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 〉.
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 〉.
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, axial (vector) Raa, scalar-tensor RsT=RTs, and tensor-tensor 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). The polynomial presentations in powers of the Rydberg-state principal quantum number n 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,n2D3/2,5/2, and n2F5/2,7/2 Rydberg states of arbitrary high n. The transformation of the van der Waals interaction law $\ensuremath{-}{C}_{6}/{R}^{6}$ into the dipole-dipole law 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 M and on the angle $\ensuremath{\theta}$ of the constant ${C}_{3}(\ensuremath{\theta};nlJM)$ are determined together with the ranges of interatomic distances R, where the transformation appears.
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 .
Rydberg atoms are used in a wide range of applications due to their peculiar properties like strong dipolar and van der Waals interactions. The choice of Rydberg state has a huge impact on the strength and angular dependence of the interactions, and so a detailed understanding of the underlying processes and resulting properties of the interactions is therefore key to select the most suitable states for experiments. We study the van der Waals interactions in alkali-metal atoms in detail and highlight the structures which allow an understanding and exploitation of the various interaction properties. A particular theme is the identification of Förster resonances with n 1 ≠ n 2 , which offer interaction potentials with a wide range of properties that make them particularly interesting for experimental applications. A second theme is a focus on the underlying structures that shape the angular dependency and sign of the interactions. This understanding — instead of brute-force calculations — allows for a much simpler and more systematic search for suitable pair states. These insights can be used for the selection of tailored interaction potentials subject to experimental constraints and requirements. We use rubidium as an example species in this work and also provide data for cesium and pair states that are coupled via two- or three-photon transitions, i.e. up to F states, in the Appendix.
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 ∣nsrangle and ∣nprangle 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 far from these resonances the van der Waals coefficients, C6sp, can be defined. We calculate the C6 coefficients for alkali atoms and present the results for lithium by applying perturbation theory. At short interatomic distances of several mum, 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.
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.