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Physical Review A

MF -dependent hyperfine-induced 5s5pP0o3–5s2S01 clock transition rates in an external magnetic field for Sr87

Xiaotong Lu, Feng Guo, Yebing Wang, Min Feng, Ting Liang, Benquan Lu, Hong Chang

Published Jul 28, 2023• 5 citations
Publisher

Abstract

In this paper, we theoretically calculate the natural lifetime of the $5s5p\phantom{\rule{4pt}{0ex}}^{3}\mathrm{P}_{0}^{o}$ state of the $^{87}\mathrm{Sr}$ atom and evaluate the effect of an external magnetic field on the lifetimes of Zeeman sublevels. Using the multiconfiguration Dirac-Hartree-Fock method, the hyperfine-induced $5s5p^{3}\mathrm{P}_{0}^{o}--5{s}^{2}\phantom{\rule{0.28em}{0ex}}^{1}\mathrm{S}_{0}$ clock transition rate of $^{87}\mathrm{Sr}$ is estimated as $6.39(37)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}$, corresponding to a lifetime of 156(9) s. Our result closely agrees with previous measurements after correcting the decay rate due to the black-body radiation effect. Additionally, the transition rates of Zeeman sublevels are calculated as a function of the magnetic-field strength. Our theoretical results reveal that transition rates between different excited Zeeman states are discrepant, and the lifetime of magnetic sublevel $+{M}_{F}^{e}\phantom{\rule{4pt}{0ex}}(\ensuremath{-}{M}_{F}^{e}$; the superscript $e$ refers to the excited state, i.e., the $5s5p\phantom{\rule{0.28em}{0ex}}^{3}\mathrm{P}_{0}^{o}$ state) will increase (decrease) with the magnetic-field strength. Moreover, we measure the coupling strength between the clock laser and the $^{87}\mathrm{Sr}$ atoms, and our findings align well with theoretical estimations regarding changes in normalized coupling strength with external magnetic-field strength.

