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Tune-out and Magic Wavelengths for Helium Atom with Exponential-Cosine-Screened Coulomb Potentials

Y. Qiao Wang, Zishi Jiang, Sabyasachi Kar

Published Feb 11, 2026• 0 citations
Publisher

Abstract

Abstract We investigate the effects of exponential-cosine-screened Coulomb potentials (ECSCP) on the tune-out and magic wavelengths (TMW) of helium atom using correlated exponential wavefunctions based on the pseudostate summation approach. The first tune-out wavelengths (TWs) for the \(\:{\text{}\text{n}}^{\text{}\text{1}}\text{S\:(}\text{n}\text{}\text{=}\text{}\text{1,}\text{}\text{2,}\text{}\text{3)\:}\)and \(\:{\text{n}\text{}}^{\text{3}}\text{S}\text{\:(}\text{n}\text{}\text{=}\text{}\text{2,}\text{}\text{3}\text{)\:}\)states, and the magic wavelengths (MWs) for the \(\:{\text{2}}^{\text{3}}\text{S}\text{}\text{→}{\text{}\text{3}}^{\text{3}}\text{S}\) transition along with the dynamic dipole polarizabities (DDP) for the \(\:{\text{}\text{n}}^{\text{}\text{1}}\text{S\:(}\text{n}\text{}\text{=}\text{}\text{1,}\text{}\text{2,}\text{}\text{3)\:}\) and \(\:{\text{n}\text{}}^{\text{3}}\text{S}\text{\:(}\text{n}\text{}\text{=}\text{}\text{2,}\text{}\text{3}\text{)\:}\)states of helium atom are reported as functions of screening parameter. Our results in the free-atom cases are in good accord with the available theoretical and experimental data. Moreover, the TMW exhibit interesting features as functions of screening parameters, reflecting the unique characteristics of ECSCP.

