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arXiv:2212.10743 · Physical Review Applied

Experimental and Theoretical Study of Dynamic Polarizabilities in the 5S1/2 – 5D5/2 Clock Transition in Rubidium-87 and Determination of Electric Dipole Matrix Elements

Rhona Hamilton, Benjamin M. Roberts, Sarah K. Scholten, Clayton R. Locke, Andre Nicholas Luiten, J. S. M. Ginges, Christopher Perrella

Published May 18, 2023• 9 citations
arXivHTMLPDF

Abstract

The interaction between light and an atom causes perturbations in the atom's energy levels, known as the light shift. These light shifts are a key source of inaccuracy in atomic clocks, and can also deteriorate their precision. We present a study of light shifts and associated dynamic polarizabilities for a two-photon atomic clock based on the $5{S}_{1/2}$--$5{D}_{5/2}$ transition in rubidium-87 over the range 770--800 nm. We determine experimental and theoretical values for a magic wavelength in this range and the electric dipole (E1) matrix element for the $5{P}_{3/2}$--$5{D}_{5/2}$ transition. We find a magic wavelength of 776.179(5) nm (experimental) and 776.21 nm (theoretical) in the vicinity of the $5{P}_{3/2}$--$5{D}_{5/2}$ resonance, and the corresponding reduced E1 matrix element $1.80(6)e{a}_{0}$ (experimental) and $1.96(15)e{a}_{0}$ (theoretical). These values resolve a previous discrepancy between theory and experiment.

