath−n\ensuremath′D,\mathrm{nS}\ensuremath{-}{n}^{\ensuremath{'}}S,nS\ensuremath{-}{n}^{\ensuremath{'}}P,nS\ensuremath{-}{n}^{\ensuremath{'}}D,nS\ensuremath−n\ensuremath′S,nS\ensuremath−n\ensuremath′P,nS\ensuremath−n\ensuremath′D, and nP\ensuremath−n\ensuremath′P\mathrm{nP}\ensuremath{-}{n}^{\ensuremath{'}}PnP\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 Rb(5S)\ensuremath−Cs(6S),Rb(6S)\ensuremath−Cs(6S),K(4S)\ensuremath−Rb(5P)1,3\ensuremathΣ,K(4P)\ensuremath−Cs(6P)1,3\ensuremathΣ,\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},Rb(5S)\ensuremath−Cs(6S),Rb(6S)\ensuremath−Cs(6S),K(4S)\ensuremath−Rb(5P)1,3\ensuremathΣ,K(4P)\ensuremath−Cs(6P)1,3\ensuremathΣ, and Rb(5S)\ensuremath−Na(3D)1,3\ensuremathΣ\mathrm{Rb}(5S)\ensuremath{-}\mathrm{N}\mathrm{a}(3D){ }^{1,3}\ensuremath{\Sigma}Rb(5S)\ensuremath−Na(3D)1,3\ensuremathΣ molecules to be the most attractive. The K(4S)\ensuremath−Rb(5P)\mathrm{K}(4S)\ensuremath{-}\mathrm{R}\mathrm{b}(5P)K(4S)\ensuremath−Rb(5P) dimer represents the best candidate molecule for ultracold photoassociative spectroscopy. We also find the K(4P)\ensuremath−Cs(6P)\mathrm{K}(4P)\ensuremath{-}\mathrm{C}\mathrm{s}(6P)K(4P)\ensuremath−Cs(6P) and Rb(5S)\ensuremath−K(4P)\mathrm{Rb}(5S)\ensuremath{-}\mathrm{K}(4P)Rb(5S)\ensuremath−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.
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.
We study the van der Waals interaction between Rydberg alkali-metal atoms with fine structure (n2Lj{n}^{2}{L}_{j}n2Lj​; $L\ensuremath{\le}2$) and heteronuclear alkali-metal dimers in the ground rovibrational state ($X^{1}\mathrm{\ensuremath{\Sigma}}^{+}$; v=0,J=0v=0, J=0v=0,J=0). We compute the associated C6{C}_{6}C6​ dispersion coefficients of atom-molecule pairs involving 133Cs^{133}\mathrm{Cs}133Cs and 85Rb^{85}\mathrm{Rb}85Rb atoms interacting with KRb, LiCs, LiRb, and RbCs molecules. The obtained dispersion coefficients can be accurately fitted to a state-dependent polynomial O(n7)O({n}^{7})O(n7) over the range of principal quantum numbers $40\ensuremath{\le}n\ensuremath{\le}150$. For all atom-molecule pairs considered, Rydberg states n2Sjn^{2}S_{j}n2Sj​ and n2Pjn^{2}P_{j}n2Pj​ result in attractive 1/R61/{R}^{6}1/R6 potentials. In contrast, n2Djn^{2}D_{j}n2Dj​ states can give rise to repulsive potentials for specific atom-molecule pairs. The interaction energy at the LeRoy distance approximately scales as ${n}^{\ensuremath{-}5}$ for n>40n>40n>40. For intermediate values of $n\ensuremath{\lesssim}40$, both repulsive and attractive interaction energies of the order of 10--$1000\phantom{\rule{0.28em}{0ex}}\ensuremath{\mu}\mathrm{K}$ can be achieved with specific atomic and molecular species. The accuracy of the reported C6{C}_{6}C6​ coefficients is limited by the quality of the atomic quantum defects, with relative errors $\mathrm{\ensuremath{\Delta}}{C}_{6}/{C}_{6}$ estimated to be no greater than 1% on average.
Knowledge of the long-range interaction between atoms and molecules is of fundamental importance for low-energy and low-temperature collisions. The electronic interaction between the charge distributions of two ground-state alkali-metal atoms can be expanded in inverse powers of R, the internuclear distance. The coefficients C6{\mathit{C}}_{6}C6​, C8{\mathit{C}}_{8}C8​, and C10{\mathit{C}}_{10}C10​ of, respectively, the ${\mathit{R}}^{\mathrm{\ensuremath{-}}6}$, ${\mathit{R}}^{\mathrm{\ensuremath{-}}8}$, and ${\mathit{R}}^{\mathrm{\ensuremath{-}}10}$ terms are calculated by integrating the products of the dynamic electric multipole polarizabilities of the individual atoms at imaginary frequencies, which are in turn obtained by solving two coupled inhomogeneous differential equations. Precise one-electron model potentials are developed to represent the motion of the valence electron in the field of the closed alkali-metal positive-ion core. The numerical results for the static multipole polarizabilities for the alkali-metal atoms and the coefficients C6{\mathit{C}}_{6}C6​, C8{\mathit{C}}_{8}C8​, and C10{\mathit{C}}_{10}C10​ for homonuclear and heteronuclear alkali-metal diatoms are compared with other calculations.
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 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.