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Physical Review A

Extracting transition rates from zero-polarizability spectroscopy

Marianna S. Safronova, Z. Zuhrianda, Marianna S. Safronova, Charles W. Clark

Published Oct 6, 2015• 31 citations
Publisher

Abstract

We predict a sequence of magic-zero wavelengths for the Sr excited $5s5p{\phantom{\rule{0.16em}{0ex}}}^{3}{P}_{0}$ state, and provide a general roadmap for extracting transition matrix elements using precise frequency measurements. We demonstrate that such measurements can serve as a global benchmark of the spectroscopic accuracy that is required for the development of high-precision predictive methods. These magic-zero wavelengths are also needed for state-selective atom manipulation for implementation of quantum logic operations. We also identify five magic wavelengths of the $5{s}^{2}{\phantom{\rule{0.16em}{0ex}}}^{1}{S}_{0}\text{\ensuremath{-}}5s5p{\phantom{\rule{0.16em}{0ex}}}^{3}{P}_{0}$ Sr clock transition between 350 and 500 nm which can also serve as precision benchmarks.

DOI
10.1103/physreva.92.040501

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We predict a sequence of magic-zero wavelengths for the Sr excited $5s5p~ ^3P_0$ state, and provide a general roadmap for extracting transition matrix elements using precise frequency measurements. We demonstrate that such measurements can serve as a best global benchmark of the spectroscopic accuracy that is required for the development of high-precision predictive methods. These magic-zero wavelengths are also needed for state-selective atom manipulation for implementation of quantum logic operations. We also identify five magic wavelengths of the $5s^2\ ^1S_0 - 5s5p\ ^3P_0$ Sr clock transition between 350 nm and 500 nm which can also serve as precision benchmarks.

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We report the experimental measurement of a magic wavelength at 476.823 62(8) nm for the ${}^{88}\mathrm{Sr}\phantom{\rule{0.2em}{0ex}}5{s}^{2}{\phantom{\rule{0.1em}{0ex}}}^{1}{\mathrm{S}}_{0}\ensuremath{\rightarrow}5s5p{\phantom{\rule{0.1em}{0ex}}}^{3}{\mathrm{P}}_{0}$ clock transition. The magic wavelength is determined through an ac Stark shift spectroscopy of atoms in an optical dipole trap. The value is in reasonable agreement with the theoretical prediction, differing by 0.061(54) nm. This magic wavelength, being shorter than the commonly used one at 813 nm, will be important for applications such as Bragg-pulse matter-wave interferometry involving both clock states. This work also paves the way for quantum simulation with a shorter lattice constant.

