long-range coefficients. We have established the range of validity of the Casimir-Polder potential by analyzing the first-order correction to the energy given by the matrix element of the contact interaction term between the atomic charge distributions. Numerical values for the Casimir-Polder long-range interaction potentials for both homonuclear and heteronuclear alkali-metal dimers are presented. Comparisons with previous analytical and numerical results are included.

DOI: 10.1103/physreva.59.1936

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Casimir-Polder long-range interaction potentials between alkali-metal atoms

M. Marinescu, Li You

A complete description of the multipolar Casimir-Polder effects between two alkali-metal atoms in the ground state is presented. We give both general and particular expressions for the electric-electric, electric-magnetic, magnetic-magnetic, diamagnetic-electric, and diamagnetic-magnetic two-photon multipole processes, many of which have not been discussed previously in literature, to our knowledge. The asymptotic behavior of the Casimir-Polder potential is studied in terms of the ${K}_{7}, {K}_{9},$ and ${K}_{11}$ long-range coefficients. We have established the range of validity of the Casimir-Polder potential by analyzing the first-order correction to the energy given by the matrix element of the contact interaction term between the atomic charge distributions. Numerical values for the Casimir-Polder long-range interaction potentials for both homonuclear and heteronuclear alkali-metal dimers are presented. Comparisons with previous analytical and numerical results are included.A complete description of the multipolar Casimir-Polder effects between two alkali-metal atoms in the ground state is presented. We give both general and particular expressions for the electric-electric, electric-magnetic, magnetic-magnetic, diamagnetic-electric, and diamagnetic-magnetic two-photon multipole processes, many of which have not been discussed previously in literature, to our knowledge. The asymptotic behavior of the Casimir-Polder potential is studied in terms of the ${K}_{7}, {K}_{9},$ and ${K}_{11}$ long-range coefficients. We have established the range of validity of the Casimir-Polder potential by analyzing the first-order correction to the energy given by the matrix element of the contact interaction term between the atomic charge distributions. Numerical values for the Casimir-Polder long-range interaction potentials for both homonuclear and heteronuclear alkali-metal dimers are presented. Comparisons with previous analytical and numerical results are included.

APSMay 1, 19980 citations

Dispersion coefficients for alkali-metal dimers

M. Marinescu, H. R. Sadeghpour, A. Dalgarno

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 ${\mathit{C}}_{6}$ , ${\mathit{C}}_{8}$ , and ${\mathit{C}}_{10}$ 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 ${\mathit{C}}_{6}$ , ${\mathit{C}}_{8}$ , and ${\mathit{C}}_{10}$ for homonuclear and heteronuclear alkali-metal diatoms are compared with other calculations.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 ${\mathit{C}}_{6}$, ${\mathit{C}}_{8}$, and ${\mathit{C}}_{10}$ 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 ${\mathit{C}}_{6}$, ${\mathit{C}}_{8}$, and ${\mathit{C}}_{10}$ for homonuclear and heteronuclear alkali-metal diatoms are compared with other calculations.

Physical Review AFeb 1, 1994377 citations

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

M. Marinescu, H. R. Sadeghpour

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.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.

Physical Review AJan 1, 1999145 citations

van der Waals interaction in excited alkali-metal dimers

Mladen Movre, Robert Beuc

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 ( ${n}_{A}$ P) and the other is in the ground state ( ${n}_{B}$ 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 ${C}_{6}$ constants for all heteronuclear alkali-metal-atom pairs. In the case of heavier perturbers (Rb and Cs) the ${C}_{6}$ 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 ${C}_{6}$ 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 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 (${n}_{A}$P) and the other is in the ground state (${n}_{B}$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 ${C}_{6}$ constants for all heteronuclear alkali-metal-atom pairs. In the case of heavier perturbers (Rb and Cs) the ${C}_{6}$ 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 ${C}_{6}$ 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.

Physical Review AMay 1, 198530 citations

Three-body dispersion coefficients for alkali-metal atoms

Mircea Marinescu, Anthony F. Starace

We study the nonadditive part of the long-range interaction among three alkali-metal atoms in their ground states. Using nondegenerate perturbation theory, up to the third order, we have computed the dispersion coefficients C for three alkali-metal atoms interacting via their electric dipole moments. Both heteronuclear and homonuclear cases are considered. The numerical values for the C coefficients suggest that such three-body dipole interaction effects may not be neglected in the description of the long-range surface potential interaction for alkali-metal atoms. Furthermore, we show that approximate formulas for C of Midzuno and Kihara [J. Phys. Soc. Jpn. 11, 1045 (1956)] give results in excellent agreement with results of our calculations. Comparisons are given of our results with results of others and with results of other approximate formulas. The effect on our results for the C coefficient of uncertainties in the experimental values of the static dipole polarizabilities, on which our results depend, is also analyzed.We study the nonadditive part of the long-range interaction among three alkali-metal atoms in their ground states. Using nondegenerate perturbation theory, up to the third order, we have computed the dispersion coefficients C for three alkali-metal atoms interacting via their electric dipole moments. Both heteronuclear and homonuclear cases are considered. The numerical values for the C coefficients suggest that such three-body dipole interaction effects may not be neglected in the description of the long-range surface potential interaction for alkali-metal atoms. Furthermore, we show that approximate formulas for C of Midzuno and Kihara [J. Phys. Soc. Jpn. 11, 1045 (1956)] give results in excellent agreement with results of our calculations. Comparisons are given of our results with results of others and with results of other approximate formulas. The effect on our results for the C coefficient of uncertainties in the experimental values of the static dipole polarizabilities, on which our results depend, is also analyzed.