DOI
10.21203/rs.3.rs-8770762/v1

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Abstract We investigate the effects of exponential-cosine-screened Coulomb potentials (ECSCP) on the tune-out and magic wavelengths (TMW) of helium atom using correlated exponential wavefunctions based on the pseudostate summation approach. The first tune-out wavelengths (TWs) for the (:{\text{}\text{n}}^{\text{}\text{1}}\text{S:(}\text{n}\text{}\text{=}\text{}\text{1,}\text{}\text{2,}\text{}\text{3):})and (:{\text{n}\text{}}^{\text{3}}\text{S}\text{:(}\text{n}\text{}\text{=}\text{}\text{2,}\text{}\text{3}\text{):})states, and the magic wavelengths (MWs) for the (:{\text{2}}^{\text{3}}\text{S}\text{}\text{→}{\text{}\text{3}}^{\text{3}}\text{S}) transition along with the dynamic dipole polarizabities (DDP) for the (:{\text{}\text{n}}^{\text{}\text{1}}\text{S:(}\text{n}\text{}\text{=}\text{}\text{1,}\text{}\text{2,}\text{}\text{3):}) and (:{\text{n}\text{}}^{\text{3}}\text{S}\text{:(}\text{n}\text{}\text{=}\text{}\text{2,}\text{}\text{3}\text{):})states of helium atom are reported as functions of screening parameter. Our results in the free-atom cases are in good accord with the available theoretical and experimental data. Moreover, the TMW exhibit interesting features as functions of screening parameters, reflecting the unique characteristics of ECSCP.
Abstract The tune‐out and magic wavelengths (TMWs) for the hydrogenlike and screened‐hydrogenlike atoms (Z = 1‐8) are calculated using a variational method. The first three tune‐out wavelengths for each of the nS (n = 1, 2, 3) states and the first three magic wavelengths for each of the 1S‐nS (n = 2, 3) transitions of the hydrogenlike atoms in terms of nuclear charge Z, and the first TMWs for the above mentioned states or transitions of the screened‐hydrogenlike atoms in terms of the screening parameter μ and Z are presented. All the results for the TMWs are reported for the first time in the literature except the magic wavelengths for the hydrogen atom. The magic wavelengths for hydrogen atom are in agreement with the available data. The precise estimation of the TMWs would be helpful for the design of the ion‐trapping experiment of the hydrogenic systems. Our study is expected to be useful for the measurements of the hydrogenic 1S‐2S and 1S‐3S transitions spectroscopy for the improved accuracy.
A large-scale full-configuration-interaction calculation based on the Dirac-Coulomb-Breit (DCB) Hamiltonian is performed for the 21S02{\phantom{\rule{0.16em}{0ex}}}^{1}{S}_{0}21S0​ and 23S12{\phantom{\rule{0.16em}{0ex}}}^{3}{S}_{1}23S1​ states of helium. The operators of the normal and specific mass shifts are directly included in the DCB framework to take the finite nuclear mass correction into account. High-accuracy energies and matrix elements involved nnn (the main quantum number) up to 13 are obtained from one diagonalization of Hamiltonian. The dynamic dipole polarizabilities are calculated by using the sum rule of intermediate states, and a series of magic wavelengths with QED and hyperfine effects included for the $2{\phantom{\rule{0.16em}{0ex}}}^{3}{S}_{1}\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{1}{S}_{0}$ transition of helium are identified. In addition, the high-order ac Stark shift determined by the dynamic hyperpolarizabilities at the magic wavelengths are also evaluated. Since the most promising magic wavelength for application in experiment is 319.8 nm, the high-accuracy magic wavelength of 319.815 3(6) nm of 4He^{4}\mathrm{He}4He is in good agreement with the recent measurement value of 319.815 92(15) nm [R. J. Rengelink et al., arXiv:1804.06693], and the present magic wavelength of 319.830 2(7) nm for 3He^{3}\mathrm{He}3He would provide theoretical support for experimentally designing an optical dipole trap to precisely determine the nuclear charge radius of helium in the future.
The dynamic dipole polarizabilities for the four lowest triplet states (23S,33S,23P2{\phantom{\rule{0.16em}{0ex}}}^{3}S, 3{\phantom{\rule{0.16em}{0ex}}}^{3}S, 2{\phantom{\rule{0.16em}{0ex}}}^{3}P23S,33S,23P, and 33P3{\phantom{\rule{0.16em}{0ex}}}^{3}P33P) of helium are calculated using the BBB-spline configuration interaction method. Present values of the static dipole polarizabilities in the length, velocity, and acceleration gauges are in good agreement with the best Hylleraas results. Also the tune-out wavelengths in the range 400 nm--4.2 $\ensuremath{\mu}\mathrm{m}$ for the four lowest triplet states are identified, and the magic wavelengths in the range 460 nm--3.5 $\ensuremath{\mu}\mathrm{m}$ for the $2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}3{\phantom{\rule{0.16em}{0ex}}}^{3}S, 2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{3}P$, and $2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}3{\phantom{\rule{0.16em}{0ex}}}^{3}P$ transitions are determined. We show that the tune-out wavelength of 23S2{\phantom{\rule{0.16em}{0ex}}}^{3}S23S state is 413.03828(3)413.038\phantom{\rule{0.16em}{0ex}}28(3)413.03828(3) nm, which corroborates the value of Mitroy and Tang [Phys. Rev. A 88, 052515 (2013)], and the magic wavelength around 1066 nm for the $2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}3{\phantom{\rule{0.16em}{0ex}}}^{3}P$ transition can be expected for precision measurement to determine the ratio of transition matrix elements $(2{\phantom{\rule{0.16em}{0ex}}}^{3}S\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{3}P)/(3{\phantom{\rule{0.16em}{0ex}}}^{3}P\ensuremath{\rightarrow}6{\phantom{\rule{0.16em}{0ex}}}^{3}S)$.
The frequency dependent polarizabilities of the francium atom are calculated from the available data of energy levels and transition rates. Magic wavelengths for the state insensitive optical dipole trapping are identified from the calculated light shifts of the 7s7s7s 2S1/2^{2}{S}_{1/2}2S1/2​, 7p7p7p 2P1/2,3/2^{2}{P}_{1/2,3/2}2P1/2,3/2​, and 8s8s8s 2S1/2^{2}{S}_{1/2}2S1/2​ levels of the 7s7s7s $^{2}{S}_{1/2}\ensuremath{-}7p$ 2P1/2,3/2^{2}{P}_{1/2,3/2}2P1/2,3/2​ and 7s7s7s $^{2}{S}_{1/2}\ensuremath{-}8s$ 2S1/2^{2}{S}_{1/2}2S1/2​ transitions, respectively. Wavelengths in the ultraviolet, visible, and near infrared region are identified that are suitable for cooling and trapping. Magic wavelengths between the 600--700 nm and 700--1000 nm regions, which are blue and red detuned with the $7s\ensuremath{-}7p$ and $7s\ensuremath{-}8s$ transitions, are feasible to implement as lasers with sufficient power are available. In addition, we calculated the tune-out wavelengths where the ac polarizability of the ground 7s7s7s 2S1/2^{2}{S}_{1/2}2S1/2​ state in francium is zero. These results are beneficial as laser cooled and trapped francium has been in use for fundamental symmetry investigations like searches for an electron permanent electric dipole moment in an atom and for atomic parity nonconservation.
A series of magic wavelengths for the 23S1 → 23PJ (J = 0, 1, 2) transitions in helium is determined with high precision using the relativistic configuration interaction combined with the radiative potential method. The calculation explicitly incorporates relativistic, finite nuclear mass, and quantum electrodynamic corrections through the inclusion of mass shift and radiative potential operators into the Dirac-Coulomb-Breit Hamiltonian. The resulting magic wavelengths lie entirely within the visible spectrum (400-700 nm) and converge to seven significant digits, making them directly accessible for experimental verification. These accurate predictions serve as benchmarks for testing atomic structure theory and support the development of state-insensitive optical trapping schemes in precision spectroscopy.
A series of magic wavelengths for the forbidden transition of $2{\phantom{\rule{0.16em}{0ex}}}^{3}{S}_{1}\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{1}{P}_{1}$ in helium are determined by performing large-scale full relativistic configuration interaction calculations based on Dirac-Coulomb-Breit Hamiltonian with the mass shift operator included. QED corrections to magic wavelengths are taken into account by using the perturbation theory in nonrelativistic configuration interaction calculations. The 1335.55(2)-nm magic wavelength for the $2{\phantom{\rule{0.16em}{0ex}}}^{3}{S}_{1}(M=\ifmmode\pm\else\textpm\fi{}1)\ensuremath{\rightarrow}2{\phantom{\rule{0.16em}{0ex}}}^{1}{P}_{1}(M=0)$ transition is found to be sensitive to the finite nuclear mass, relativistic, and QED effects, and its measurement might provide a test of atomic structure theory.