arXiv:2507.04532•Physical Review Applied•Nov 10, 2025•1 citation
We predict a sequence of magic-zero wavelengths for the Sr excited 5s5p3P05s5p{\phantom{\rule{0.16em}{0ex}}}^{3}{P}_{0}5s5p3P0​ state, and provide a general roadmap for extracting transition matrix elements using precise frequency measurements. We demonstrate that such measurements can serve as a global benchmark of the spectroscopic accuracy that is required for the development of high-precision predictive methods. These magic-zero wavelengths are also needed for state-selective atom manipulation for implementation of quantum logic operations. We also identify five magic wavelengths of the 5s21S0\ensuremath-5s5p3P05{s}^{2}{\phantom{\rule{0.16em}{0ex}}}^{1}{S}_{0}\text{\ensuremath{-}}5s5p{\phantom{\rule{0.16em}{0ex}}}^{3}{P}_{0}5s21S0​\ensuremath-5s5p3P0​ Sr clock transition between 350 and 500 nm which can also serve as precision benchmarks.
We predict a sequence of magic-zero wavelengths for the Sr excited 5s5p 3P05s5p~ ^3P_05s5p 3P0​ state, and provide a general roadmap for extracting transition matrix elements using precise frequency measurements. We demonstrate that such measurements can serve as a best global benchmark of the spectroscopic accuracy that is required for the development of high-precision predictive methods. These magic-zero wavelengths are also needed for state-selective atom manipulation for implementation of quantum logic operations. We also identify five magic wavelengths of the 5s2 1S0−5s5p 3P05s^2\ ^1S_0 - 5s5p\ ^3P_05s2 1S0​−5s5p 3P0​ Sr clock transition between 350 nm and 500 nm which can also serve as precision benchmarks.
Predicting magic wavelengths accurately requires precise knowledge of electric dipole matrix elements of nearby atomic transitions. As a result, measurements of magic wavelengths allow us to test theoretical predictions for the matrix elements that frequently cannot be probed by any other methods. Here, we calculate and measure a magic wavelength near 473 nm of the $5{s}^{2}\phantom{\rule{0.16em}{0ex}}^{1}{S}_{0}\ensuremath{-}5s5p\phantom{\rule{0.16em}{0ex}}^{3}{P}_{1}$ intercombination transition of 88Sr^{88}\mathrm{Sr}88Sr. Experimentally, we find 473.371(6) nm for $\mathrm{\ensuremath{\Delta}}m=0$ ($\ensuremath{\pi}$ transition) and 473.117(15) nm for $\mathrm{\ensuremath{\Delta}}m=\ensuremath{-}1$ (${\ensuremath{\sigma}}^{\ensuremath{-}}$ transition). Theoretical calculations yield 473.375(22) and 473.145(20) nm, respectively. Determining magic wavelengths nearby the dominant 461-nm probe transition of 88Sr^{88}\mathrm{Sr}88Sr holds promise for state insensitive interfacing of strontium atoms and nanophotonic devices with a narrow band of operating wavelengths. Furthermore, the 3P1^{3}{P}_{1}3P1​ polarizability is dominated by the contributions to the 5p23P5{p}^{2}\phantom{\rule{0.16em}{0ex}}^{3}P5p23P levels, and excellent agreement of theory and experiment validates both theoretical values of these matrix elements and estimates of their uncertainties.
We have carried out calculations towards the goal of reducing the inaccuracy of the Sr optical atomic clock to $1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}17}$ and below. In particular, we focused on ac polarizabilities of the 5s25{s}^{2}5s2 1S0^{1}S_{0}1S0​ and 5s5p5s5p5s5p 3P0o^{3}P_{0}^{o}3P0o​ clock states that are important for reducing the uncertainty of black-body radiation-induced frequency shifts for the $^{1}S_{0}\text{\ensuremath{-}}^{3}P_{0}^{o}$ clock transition. Four low-lying even-parity states have been identified, whose total contribution to the static polarizability of the 3P0o^{3}P_{0}^{o}3P0o​ clock state is at the level of 90%. We show that if the contribution of these states is experimentally known with 0.1% accuracy, the same accuracy can be achieved for the total polarizability of the 3P0o^{3}P_{0}^{o}3P0o​ state. The corresponding uncertainty for the blackbody shift at a fixed room temperature will be below $1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}17}$. The calculations are confirmed by a number of experimental measurements on various Sr properties.
