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arXiv:2309.16910 · Atoms

Magnetic Sublevel Independent Magic and Tune-Out Wavelengths of the Alkaline-Earth Ions

Jyoti, Harpreet Kaur, Bindiya Arora, B. K. Sahoo

Published Jul 11, 2022• 3 citations
arXivHTMLPDF

Abstract

Light shift in a state due to the applied laser in an atomic system vanishes at tune-out wavelengths (λTs). Similarly, differential light shift in a transition vanishes at the magic wavelengths (λmagics). In many of the earlier studies, values of the electric dipole (E1) matrix elements were inferred precisely by combining measurements and calculations of λmagic. Similarly, the λT values of an atomic state can be used to infer the E1 matrix element, as it involves dynamic electric dipole (α) values of only one state whereas the λmagic values require evaluation of α values for two states. However, both the λmagic and λT values depend on angular momenta and their magnetic components (M) of states. Here, we report the λmagic and λT values of many S1/2 and D3/2,5/2 states, and transitions among these states of the Mg+, Ca+, Sr+ and Ba+ ions that are independent of M values. It is possible to infer a large number of E1 matrix elements of the above ions accurately by measuring these values and combining with our calculations.

DOI
10.3390/atoms10030072

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Light shift in a state due to the applied laser in an atomic system vanishes at tune-out wavelengths (λTs). Similarly, differential light shift in a transition vanishes at the magic wavelengths (λmagics). In many of the earlier studies, values of the electric dipole (E1) matrix elements were inferred precisely by combining measurements and calculations of λmagic. Similarly, the λT values of an atomic state can be used to infer the E1 matrix element, as it involves dynamic electric dipole (α) values of only one state whereas the λmagic values require evaluation of α values for two states. However, both the λmagic and λT values depend on angular momenta and their magnetic components (M) of states. Here, we report the λmagic and λT values of many S1/2 and D3/2,5/2 states, and transitions among these states of the Mg+, Ca+, Sr+ and Ba+ ions that are independent of M values. It is possible to infer a large number of E1 matrix elements of the above ions accurately by measuring these values and combining with our calculations.
We present additional magic wavelengths (${\ensuremath{\lambda}}_{\mathrm{magic}}$) for the clock transitions in the alkaline-earth-metal ions, considering circular polarized light, aside from our previously reported values [Kaur et al., Phys. Rev. A 92, 031402(R) (2015)] for linearly polarized light. Contributions from the vector component to the dynamic dipole polarizabilities $[{\ensuremath{\alpha}}_{d}(\ensuremath{\omega})]$ of the atomic states associated with the clock transitions play major roles in the evaluation of these ${\ensuremath{\lambda}}_{\mathrm{magic}}$, hence facilitating choosing circular polarization of lasers in the experiments. Moreover, the actual clock transitions in these ions are carried out among the hyperfine levels. The ${\ensuremath{\lambda}}_{\mathrm{magic}}$ values in these hyperfine transitions are estimated and found to be different from ${\ensuremath{\lambda}}_{\mathrm{magic}}$ for the atomic transitions due to different contributions coming from the vector and tensor parts of ${\ensuremath{\alpha}}_{d}(\ensuremath{\omega})$. Importantly, we also present ${\ensuremath{\lambda}}_{\mathrm{magic}}$ values that depend only on the scalar component of ${\ensuremath{\alpha}}_{d}(\ensuremath{\omega})$ for their uses in a specially designed trap geometry for these ions so that they can be used unambiguously among any hyperfine levels of the atomic states of the clock transitions. We also present ${\ensuremath{\alpha}}_{d}(\ensuremath{\omega})$ values explicitly at 1064 nm for the atomic states associated with the clock transitions, which may be useful for creating high-field-seeking traps for the above ions using the Nd:YAG laser. The tune-out wavelengths at which the states would be free from the Stark shifts are also presented. Accurate values of the electric dipole matrix elements required for these studies are given and trends of electron correlation effects in determining them are also highlighted.