DOI
10.1103/physrevapplied.19.054059

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The interaction between light and an atom causes perturbations in the atom's energy levels, known as the light shift. These light shifts are a key source of inaccuracy in atomic clocks, and can also deteriorate their precision. We present a study of light shifts and associated dynamic polarizabilities for a two-photon atomic clock based on the 5S1/25{S}_{1/2}5S1/2​--5D5/25{D}_{5/2}5D5/2​ transition in rubidium-87 over the range 770--800 nm. We determine experimental and theoretical values for a magic wavelength in this range and the electric dipole (E1) matrix element for the 5P3/25{P}_{3/2}5P3/2​--5D5/25{D}_{5/2}5D5/2​ transition. We find a magic wavelength of 776.179(5) nm (experimental) and 776.21 nm (theoretical) in the vicinity of the 5P3/25{P}_{3/2}5P3/2​--5D5/25{D}_{5/2}5D5/2​ resonance, and the corresponding reduced E1 matrix element 1.80(6)ea01.80(6)e{a}_{0}1.80(6)ea0​ (experimental) and 1.96(15)ea01.96(15)e{a}_{0}1.96(15)ea0​ (theoretical). These values resolve a previous discrepancy between theory and experiment.
State-of-the-art optical lattice clocks require careful characterization of systematic frequency shifts, which often needs to include contributions to the coupling of atoms to the trapping light field that are beyond the electric-dipole approximation. An experimental determination of the light-shift coefficient that accounts for electric-quadrupole and magnetic-dipole transitions for the clock transition in neutral strontium is reported on, addressing the substantial discrepancy between values detailed in previous works from theory and experiment.
To operate an optical lattice clock at a fractional uncertainty below 10−1710^{-17}10−17, one must typically consider not only electric-dipole (E1) interaction between an atom and the lattice light field when characterizing the resulting lattice light shift of the clock transition but also higher-order multipole contributions, such as electric-quadrupole (E2) and magnetic-dipole (M1) interactions. However, strongly incompatible values have been reported for the E2-M1 polarizability difference of the clock states (5s5p) 3P0(5s5p)\,{}^{3}\mathrm{P}_{0}(5s5p)3P0​ and (5s2) 1S0(5s^2)\,{}^{1}\mathrm{S}_{0}(5s2)1S0​ of strontium [Ushijima et al., Phys. Rev. Lett. 121 263202 (2018); Porsev et al., Phys. Rev. Lett. 120, 063204 (2018)]. This largely precludes operating strontium clocks with uncertainties of few 10−1810^{-18}10−18, as the resulting lattice light shift corrections deviate by up to 1×10−171 \times 10^{-17}1×10−17 from each other at typical trap depths. We have measured the E2-M1 polarizability difference using our 87Sr{}^{87}\mathrm{Sr}87Sr lattice clock and find a value of Δαqm=−987−223+174  μHzΔα_{\mathrm{qm}} = -987^{+174}_{-223} \; \mathrm{μHz}Δαqm​=−987−223+174​μHz. This result is in very good agreement with the value reported by Ushijima et al.
We employ doubly resonant two-photon excitation into the 74S74S74S Rydberg state to spectroscopically measure the dynamic scalar polarizability, ${\ensuremath{\alpha}}_{0}$, and tensor polarizability, ${\ensuremath{\alpha}}_{2}$, of rubidium 5P3/25{P}_{3/2}5P3/2​. A cavity-generated 1064-nm optical lattice allows us to reach intensities near $2\ifmmode\times\else\texttimes\fi{}{10}^{11}\phantom{\rule{0.28em}{0ex}}\mathrm{W}/{\mathrm{m}}^{2}$, where the atom field is larger than the hyperfine interaction, and magnetic sublevels are well resolved. In the evaluation of the data we use a self-referencing method that renders the polarizability measurement largely free from the intensity calibration of the laser light field. We obtain experimental values ${\ensuremath{\alpha}}_{0}=\ensuremath{-}1149(\ifmmode\pm\else\textpm\fi{}2.5%)$ and ${\ensuremath{\alpha}}_{2}=563(\ifmmode\pm\else\textpm\fi{}4.2%)$, in atomic units. Methods and results are supported by simulations. The results provide an experimental test of atomic-structure calculations used to determine systematic shifts in atomic clocks and to interpret fundamental-physics experiments.
The zero crossing of the dynamic differential scalar polarizability of the ${S}_{1/2}\ensuremath{-}{D}_{5/2}$ clock transition in 138Ba+^{138}\mathrm{Ba}^{+}138Ba+ has been determined to be 459.1614(28)THz459.1614(28)\phantom{\rule{0.28em}{0ex}}\mathrm{THz}459.1614(28)THz. Together with previously determined matrix elements and branching ratios, this tightly constrains the dynamic differential scalar polarizability of the clock transition over a large wavelength range ($\ensuremath{\gtrsim}700\phantom{\rule{0.28em}{0ex}}\mathrm{nm}$). In particular, it allows an estimate of the blackbody radiation shift of the clock transition at room temperature.
Higher-order perturbation theory is used to study frequency shifts of the spin-forbidden 3 P J - 1 S 0 transitions for Mg, Sr and Yb atoms in an external laser field. Calculations of the higher-order dynamic Stark effect have been performed in connection with precise measurements on cooled and trapped atoms and the design of ultra stable optical clock in an engineered light shift trap. The form of the model potential of the valence electrons suggested by Simons is used for the calculation of atomic dynamic characteristics given by second- and fourth-order perturbation theory for the interaction of the atoms with the electromagnetic field. Dispersion properties of the polarizability and hyperpolarizability components and their behavior in the vicinity of both resonances and so-called magic wavelengths for the upper and lower atomic states, are analyzed. Calculations of ac multipole polarizabilities and dipole hyperpolarizabilities for the 3 P 0 - 1 S 0 clock transition in 87 Sr indicate that the contribution of the higher- order light shifts can be reduced to less than 1 mHz, allowing for a projected accuracy of better than 10 -17 .
We carry out relativistic many-body calculations of the static and dynamic dipole polarizabilities of the ground 6s 2 1 S 0 and the first excited 6s6p 3 P o 0 states of Yb. With these polarizabilities, we compute several properties of Yb relevant to optical lattice clocks operating on the 6s 2 1 S 0 –6s6p 3 P o 0 transition. We determine (i) the first four magic wavelengths of the laser field for which the frequency of the clock transition is insensitive to the laser intensity. While the first magic wavelength is known, we predict the second, the third and the fourth magic wavelengths to be 551 nm, 465 nm and 413 nm. (ii) We re-evaluate the effect of black-body radiation on the frequency of the clock transition, the resulting clock shift at T = 300 K being −1.41(17) Hz. (iii) We compute long-range interatomic van der Waals coefficients (in a.u.) C 6 (6s 2 1 S 0 + 6s 2 1 S 0 ) = 1909(160), C 6 (6s 2 1 S 0 + 6s6p 3 P 0 ) = 2709(338) and C 6 (6s6p 3 P 0 + 6s6p 3 P 0 ) = 3886(360). Finally, we determine the atom-wall interaction coefficients (in a.u.), C 3 (6s 2 1 S 0 ) = 3.34 and C 3 (6s6p 3 P 0 ) = 3.68. We also address and resolve a disagreement between previous calculations of the static polarizability of the ground state.