We present measurements of three distinctive state-dependent wavelengths for the $^{1}S_{0}\text{\ensuremath{-}}^{3}P_{0}$ clock transition in 174Yb{}^{\text{174}}\text{Yb}174Yb atoms. Specifically, we determine two magic wavelengths at 652.281(21) and 542.50205(19)THz542.502\phantom{\rule{0.16em}{0ex}}05(19)\phantom{\rule{0.16em}{0ex}}\text{THz}542.50205(19)THz, where the differential light shift on the $^{1}S_{0}\text{\ensuremath{-}}^{3}P_{0}$ clock transition vanishes, and one tune-out wavelength at 541.8325(5)THz541.8325(5)\phantom{\rule{0.16em}{0ex}}\text{THz}541.8325(5)THz, where the polarizability of the 1S0^{1}S_{0}1S0​ ground state exhibits a zero crossing. The two magic wavelengths are identified by spectroscopically interrogating cold 174Yb{}^{\text{174}}\text{Yb}174Yb atoms on the clock transition in a one-dimensional optical lattice. The ground-state tune-out wavelength is determined via a parametric heating scheme. With a simple empirical model, we then extrapolate the ground- and excited-state polarizability over a broad range of wavelengths in the visible spectrum.
Magic wavelengths and triple magic trapping conditions for 5s2 1 ⁣S05s^2~^1\!S_05s2 1S0​ and 5s5p 3 ⁣P0,25s5p~^3\!P_{0,2}5s5p 3P0,2​ states of Sr atoms
The static and dynamic electric dipole polarizabilities of the 5s2 1 ⁣S05s^2~^1\!S_05s2 1S0​ and 5s5p 3 ⁣P0,25s5p~^3\!P_{0,2}5s5p 3P0,2​ states of Sr atoms are calculated using the relativistic configuration interaction plus the many-body perturbation theory (RCI+MBPT) method. Magic wavelengths are determined for the transitions 5s2 1 ⁣S0→5s5p 3 ⁣P05s^2~^1\!S_0\rightarrow 5s5p~^3\!P_{0}5s2 1S0​→5s5p 3P0​, 5s2 1 ⁣S0→5s5p 3 ⁣P25s^2~^1\!S_0\rightarrow 5s5p~^3\!P_{2}5s2 1S0​→5s5p 3P2​, and 5s5p 3 ⁣P0→5s5p 3 ⁣P25s5p~^3\!P_0\rightarrow 5s5p~^3\!P_{2}5s5p 3P0​→5s5p 3P2​. A comprehensive study is conducted on the dependence of magic wavelengths on the angle between the laser polarization and the magnetic field. Furthermore, the conditions for realizing triple magic trapping at 813.4~nm for the 5s2 1 ⁣S05s^2~^1\!S_05s2 1S0​, 5s5p 3 ⁣P05s5p~^3\!P_{0}5s5p 3P0​ and 5s5p 3 ⁣P25s5p~^3\!P_{2}5s5p 3P2​ states are investigated. In the case of linearly polarized light, when the angle (θpθ_pθp​) between the laser polarization direction and the magnetic field is 79.1(0.7)∘79.1(0.7)^\circ79.1(0.7)∘, triple magic trapping for the 5s2 1 ⁣S05s^2~^1\!S_05s2 1S0​, 5s5p 3 ⁣P05s5p~^3\!P_{0}5s5p 3P0​, and 5s5p 3 ⁣P2 M=05s5p~^3\!P_{2}~M=05s5p 3P2​ M=0 states can be achieved. This result agrees well with the recent experimental measurement (78.49(3)∘^\circ∘)[Phys. Rev. Lett. 135, 143401 (2025)]. Meanwhile, triple magic trapping involving the 5s5p 3 ⁣P2 M=25s5p~^3\!P_{2}~M=25s5p 3P2​ M=2 state can be achieved when θp=37.4(0.3)∘θ_p= 37.4(0.3)^\circθp​=37.4(0.3)∘. The conditions for achieving triple magic trapping with circularly and arbitrarily elliptically polarized light are also presented.
We report the experimental measurement of a magic wavelength at 476.823 62(8) nm for the ${}^{88}\mathrm{Sr}\phantom{\rule{0.2em}{0ex}}5{s}^{2}{\phantom{\rule{0.1em}{0ex}}}^{1}{\mathrm{S}}_{0}\ensuremath{\rightarrow}5s5p{\phantom{\rule{0.1em}{0ex}}}^{3}{\mathrm{P}}_{0}$ clock transition. The magic wavelength is determined through an ac Stark shift spectroscopy of atoms in an optical dipole trap. The value is in reasonable agreement with the theoretical prediction, differing by 0.061(54) nm. This magic wavelength, being shorter than the commonly used one at 813 nm, will be important for applications such as Bragg-pulse matter-wave interferometry involving both clock states. This work also paves the way for quantum simulation with a shorter lattice constant.