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Physical Review A

Rydberg atoms in far-infrared radiation fields. I. Dipole matrix elements of H, Li, and Rb

J. H. Hoogenraad, Lambertus D. Noordam

Published Jun 1, 1998• 34 citations
Publisher

Abstract

The radial electric dipole moments of bound-bound, bound-free, and free-free transitions of high-lying $(n>15)$ 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-$l$ $(s$, $p$, $d$, and $f)$ 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.

DOI
10.1103/physreva.57.4533

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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 (\ensuremathϵ<400(\ensuremath{\epsilon}<400(\ensuremathϵ<400 cm\ensuremath−1){\mathrm{cm}}^{\ensuremath{-}1})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.
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.
The structure of the Cooper minima in the transition probabilities and photoionization cross-sections for low-excited and Rydberg nS, nP, nD and nF states of alkali-metal atoms has been studied using a Coulomb approximation and a quasiclassical model. The range of applicability of the quasiclassical model has been defined from comparison with available experimental and theoretical data on dipole moments, oscillator strengths, and photoionization cross-sections. A new Cooper minimum for transitions between rubidium Rydberg states has been found.
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.
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.
We have studied ionization of alkali-metal Rydberg atoms by blackbody radiation (BBR). The results of theoretical calculations of ionization rates of Li, Na, K, Rb and Cs Rydberg atoms are presented. The calculations have been performed for n S, n P and n D states for principal quantum numbers n =8–65 at ambient temperatures of 77, 300 and 600 K. The calculations take into account the contributions of BBR-induced redistribution of population between Rydberg states prior to photoionization and field ionization by extraction electric field pulses. The obtained results show that these phenomena affect both the magnitude of the measured ionization rates and their n dependence. A Cooper minimum for BBR-induced transitions between bound Rydberg states of Li has been found. The calculated ionization rates are compared with our earlier measurements of BBR-induced ionization rates of Na n S and n D Rydberg states with n =8–20 at 300 K. Good agreement for all states except n S with n >15 is observed. Useful analytical formulae for quick estimates of BBR ionization rates of Rydberg atoms are presented. Application of BBR-induced ionization signal to measurements of collisional ionization rates is demonstrated.