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.
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.
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 .
We calculate the interaction strength of the van der Waals force between two Rydberg atoms by applying the applications of quantum information processing using for Rydberg blockade. The alkali metals (Cs and K) in states of principal quantum number n were used to calculate the interaction strength. We use some possible angular momentum channels involving s, p, and d states for measuring interaction strength. The obtained results were then generalized for all angular momentum channels which mostly have small interactions and therefore a poor candidate for blockade experiments. The interaction strength in the atoms of Cs and K dipole matrix elements was calculated first and then relevant energy levels were determined by using the quantum defects theory. The radial wave function was calculated numerically with the integration of the radial Schrodinger equation. Also, the interaction coefficients of van der Waals for various channels were calculated here.
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 〉.
We study the van der Waals interaction between Rydberg alkali-metal atoms with fine structure (
n2Lj; $L\ensuremath{\le}2$) and heteronuclear alkali-metal dimers in the ground rovibrational state ($X^{1}\mathrm{\ensuremath{\Sigma}}^{+}$;
v=0,J=0). We compute the associated
C6 dispersion coefficients of atom-molecule pairs involving
133Cs and
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) over the range of principal quantum numbers $40\ensuremath{\le}n\ensuremath{\le}150$. For all atom-molecule pairs considered, Rydberg states
n2Sj and
n2Pj result in attractive
1/R6 potentials. In contrast,
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>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 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.
We report the direct measurement of the van der Waals interaction between two isolated, single Rydberg atoms separated by a controlled distance of a few micrometers. Working in a regime where the single-atom Rabi frequency for excitation to the Rydberg state is comparable to the interaction, we observe partial Rydberg blockade, whereby the time-dependent populations of the various two-atom states exhibit coherent oscillations with several frequencies. Quantitative comparison of the data with a simple model based on the optical Bloch equations allows us to extract the van der Waals energy, and observe its characteristic C6/R6 dependence. The measured C6 coefficients agree well with ab initio calculations, and we observe their dramatic increase with the principal quantum number n of the Rydberg state.