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Journal of Physics Communications

Asymptotic approximations to the energy of dispersion interaction between rubidium atoms in Rydberg states

Aleksandr A. Kamenski, Sergey N. Mokhnenko, Vitalii D Ovsiannikov

Published Aug 22, 2017• 1 citation
Publisher

Abstract

Irreducible components determining the dependence of the van der Waals coefficient C 6 ( n l J M ) on the angle θ between the interatomic and the quantisation axes of two Rb atoms in their identical Rydberg states | n l J M ⟩ are evaluated with account of the most contributing terms of the spectral resolution for the bi-atomic Green’s function. Asymptotic polynomials in powers of the Rydberg-state principal quantum number n are derived for the C 6 irreducible components. Numerical values of the polynomial coefficients are determined for Rb atoms in their n 2 S 1 / 2 , n 2 P 1 / 2 , 3 / 2 , n 2 D 3 / 2 , 5 / 2 and n 2 F 5 / 2 , 7 / 2 Rydberg states of arbitrary high n . The transformation of the van-der-Waals interaction law − C 6 / R 6 into the dipole–dipole law C 3 / R 3 in the case of close two-atomic states (the Förster resonance) is considered. Numerical values, the dependences on the magnetic quantum numbers M and on the angle θ of the constant C 3 are determined together with the ranges of interatomic distances R , where the interaction law R − 6 transforms into the law of R − 3 .

