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arXiv:2608.22798

Exchange and core-polarization effects on Rydberg-transition electric-dipole matrix elements in Rb and Cs

Yu Shi-Cheng, Zhen-Xiang Zhong, Cheng-Bin Li

Published Aug 24, 2026• 0 citations
arXivHTMLPDF

Abstract

Microwave electric-field measurements with Rydberg atoms require accurate Rydberg-transition electric-dipole matrix elements. We calculate these matrix elements for Rb and Cs using a Dirac-Fock plus core-polarization method, which core-valence exchange is treated explicitly and core-valence correlation is represented by a cutoff core-polarization potential. A smooth cutoff-radius model fitted to quantum-defect energies reproduces the target energies with MHz-level residuals for states up to $n=90$. Comparisons with model-potential matrix elements and internal length-velocity consistency show that the $s$-$p$ transitions are generally the most robust, whereas the $p$-$d$ and $d$-$f$ transitions show stronger atom- and branch-dependent sensitivity to short-range modeling. The largest relative differences arise from radial-integral cancellation, where short-range phase changes are amplified by the small final matrix element. These results provide a practical reliability assessment for Rydberg-transition electric-dipole matrix elements used in field-sensing applications.

DOI
10.48550/arxiv.2608.22798

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Microwave electric-field measurements with Rydberg atoms require accurate Rydberg-transition electric-dipole matrix elements. We calculate these matrix elements for Rb and Cs using a Dirac-Fock plus core-polarization method, which core-valence exchange is treated explicitly and core-valence correlation is represented by a cutoff core-polarization potential. A smooth cutoff-radius model fitted to quantum-defect energies reproduces the target energies with MHz-level residuals for states up to n=90n=90n=90. Comparisons with model-potential matrix elements and internal length-velocity consistency show that the sss-ppp transitions are generally the most robust, whereas the ppp-ddd and ddd-fff transitions show stronger atom- and branch-dependent sensitivity to short-range modeling. The largest relative differences arise from radial-integral cancellation, where short-range phase changes are amplified by the small final matrix element. These results provide a practical reliability assessment for Rydberg-transition electric-dipole matrix elements used in field-sensing applications.
Accurate data on electric-dipole transition matrix elements (EDTMs) for bound-bound Rydberg-atom transitions have become increasingly important in science and technology. Here we compute radial EDTMs of rubidium in length, velocity, and acceleration forms for electric-dipole-allowed transitions between states with principal and angular-momentum quantum numbers nnn and $\ensuremath{\ell}$ ranging from 15 to 100. Wave functions are computed based upon model potentials from [Phys. Rev. A 49, 982 (1994)]. Length-gauge EDTMs of strong low-$\ensuremath{\ell}$ Rydberg transitions, often employed in research and technology, are found to deviate from the fundamentally more accurate velocity-gauge form by approximately-nnn-independent and series-dependent shifts. The shift amount peaks at about 0.34ea00.34\phantom{\rule{4pt}{0ex}}e{a}_{0}0.34ea0​ for the Rb $nD\ensuremath{\leftrightarrow}\phantom{\rule{4pt}{0ex}}(n+1)P$ series, a particularly strong Rydberg-transition series in Rb. The shift corresponds to relative EDTM corrections of up to $\ensuremath{\approx}{10}^{\ensuremath{-}3}$, which can be of concern in high-precision applications. We discuss the physical reasons for the observed gauge differences, explain the conditions for applicability of the velocity- and length-gauge forms for different transition series, and present a decision tree of how to choose EDTMs.
The radial electric dipole moments of bound-bound, bound-free, and free-free transitions of high-lying (n>15)(n>15)(n>15) Rydberg states and low-lying continuum states $(\ensuremath{\epsilon}<400$ ${\mathrm{cm}}^{\ensuremath{-}1})$ are calculated for hydrogen and the alkali-metal atoms. All dipole matrix elements for the transitions between low-lll (s(s(s, ppp, ddd, and f)f)f) states of H, Li, and Rb are presented. Results of similar calculations for the other alkali-metal atoms (Na, K, and Cs) are summarized. These numerical results, using quantum defect theory, are compared with semiclassical approximations. The dipole matrix elements cross zero at certain combinations of initial and final states. The matrix elements at these so-called Cooper minima in Li, K, Rb, and Cs are presented in detail.
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.
The high-precision determination of microwave radiation parameters may be based on measurements of the spectral characteristics of radiation transitions between the Rydberg states of atoms. Frequencies and matrix elements are calculated for dipole transitions from even-parity nS1/2 and nD5/2 to odd-parity n′P3/2 and n′F7/2 (where n′ = n, n ± 1, n ± 2) for the Rydberg states of alkali-metal atoms. The matrix elements determine the splitting of Rydberg-state energy levels in the field of a resonance microwave (μw) radiation, which results in the splitting of the resonance in electromagnetic induced transparency (EIT). Numerical computations based on the single-electron quantum defect method (QDM) and the Fues’ model potential (FMP) approach with the use of the most reliable data of the current literature on quantum defect values were performed for the 2S, 2P, 2D and 2F series of the Rydberg states of Li, Na, K, Rb and Cs atoms. The calculated data were approximated by quadratic polynomials of the principal quantum number. The polynomial coefficients were determined with the use of a standard curve-fitting interpolation polynomial procedure for numerically presented functions. The approximation equations may be used for the accurate evaluation of the frequencies and matrix elements of μw transitions in wide ranges of the Rydberg-state quantum numbers n >> 1.
We develop practical formulas for the calculation of the matrix elements of the interaction of the electromagnetic field with an atomic state, beyond the long-wavelength approximation. The atom-plus-field Hamiltonian is chosen to have the multipolar form, containing the electric, paramagnetic, and diamagnetic operators. The final workable expressions include the interactions to all orders and are derived by first expanding the fields in partial waves. The electric-field operator reaches a constant value as the radial variable becomes large, contrary to the result of the electric-dipole approximation (EDA) where the value of the corresponding operator increases indefinitely. Applications are given for Rydberg states of hydrogen up to n=50n=50n=50 and for free-free transitions in a Coulomb potential. Such matrix elements are relevant to a number of real and virtual processes occurring during laser-atom interactions. The computation is done numerically, using a combination of analytic with numerical techniques. By comparing the results of the EDA with those of the exact treatment, it is shown that the former is inadequate in such cases. This finding has repercussions on the theory and understanding of the physics of quantum systems in high-lying Rydberg levels and wave packets or in scattering states.
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