Physical Review AMar 1, 199718 citations

Long-range interactions between the alkali-metal atoms and alkaline earth ions

Jasmeet Kaur, D. K. Nandy, Bindiya Arora +1 more

Accurate knowledge of interaction potentials among the alkali atoms and alkaline earth ions is very useful in the studies of cold atom physics. Here we carry out theoretical studies of the long-range interactions among the Li, Na, K, and Rb alkali atoms with the Ca $^+$ , Ba $^+$ , Sr $^+$ , and Ra $^+$ alkaline earth ions systematically which are largely motivated by their importance in a number of applications. These interactions are expressed as a power series in the inverse of the internuclear separation $R$ . Both the dispersion and induction components of these interactions are determined accurately from the algebraic coefficients corresponding to each power combination in the series. Ultimately, these coefficients are expressed in terms of the electric multipole polarizabilities of the above mentioned systems which are calculated using the matrix elements obtained from a relativistic coupled-cluster method and core contributions to these quantities from the random phase approximation. We also compare our estimated polarizabilities with the other available theoretical and experimental results to verify accuracies in our calculations. In addition, we also evaluate the lifetimes of the first two low-lying states of the ions using the above matrix elements. Graphical representation of the interaction potentials versus $R$ are given among all the considered atoms and ions.Accurate knowledge of interaction potentials among the alkali atoms and alkaline earth ions is very useful in the studies of cold atom physics. Here we carry out theoretical studies of the long-range interactions among the Li, Na, K, and Rb alkali atoms with the Ca$^+$, Ba$^+$, Sr$^+$, and Ra$^+$ alkaline earth ions systematically which are largely motivated by their importance in a number of applications. These interactions are expressed as a power series in the inverse of the internuclear separation $R$. Both the dispersion and induction components of these interactions are determined accurately from the algebraic coefficients corresponding to each power combination in the series. Ultimately, these coefficients are expressed in terms of the electric multipole polarizabilities of the above mentioned systems which are calculated using the matrix elements obtained from a relativistic coupled-cluster method and core contributions to these quantities from the random phase approximation. We also compare our estimated polarizabilities with the other available theoretical and experimental results to verify accuracies in our calculations. In addition, we also evaluate the lifetimes of the first two low-lying states of the ions using the above matrix elements. Graphical representation of the interaction potentials versus $R$ are given among all the considered atoms and ions.

arXiv:1409.7154Sep 25, 20140 citations

{K}_{7}, {K}_{9},

and

{K}_{11}

{K}_{7}, {K}_{9},

and

{K}_{11}

{\mathit{C}}_{6}

{\mathit{C}}_{8}

, and

{\mathit{C}}_{10}

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

{\mathit{C}}_{6}

{\mathit{C}}_{8}

, and

{\mathit{C}}_{10}

for homonuclear and heteronuclear alkali-metal diatoms are compared with other calculations.

{1/R}^{6}

van der Waals coefficient for the interaction between

\mathrm{Cs}(6S)

and

\mathrm{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 (

{n}_{A}

P) and the other is in the ground state (

{n}_{B}

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

{C}_{6}

constants for all heteronuclear alkali-metal-atom pairs. In the case of heavier perturbers (Rb and Cs) the

{C}_{6}

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

{C}_{6}

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.

^+

, Ba

^+

, Sr

^+

, and Ra

^+

alkaline earth ions systematically which are largely motivated by their importance in a number of applications. These interactions are expressed as a power series in the inverse of the internuclear separation

R

. Both the dispersion and induction components of these interactions are determined accurately from the algebraic coefficients corresponding to each power combination in the series. Ultimately, these coefficients are expressed in terms of the electric multipole polarizabilities of the above mentioned systems which are calculated using the matrix elements obtained from a relativistic coupled-cluster method and core contributions to these quantities from the random phase approximation. We also compare our estimated polarizabilities with the other available theoretical and experimental results to verify accuracies in our calculations. In addition, we also evaluate the lifetimes of the first two low-lying states of the ions using the above matrix elements. Graphical representation of the interaction potentials versus

R

are given among all the considered atoms and ions.