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International Conference on Optics, Physics, and Electronic Information (OPEI)

Parametric approximation of Stark energy shift in the highly excited Rydberg state of Cs atoms

Guo Ma, Chao Li

Published May 7, 2026• 0 citations
Publisher

Abstract

This paper systematically studies the energy level shift (Stark effect) of the highly excited Rydberg state of cesium (Cs) atoms under the action of an electrostatic field. In view of the limitations of the traditional hydrogen atom Stark perturbation theory in describing alkali metal atoms, we propose a complete set of parametric approximation methods, which fully consider the correction of energy level structure by quantum number deficiency. By introducing the effective principal quantum number n* and the correction coefficient 𝛽l, we extend the fourth-order Stark perturbation formula for hydrogen atoms to the Cs atomic system. The study first reviewed the analytical theory of the Stark effect of hydrogen atoms and derived the fourth-order energy shift expression based on parabolic quantum numbers. Then, the quantum number depletion characteristics of Cs atoms and their influence on Stark matrix elements were analyzed; On this basis, the parametric correction rules of the system were established, including the replacement of the effective quantum number, the determination of the polarizability correction coefficient, and the conditional use of the linear term. We have developed corresponding computing programs that can quickly estimate the energy shift of different quantum states under any electric field. By comparing with the existing experimental data and precise numerical calculation results, the reliability and application scope of this method were verified. Research shows that for the high-Rydberg state with n>20, the relative error of this parametric method under moderate electric field intensity (10-100 V/cm) is very small, making it particularly suitable for rapid estimation in large-scale parameter scanning and experimental design. The work of this paper provides practical tools and theoretical basis for the theoretical research and experimental analysis of the Stark effect of alkali metal atoms.

