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Physical Review A

Polarizability of Rydberg atoms and the dominant long-range interactions

Yukap Hahn

Published Sep 11, 2000• 7 citations
Publisher

Abstract

The dominant long-range interaction of an alkali-metal atom with charged particles at low energies is given for large separation R by the dispersive potential $W\ensuremath{\simeq}\ensuremath{-}{\ensuremath{\alpha}}_{d}{/R}^{4}$ in terms of the dipole polarizability ${\ensuremath{\alpha}}_{d}.$ For atoms prepared initially in Rydberg states of quantum numbers (n, l), the potential assumes a more complicated form due to the complete or near degeneracy of the n manifold. Contributions to the polarizability are treated in two parts, (a) one for the nondegenerate states and (b) the other for the degenerate or near degenerate cases. It is shown that ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\alpha}}}_{d}^{(b)}$ for case (b) is in general R dependent, and in the limit of complete degeneracy, diverges as ${R}^{2}.$ That is, for a small energy gap \ensuremath{\Delta} between a pair of nearly degenerate states which are dipole coupled, the dispersion potential ${W}^{(b)}\ensuremath{\simeq}{D/R}^{2}$ for $R<{R}_{x}$ and ${W}^{(b)}\ensuremath{\simeq}\ensuremath{-}{\ensuremath{\alpha}}_{d}^{(b)}{/R}^{4}$ for $R>{R}_{x},$ where D is the dipole moment, ${R}_{x}=[2|D/\ensuremath{\Delta}|{]}^{1/2},$ and ${\ensuremath{\alpha}}_{d}^{(b)}\ensuremath{\simeq}{D}^{2}/\ensuremath{\Delta}.$ They may also compete with a ${1/R}^{3}$ potential for Rydberg atoms with $l>0.$ The total ${\ensuremath{\alpha}}_{d}$ can be very large in magnitude for small \ensuremath{\Delta} and even assume negative values, but the corresponding ${R}_{x}$ also increases as \ensuremath{\Delta} decreases. The validity region in R of the ${R}^{\ensuremath{-}4}$ behavior of the potential recedes to larger R as the polarizability grows. A general formula for ${\ensuremath{\alpha}}_{d}$ is given, taking into account the effects of fine-structure splitting, the Lamb shift, and quantum defects.