DOI
10.1088/2399-6528/aa76c8

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Irreducible components determining the dependence of the van der Waals coefficient C 6 ( n l J M ) on the angle θ between the interatomic and the quantisation axes of two Rb atoms in their identical Rydberg states | n l J M ⟩ are evaluated with account of the most contributing terms of the spectral resolution for the bi-atomic Green’s function. Asymptotic polynomials in powers of the Rydberg-state principal quantum number n are derived for the C 6 irreducible components. Numerical values of the polynomial coefficients are determined for Rb atoms in their n 2 S 1 / 2 , n 2 P 1 / 2 , 3 / 2 , n 2 D 3 / 2 , 5 / 2 and n 2 F 5 / 2 , 7 / 2 Rydberg states of arbitrary high n . The transformation of the van-der-Waals interaction law − C 6 / R 6 into the dipole–dipole law C 3 / R 3 in the case of close two-atomic states (the Förster resonance) is considered. Numerical values, the dependences on the magnetic quantum numbers M and on the angle θ of the constant C 3 are determined together with the ranges of interatomic distances R , where the interaction law R − 6 transforms into the law of R − 3 .
The van der Waals coefficient ${C}_{6}(\ensuremath{\theta};nlJM)$ of two like Rydberg atoms in their identical Rydberg states $|nlJM\ensuremath{\rangle}$ is resolved into four irreducible components called scalar Rss{R}_{ss}Rss​, axial (vector) Raa{R}_{aa}Raa​, scalar-tensor RsT=RTs{R}_{sT}={R}_{Ts}RsT​=RTs​, and tensor-tensor RTT{R}_{TT}RTT​ parts in analogy with the components of dipole polarizabilities. The irreducible components determine the dependence of ${C}_{6}(\ensuremath{\theta};nlJM)$ on the angle $\ensuremath{\theta}$ between the interatomic and the quantization axes of atoms. The spectral resolution for the biatomic Green's function with account of the most contributing terms is used for evaluating the components ${R}_{\ensuremath{\alpha}\ensuremath{\beta}}$ of atoms in their Rydberg series of doublet states of the low angular momenta (2S,2P,2D,2F^{2}S, ^{2}P, ^{2}D, ^{2}F2S,2P,2D,2F). The polynomial presentations in powers of the Rydberg-state principal quantum number nnn taking into account the asymptotic dependence ${C}_{6}(\ensuremath{\theta};nlJM)\ensuremath{\propto}{n}^{11}$ are derived for simplified evaluations of irreducible components. Numerical values of the polynomial coefficients are determined for Rb atoms in their n2S1/2,n2P1/2,3/2,n2D3/2,5/2n{\phantom{\rule{0.222222em}{0ex}}}^{2}{S}_{1/2}, n{\phantom{\rule{0.222222em}{0ex}}}^{2}{P}_{1/2,3/2}, n{\phantom{\rule{0.222222em}{0ex}}}^{2}{D}_{3/2,5/2}n2S1/2​,n2P1/2,3/2​,n2D3/2,5/2​, and n2F5/2,7/2n{\phantom{\rule{0.222222em}{0ex}}}^{2}{F}_{5/2,7/2}n2F5/2,7/2​ Rydberg states of arbitrary high nnn. The transformation of the van der Waals interaction law $\ensuremath{-}{C}_{6}/{R}^{6}$ into the dipole-dipole law C3/R3{C}_{3}/{R}^{3}C3​/R3 in the case of close dipole-connected two-atomic states (the F\"orster resonance) is considered and the dependencies on the magnetic quantum numbers MMM and on the angle $\ensuremath{\theta}$ of the constant ${C}_{3}(\ensuremath{\theta};nlJM)$ are determined together with the ranges of interatomic distances RRR, where the transformation appears.
With the use of second-order perturbation theory in the long-range interatomic interaction for the degenerate states of two Rydberg atoms we have obtained a general formula for the dependence of atomic interaction energy on the interatomic distance R in the presence of the Förster resonance. Inside of the ‘Förster sphere’ ( R < R F ) this dependence transforms to the formula for electric dipole interaction energy Δ E d − d = C 3 / R 3 and for R > R F it transforms to the formula for the van der Waals interaction energy Δ E VdW = − C 6 / R 6 . The van der Waals constant C 6 is represented as an expansion in terms of irreducible components which define the dependence on the interatomic axis orientation relative to the quantisation axis of projections M of the total angular momentum J . The numerical values of the irreducible components of tensor C 6 were calculated for rubidium atoms in the same Rydberg states | nlJM 〉 with large quantum numbers n . We present the calculated resonance interaction energy of two rubidium atoms in the states |43D 5/2 M 〉, whose total energy exceeds by only 8 MHz the total energy of one of the atoms in the state |45P 3/2 M 〉 and of the other in the state |41F 7/2 M 〉.
Abstract The numerical values of the van der Waals interaction constant C6 are determined between alkaline earth atoms in singlet Rydberg |n1S0〉 states with large principal quantum numbers, n > 20. The results of calculations using the semiempirical quantum defect theory (QDT) method are approximated by quadratic polynomials. The polynomial coefficients are tabulated and can be used for simplified quantitative estimates of interatomic interaction-induced energy shifts, determination of the Rydberg state excitation blockade radius, and limitations on the wavelengths of radiation forming Rydberg lattices for group IIA atoms. The numerical values of the contributions of various diatomic states to the constant C6 are calculated. States providing the main contribution, analogous to that of the Förster resonance, are identified.
Abstract Analytical expressions are obtained for the constants determining the energy of the long-range (at large distances exceeding the LeRoy radius R LR ≈ 4n 2) polarisation interaction of the multipole electric moments of two atoms in identical Rydberg states with large principal (n) and maximum orbital (l) and magnetic (m) quantum numbers (|m| = l = n − 1). Asymptotic expansions in powers of n are found for the components of the van der Waals interaction tensor, which determine the dependence of the dispersion energy on the orientation of the interatomic axis. The spontaneous and thermally induced broadenings of the energy levels of circular states are represented analytically as functions of the principal quantum number and temperature. It is found that analytical formulae for the sum of spontaneous and thermally induced line widths in the region of high temperatures and large principal quantum numbers (assuming n 3 k B T >> 1) do not contain a temperature-independent contribution of the spontaneous part of the broadening. Closed analytical expressions are also obtained for the Farley − Wing function and for the n- and T-dependent corrections to the asymptotic behaviour of the thermally induced shift.
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.
Abstract We investigate the influence of the appearance of excitonic states on van der Waals interactions among two Rydberg atoms. The atoms are assumed to be in different Rydberg states, e.g., in the and states. The resonant dipole-dipole interactions yield symmetric and antisymmetric excitons, with energy splitting that give rise to new resonances as the atoms approach each other. Only away from these resonances can the van der Waals coefficients, C 6 sp , be defined. We calculate the C 6 coefficients for alkali atoms and present the results for lithium by applying perturbation theory. At short interatomic distances of several , we show that the widely used simple model of two-level systems for excitons in Rydberg atoms breaks down, and the correct representation implies multi-level atoms. Even though, at larger distances one can keep the two-level systems but in including van der Waals interactions among the atoms .