DOI
10.1117/12.3111185

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This paper systematically studies the energy level shift (Stark effect) of the highly excited Rydberg state of cesium (Cs) atoms under the action of an electrostatic field. In view of the limitations of the traditional hydrogen atom Stark perturbation theory in describing alkali metal atoms, we propose a complete set of parametric approximation methods, which fully consider the correction of energy level structure by quantum number deficiency. By introducing the effective principal quantum number n* and the correction coefficient 𝛽l, we extend the fourth-order Stark perturbation formula for hydrogen atoms to the Cs atomic system. The study first reviewed the analytical theory of the Stark effect of hydrogen atoms and derived the fourth-order energy shift expression based on parabolic quantum numbers. Then, the quantum number depletion characteristics of Cs atoms and their influence on Stark matrix elements were analyzed; On this basis, the parametric correction rules of the system were established, including the replacement of the effective quantum number, the determination of the polarizability correction coefficient, and the conditional use of the linear term. We have developed corresponding computing programs that can quickly estimate the energy shift of different quantum states under any electric field. By comparing with the existing experimental data and precise numerical calculation results, the reliability and application scope of this method were verified. Research shows that for the high-Rydberg state with n>20, the relative error of this parametric method under moderate electric field intensity (10-100 V/cm) is very small, making it particularly suitable for rapid estimation in large-scale parameter scanning and experimental design. The work of this paper provides practical tools and theoretical basis for the theoretical research and experimental analysis of the Stark effect of alkali metal atoms.
The Stark effect in Rydberg atoms has potential applications in the areas of dipole-dipole interaction, quantum information, quantum control, and so on. Many reflevant theoretical calculations and experimental studies about the Stark effect of alkali metal and alkali earth metals have been reported, but the other atom’s Stark effect is studied still relatively less. Our goal in this paper is to reflearch the third main group atom’s Stark effect in a large electric field. First, according to the level data of gallium atom in zero-field, we obtain the quantum defects from the modified Ritz formula in each state by using a nonlinear least-squares-fitting algorithm. The quantum defects as a function of the principal quantum number are analyzed in detail. Influences of both the core polarization and the penetrating valence electron on the quantum defect are discussed according to the fitting results. Then we use the Numerov algorithm to calculate the radial wave functions of atomic gallium. Finally, the Stark structures of Rydberg states around n=7 and n=18 are numerically calculated by matrix diagonalization. Results show that at the levels above n=7 manifold states, (n+1)P is higher than nD state, and it is in contrast to the levels below the n=7 manifold states. This phenomenon is different from the usual Stark structure of alkali metal atoms, the level’s order of which does not change with the principal quantum number. The Stark levels with the identical |m| anti-cross each other, and those with different |m| cross. Our results give an important reflerence for related reflearches, and are of great significance for insight into the atomic structure and the interaction between the atomic core and the highly excited electrons.
We report on the measurement of Stark shifted energy levels of 87 Rb Rydberg atoms in static electric fields by means of electromagnetically induced transparency (EIT). Electric field strengths of up to 500 V cm −1 , ranging beyond the classical ionization threshold, were applied using electrodes inside a glass cell with rubidium vapour. Stark maps for principal quantum numbers n = 35 and n = 70 have been obtained with high signal-to-noise ratio for comparison with results from ab initio calculations following the method described in (Zimmerman et al 1979 Phys. Rev. A 20 2251 ), which was originally only verified for states around n = 15. We also calculate the dipole matrix elements between low-lying states and Stark shifted Rydberg states to give a theoretical estimate of the relative strength of the EIT signal. The present work significantly extends the experimental verification of this numerical method in the range of both high principal quantum numbers and high electric fields with an accuracy of up to 2 MHz.
A highly excited (Rydberg) atom bathed in blackbody radiation is perturbed in two ways. A dynamic Stark shift is induced by the off-resonant components of the blackbody radiation. Additionally, electric-dipole transitions to other atomic energy levels are induced by the resonant components of the blackbody radiation. This depopulation effect shortens the Rydberg-state lifetime, thereby broadening the energy level. Calculations of these two effects in many states of hydrogen, helium, and the alkali-metal atoms Li, Na, K, Rb, and Cs are presented for T=300T=300T=300 K. Contributions from the entire blackbody spectrum and from both discrete and continuous perturbing states are included. The accuracy is considerably greater than that of previous estimates.
We present the results of studying the characteristics of Rydberg alkali atomic systems in a black-body (BBR; thermal) radiation field, in particular, BBR induced Stark shift coefficient k. As theoretical approach the combined generalized relativistic energy approach and relativistic many-body perturbation theory (PT) with ab initio Dirac zeroth approximation is applied. Application of theory to computing the spectral parameters of studied atomic systems have demonstrated physically reasonable agreement between the theoretical and experimental data. carefully. It should be noted that our method takes carefully into account such important factors as the implementation of a gauge invariance principle when calculating the corresponding matrix elements, the correct degree of consideration of complex exchange-correlation effects (primarily, the effect of polarization of the core), and also generates a fairly optimal one-quasiparticle representation in within the limits of many-body relativistic PT with ab initio zeroth Dirac-Fock (Kohn-Sham) approximation. In any case, the formalism developed in this work, as it follows from the given results, can be used for precise calculations of the Stark shift coefficient k, the BBR shift parameter β, and other parameters.
The Stark effect of the fine structure of hydrogenic atoms is discussed in the light of the quantum mechanics. Explicit formulas are given for the energy levels in either a weak or a strong electric field, including in each case both the relativity corrections and the Stark effect. (The conventional treatment neglects either one or the other.) The results are, except in form, similar to those recently given by Schlapp. In the new mechanics, in contradistinction to the old Bohr theory, there is a linear Stark effect even in weak fields, because of the identity of energy for the ns\mathrm{ns}ns and np1n{p}_{1}np1​ levels in the absence of external fields. This degeneracy is shown to account for the results of the recent experiments of Ornstein, Zernike, and Snoek, who found that the 2s2s2s level in HHH is not metastable.The coefficients are found for the development of the "parabolic" eigenfunctions in terms of the "polar" ones, and a set of numerical values of these coefficients is tabulated.Methods are given for the calculation of the relative intensities in the Stark effect in terms of the already available theoretical fine structure intensities in the absence of fields.
The adiabatic \ensuremath{\Sigma} electronic terms of highly excited states of the He--alkali-metal atom systems are determined as the roots of a transcendental equation whose only required inputs are the alkali-metal quantum defects, the ${e}^{\mathrm{\ensuremath{-}}}$-He s-wave phase shifts and, of course, the internuclear distance R. The results obtained for relatively low excited states are in reasonable agreement with more elaborate calculations. It is found that the notion of adiabatic quantum defects requires R(a.u.)\ensuremath{\lesssim}2n, n being the principal quantum number. The method is generalized to the case where a small external electric field scrE is present. Then the energy levels strongly depend on the angle between scrE and the internuclear axis. Numerical applications to He-H are presented.