DOI
10.1103/physreva.62.042703

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The dominant long-range interaction of an alkali-metal atom with charged particles at low energies is given for large separation R by the dispersive potential W\ensuremath≃\ensuremath−\ensuremathαd/R4W\ensuremath{\simeq}\ensuremath{-}{\ensuremath{\alpha}}_{d}{/R}^{4}W\ensuremath≃\ensuremath−\ensuremathαd​/R4 in terms of the dipole polarizability \ensuremathαd.{\ensuremath{\alpha}}_{d}.\ensuremathαd​. For atoms prepared initially in Rydberg states of quantum numbers (n, l), the potential assumes a more complicated form due to the complete or near degeneracy of the n manifold. Contributions to the polarizability are treated in two parts, (a) one for the nondegenerate states and (b) the other for the degenerate or near degenerate cases. It is shown that \ensuremathα\ifmmode~\else˜\fid(b){\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\alpha}}}_{d}^{(b)}\ensuremathα\ifmmode~\else˜\fi​d(b)​ for case (b) is in general R dependent, and in the limit of complete degeneracy, diverges as R2.{R}^{2}.R2. That is, for a small energy gap \ensuremath{\Delta} between a pair of nearly degenerate states which are dipole coupled, the dispersion potential W(b)\ensuremath≃D/R2{W}^{(b)}\ensuremath{\simeq}{D/R}^{2}W(b)\ensuremath≃D/R2 for R<RxR<{R}_{x}R<Rx​ and W(b)\ensuremath≃\ensuremath−\ensuremathαd(b)/R4{W}^{(b)}\ensuremath{\simeq}\ensuremath{-}{\ensuremath{\alpha}}_{d}^{(b)}{/R}^{4}W(b)\ensuremath≃\ensuremath−\ensuremathαd(b)​/R4 for R>Rx,R>{R}_{x},R>Rx​, where D is the dipole moment, Rx=[2∣D/\ensuremathΔ∣]1/2,{R}_{x}=[2|D/\ensuremath{\Delta}|{]}^{1/2},Rx​=[2∣D/\ensuremathΔ∣]1/2, and \ensuremathαd(b)\ensuremath≃D2/\ensuremathΔ.{\ensuremath{\alpha}}_{d}^{(b)}\ensuremath{\simeq}{D}^{2}/\ensuremath{\Delta}.\ensuremathαd(b)​\ensuremath≃D2/\ensuremathΔ. They may also compete with a 1/R3{1/R}^{3}1/R3 potential for Rydberg atoms with l>0.l>0.l>0. The total \ensuremathαd{\ensuremath{\alpha}}_{d}\ensuremathαd​ can be very large in magnitude for small \ensuremath{\Delta} and even assume negative values, but the corresponding Rx{R}_{x}Rx​ also increases as \ensuremath{\Delta} decreases. The validity region in R of the R\ensuremath−4{R}^{\ensuremath{-}4}R\ensuremath−4 behavior of the potential recedes to larger R as the polarizability grows. A general formula for \ensuremathαd{\ensuremath{\alpha}}_{d}\ensuremathαd​ is given, taking into account the effects of fine-structure splitting, the Lamb shift, and quantum defects.
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 〉.
The theory is developed for one- and two-atom interactions when the atom has a Rydberg electron attached to a hyperfine-split core state. This situation is relevant for some of the rare-earth and alkaline-earth atoms that have been proposed for experiments on Rydberg-Rydberg interactions. For the rare-earth atoms, the core electrons can have a very substantial total angular momentum JJJ and a nonzero nuclear spin III. In the alkaline-earth atoms there is a single (sss) core electron whose spin can couple to a nonzero nuclear spin for odd isotopes. The resulting hyperfine splitting of the core state can lead to substantial mixing between the Rydberg series attached to different thresholds. Compared to the unperturbed Rydberg series of the alkali-metal atoms, the series perturbations and near degeneracies from the different parity states could lead to qualitatively different behavior for single-atom Rydberg properties (polarizability, Zeeman mixing and splitting, etc.) as well as Rydberg-Rydberg interactions (C5{C}_{5}C5​ and C6{C}_{6}C6​ matrices).
The work presented in this dissertation represents a comprehensive study on long-range interactions between alkali Rydberg atoms. As a consequence of their size, Rydberg atoms possess very large dipole moments, making them highly sensitive to external fields and to other Rydberg atoms. The strong dipole-dipole interaction between two Rydberg atoms results in a mixing of their angular momentum characters, which is the cornerstone of the work presented here. We describe the ℓ-mixing between various local molecular asymptotes through a residual Coulomb potential, assuming no electron cloud overlap. We then directly diagonalize an interaction Hamiltonian consisting of these Rydberg-Rydberg interactions and spin-orbit (fine structure) coupling in the Hund's case (c). In this manner, we calculate potential energy curves for various molecular symmetries and find that some of these curves exhibit deep potential wells, capable of supporting very extended bound molecular states ( macrodimers). We analyze the specific structure of the ℓ-mixing for these potential wells and derive convenient n-scaling relations for both their equilibrium separations Re and well depths De. We also explore various properties of the macrodimers, including their response to small electric fields and stability with respect to predissociation. We note throughout the dissertation that the detection of such extended dimers could help progress studies in a variety of areas such as exotic, ultracold chemistry and quantum information and computing. To facilitate detection, we propose a scheme to form and study the macrodimers via photoassociation (PA). We present calculated PA lineshapes from two different electronic ground states based on a harmonic trapping model. We find that both PA signals vary significantly with the laser detunings, indicating that different ℓ-characters could be probed at different equilibrium lengths. Finally, we extend the two-body formalism to the case of three interacting Rydberg atoms and present preliminary results for triply excited 58p rubidium atoms along a common axis. We provide a few examples of surface plots corresponding to different local asymptotes, each with a unique three-dimensional "landscape" and discuss the relevance of the physical features. The highlight of our analysis is the appearance of a potential well, which indicates the formation of a linear trimer. ^
The most attractive and the most repulsive potential-energy curves for interaction between two Rydberg atoms in a broad superposition of internal angular momentum states are studied. The extreme Stark states have the largest dipole moments and provide the dominant contribution to the interaction which is then expressed as a long-range expansion involving the permanent multipole moments Q j of each polar atom. Analytical expressions are obtained for all Q j associated with principal quantum number n of H( n ) and permit the long-range expansion for the H( n )–H( n ') first-order interaction to be explicitly expressed analytically in terms of n , n ' and internuclear distance R . Possible quasi-molecular formation is investigated. Direct calculations show that the H( n = 2)–H( n ' = 2) interaction is capable of supporting 47 bound vibrational levels. As n increases, the long-range interaction becomes increasingly attractive so that molecular formation at large internuclear distances is expected to be scarcely possible for these extreme Stark levels.
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.
Calculations of the static electric-dipole scalar and tensor polarizabilities are presented for two alkali-metal atoms, Rb and Cs, for the $nS, n{P}_{\textonehalf{},\phantom{\rule{0.16em}{0ex}}3/2}$, and nD3/2,5/2n{D}_{3/2,\phantom{\rule{0.16em}{0ex}}5/2}nD3/2,5/2​ states with large principal quantum numbers up to n=50n=50n=50. The calculations are performed within an effective one-electron approximation, based on the Dirac-Fock Hamiltonian with a semiempirical core-polarization potential. The obtained results are compared with those from a simpler semiempirical approach and with available experimental data.