The lowest three tune-out wavelengths of the four alkaline-earth-metal atoms Be, Mg, Ca, and Sr are determined from tabulations of matrix elements produced from large first-principles calculations. The tune-out wavelengths are located near the wavelengths for the 3P1o{}^{3}{P}_{1}^{o}3P1o​ and 1P1o{}^{1}{P}_{1}^{o}1P1o​ excitations. The measurement of the tune-out wavelengths could be used to establish a quantitative relationship between the oscillator strength of the transition leading to existence of the tune-out wavelength and the dynamic polarizability of the atom at the tune-out frequency. The longest tune-out wavelengths for Be, Mg, Ca, Sr, Ba, and Yb are 454.9813, 457.2372, 657.446, 689.200, 788.875, and 553.00 nm, respectively.
In this paper, we investigate light shifts of heavy alkaline earth elements barium (Ba) and radium (Ra), which are interesting for optical lattice clocks and for permanent electric dipole moment searches. Detailed knowledge is required in the design of efficient loading of atoms from a magneto-optical trap into an optical dipole trap, to facilitate Doppler cooling while trapping and to achieve longer lifetimes of the trapped atoms with reduced heating rates. The wavelength dependence of light shifts of the n s 2 1 S 0 ground state, the n s n p 3 P 1 and n s( n − 1)d 1 D 2 excited states in barium ( n = 6) and the n s 2 1 S 0 ground state, the n s n p 3 P 1 and n s( n − 1)d 3 D 2 excited states in radium ( n = 7) are calculated. Several magic wavelengths in the visible and infrared regions accessible with commercial lasers for optical dipole trapping of Ba and Ra are identified.
We present magic wavelengths for the $n{S}_{1/2}\text{\ensuremath{-}}n{P}_{1/2,3/2},$ and $n{S}_{1/2}\text{\ensuremath{-}}m{D}_{3/2,5/2}$ transitions in Mg+,{\mathrm{Mg}}^{+},Mg+, Ca+,{\mathrm{Ca}}^{+},Ca+, Sr+,{\mathrm{Sr}}^{+},Sr+, and Ba+{\mathrm{Ba}}^{+}Ba+ alkaline-earth-metal ions, for the principal quantum numbers nnn and mmm for the ground and first excited DDD states, respectively. These wavelengths are presented for linearly polarized light by plotting the dynamic polarizabilities of the nS,nP1/2,3/2,nS,n{P}_{1/2,3/2},nS,nP1/2,3/2​, and mD3/2,5/2m{D}_{3/2,5/2}mD3/2,5/2​ states of the considered ions. Required dynamic polarizabilities are evaluated by employing a relativistic all-order perturbative method and their accuracies are ratified by comparing their static values with available high-precision experimental or other theoretical results. Moreover, some of the magic wavelengths identified by us in Ca+{\mathrm{Ca}}^{+}Ca+ concur with the recent measurements reported by Liu et al. [Phys. Rev. Lett. 114, 223001 (2015)]. Knowledge of these magic wavelengths is propitious to carry out many proposed high-precision measurements trapping the above ions in electric fields with the corresponding frequencies.
Using first-principles calculations, we identify magic wavelengths ${\ensuremath{\lambda}}_{\mathrm{magic}}$ for the 2s2s2s-2p2p2p and 2s2s2s-3p3p3p transitions in lithium. The nsnsns and npnpnp atomic levels have the same ac Stark shifts at the corresponding magic wavelength, which facilitates state-insensitive optical cooling and trapping. Tune-out wavelengths for which the ground-state frequency-dependent polarizability vanishes are also calculated. Differences of these wavelengths between 6{}^{6}6Li and 7{}^{7}7Li are reported. Our approach uses high-precision, relativistic all-order methods in which all single, double, and partial triple excitations of the Dirac-Fock wave functions are included to all orders of perturbation theory. Recommended values are provided for a large number of Li electric-dipole matrix elements. Static polarizabilities for the 2s2s2s, 2p2p2p, 3s3s3s, 3p3p3p, and 3d3d3d levels are compared with other theory and experiment where available. Uncertainties of all recommended values are estimated. The magic wavelengths for the uv 2s2s2s-3p3p3p transition are of particular interest for the production of a quantum gas of lithium [Duarte et al., Phys. Rev. A 84, 061406R (2011)].
Extensive calculations of the electric-dipole matrix elements in alkali-metal atoms are conducted using the relativistic all-order method. This approach is a linearized version of the coupled-cluster method, which sums infinite sets of many-body perturbation theory terms. All allowed transitions between the lowest nsnsns, np1∕2n{p}_{1∕2}np1∕2​, np3∕2n{p}_{3∕2}np3∕2​ states and a large number of excited states are considered in these calculations and their accuracy is evaluated. The resulting electric-dipole matrix elements are used for the high-precision calculation of frequency-dependent polarizabilities of the excited states of alkali-metal atoms. We find ``magic'' wavelengths in alkali-metal atoms for which the nsnsns and np1∕2n{p}_{1∕2}np1∕2​ and np3∕2n{p}_{3∕2}np3∕2​ atomic levels have the same ac Stark shifts, which facilitates state-insensitive optical cooling and trapping.