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Physical Review A

Energy of van der Waals and dipole-dipole interactions between atoms in Rydberg states

Aleksandr A. Kamenski, Nikolai L. Manakov, Sergey N. Mokhnenko, Vitalii D Ovsiannikov

Published Sep 21, 2017• 13 citations
Publisher

Abstract

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 ${R}_{ss}$, axial (vector) ${R}_{aa}$, scalar-tensor ${R}_{sT}={R}_{Ts}$, and tensor-tensor ${R}_{TT}$ 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 ($^{2}S, ^{2}P, ^{2}D, ^{2}F$). 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 $n{\phantom{\rule{0.222222em}{0ex}}}^{2}{S}_{1/2}, n{\phantom{\rule{0.222222em}{0ex}}}^{2}{P}_{1/2,3/2}, n{\phantom{\rule{0.222222em}{0ex}}}^{2}{D}_{3/2,5/2}$, and $n{\phantom{\rule{0.222222em}{0ex}}}^{2}{F}_{5/2,7/2}$ 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 ${C}_{3}/{R}^{3}$ 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.

DOI
10.1103/physreva.96.032716

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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.

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The van der Waals coefficient C6(\ensuremathθ;nlJM){C}_{6}(\ensuremath{\theta};nlJM)C6​(\ensuremathθ;nlJM) of two like Rydberg atoms in their identical Rydberg states ∣nlJM\ensuremath⟩|nlJM\ensuremath{\rangle}∣nlJM\ensuremath⟩ 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 C6(\ensuremathθ;nlJM){C}_{6}(\ensuremath{\theta};nlJM)C6​(\ensuremathθ;nlJM) on the angle \ensuremathθ\ensuremath{\theta}\ensuremathθ 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\ensuremathα\ensuremathβ{R}_{\ensuremath{\alpha}\ensuremath{\beta}}R\ensuremathα\ensuremathβ​ 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 C6(\ensuremathθ;nlJM)\ensuremath∝n11{C}_{6}(\ensuremath{\theta};nlJM)\ensuremath{\propto}{n}^{11}C6​(\ensuremathθ;nlJM)\ensuremath∝n11 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−C6/R6\ensuremath{-}{C}_{6}/{R}^{6}\ensuremath−C6​/R6 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 \ensuremathθ\ensuremath{\theta}\ensuremathθ of the constant C3(\ensuremathθ;nlJM){C}_{3}(\ensuremath{\theta};nlJM)C3​(\ensuremathθ;nlJM) are determined together with the ranges of interatomic distances RRR, 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 .
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{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.
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|ns\\rangle∣nsrangle and ∣nprangle|np\\rangle∣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, C6spC_6^{sp}C6sp​, can be defined. We calculate the C6C_6C6​ coefficients for alkali atoms and present the results for lithium by applying perturbation theory. At short interatomic distances of several mum\\mu mmum, 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.
Mesure de l’interaction de van der Waals entre deux atomes de Rydberg Les atomes neutres sont des candidats prometteurs pour la réalisation et l’étude d’états intriqués à quelques dizaines de particules. Pour générer de tels états, une approche consiste à utiliser le mécanisme de blocage dipolaire résultant des fortes interactions dipôle-dipôle entre atomes de Rydberg.Suivant cette approche, cette thèse présente la conception et la caractérisation d’un dispositif expérimental permettant de manipuler des atomes de 87Rb individuels piégés dans des pinces op- tiques microscopiques, et à les exciter vers des états de Rydberg. Un environnement électrostatique stable et des électrodes de contrôle permettent une manipulation fine de ces états. Avec deux pinces optiques séparées de quelques microns, nous démontrons le blocage de Rydberg entre deux atomes, et nous observons leur excitation collective.Enfin, en opérant en régime de blocage partiel, nous développons une méthode permettant de mesurer l’interaction de van der Waals ∆E = C6 /R6 entre deux atomes séparés par une distance R contrôlée. Les coefficients C6 obtenus pour différents états de Rydberg sont en bon accord avec des calculs théoriques ab initio, et nous observons l’augmentation spectaculaire de l’interaction en fonction du nombre quantique principal n de l’état de Rydberg.
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.