DOI
10.1103/physreva.108.012820

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In this paper, we theoretically calculate the natural lifetime of the 5s5p3P0o5s5p\phantom{\rule{4pt}{0ex}}^{3}\mathrm{P}_{0}^{o}5s5p3P0o​ state of the 87Sr^{87}\mathrm{Sr}87Sr atom and evaluate the effect of an external magnetic field on the lifetimes of Zeeman sublevels. Using the multiconfiguration Dirac-Hartree-Fock method, the hyperfine-induced 5s5p3P0o−−5s21S05s5p^{3}\mathrm{P}_{0}^{o}--5{s}^{2}\phantom{\rule{0.28em}{0ex}}^{1}\mathrm{S}_{0}5s5p3P0o​−−5s21S0​ clock transition rate of 87Sr^{87}\mathrm{Sr}87Sr is estimated as 6.39(37)\ifmmode×\else\texttimes\fi10\ensuremath−3s\ensuremath−16.39(37)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}6.39(37)\ifmmode×\else\texttimes\fi10\ensuremath−3s\ensuremath−1, corresponding to a lifetime of 156(9) s. Our result closely agrees with previous measurements after correcting the decay rate due to the black-body radiation effect. Additionally, the transition rates of Zeeman sublevels are calculated as a function of the magnetic-field strength. Our theoretical results reveal that transition rates between different excited Zeeman states are discrepant, and the lifetime of magnetic sublevel +MFe(\ensuremath−MFe+{M}_{F}^{e}\phantom{\rule{4pt}{0ex}}(\ensuremath{-}{M}_{F}^{e}+MFe​(\ensuremath−MFe​; the superscript eee refers to the excited state, i.e., the 5s5p3P0o5s5p\phantom{\rule{0.28em}{0ex}}^{3}\mathrm{P}_{0}^{o}5s5p3P0o​ state) will increase (decrease) with the magnetic-field strength. Moreover, we measure the coupling strength between the clock laser and the 87Sr^{87}\mathrm{Sr}87Sr atoms, and our findings align well with theoretical estimations regarding changes in normalized coupling strength with external magnetic-field strength.
We present experimental and theoretical studies of the effect of an external magnetic field on the 3 P 0 –5 s 2 1 S 0 forbidden transition in the bosonic Sr atom. In our ultra-cold atomic system, the excitation fraction of the forbidden transition was measured under the circumstance of the different magnetic field strengths by using the normalized detection method. Based on the perturbation theory, we calculated the magnetic-field-induced 3 P 0 –5 s 2 1 S 0 transition rate as a function of the magnetic field strength. Electronic matrix elements entering the calculation of the transition rate were also used to derive the theoretical excitation fraction as a function of the magnetic field strength. A good agreement was found between the experimental measurements and the calculations. This study should be helpful in evaluating the magnetic field effects on the forbidden transition rate with higher accuracy. Moreover, it can help to understand the ultra-cold atomic interaction in the external magnetic field.
We demonstrate a spin-polarized clock transition spectrum of the 87Sr optical lattice clock. The clock transition 5s2 1S05s5p 3P0 of isotope 87Sr has a hyperfine structure due to non-zero nuclear spin, inducing ten -polarized transitions from each individual mF state under the condition of a bias magnetic field along the probing polarization axis. In this experiment, atoms are driven to a certain mF state by a circular-polarization pump light to maximize the atomic population, which is beneficial to the stability and uncertainty evaluation of the optical lattice clock. After two stages cooling and trapping, about 3.5106 atoms are trapped in the red magneto-optical trap with a temperature of 3.9 K. A grating-feedback external cavity diode laser with a tapered amplifier is used to build the optical lattice with a magic-wavelength of 813.426 nm. Both waists of the counter-propagating lattice beam along the horizontal direction are overlapped to form a one-dimensional (1D) optical lattice. The lifetime of the atoms trapped in the 1D optical lattice is 1600 ms. The clock laser at 698 nm is a grating-feedback diode laser, which is locked to an ultra-low expansion cavity by the Pound-Drever-Hall technique to stabilize the frequency and phase. As a result, the linewidth of clock laser is narrowed to Hz level. By the normalized shelving method, we obtain a resolved sideband spectrum of 87Sr 5s2 1S05s5p 3P0 transition. According to the spectrum, the lattice temperature along the longitudinal direction is approximately 4.2 K. After that a linewidth of 6.7 Hz of the degenerate clock transition is obtained at a probing time of 150 ms by utilizing a three-dimensional (3D) bias magnetic field, which is used to eliminate the stray magnetic fields. Then a small bias magnetic field of 300 mGs is applied along the polarization axis of the lattice light to achieve the spectrum of Zeeman magnetic sublevels of the clock transition. Furthermore, the mF=+9/2 and mF=-9/2 magnetic sublevels are picked to be respectively pumped by the +-polarized and --polarized light at 689 nm, a variable liquid crystal wave plate is employed to switch on both polarizations. Finally, the spin polarized clock transition spectrum is obtained at the interrogating pulse of 150 ms, and the linewidths of the mF=+9/2, mF=-9/2 magnetic sublevel transitions are 6.8 Hz and 6.2 Hz respectively.
In the weak-magnetic-field approximation, we derived an expression of quadratic Zeeman shift coefficient of 3 P 0 o clock state for 88Sr and 87Sr atoms. By using this formula and the multi-configuration Dirac–Hartree–Fock theory, the quadratic Zeeman shift coefficients were calculated. The calculated values C 2 = –23.38(5) MHz/T2 for 88Sr and the 3 P 0 o , F = 9/2, MF = ±9/2 clock states for 87Sr agree well with the other available theoretical and experimental values, especially the most accurate measurement recently. In addition, the calculated values of the 3 P 0 o , F = 9/2, MF = ±9/2 clock states were also determined in our 87Sr optical lattice clock. The consistency with measurements verifies the validation of our calculation model. Our theory is also useful to evaluate the second-order Zeeman shift of the clock transition, for example, the new proposed 1 S 0, F = 9/2, MF = ±5/2– 3 P 0 o , F = 9/2, MF = ±3/2 transitions.
The theory of how excited-state hyperfine structure can be determined using a magnetic field to decouple the nuclear and electronic angular momentum is presented. The initial excited state is assumed to have an anisotropic Zeeman sublevel population distribution, which can be conveniently created by a laser-induced multiphoton excitation of an unpolarized atom. The Zeeman sublevel populations are subsequently mixed by the hyperfine interaction. The degree of mixing is controlled by a magnetic field applied along the quantization axis. The relative Zeeman sublevel populations can be monitored by detecting the polarization of fluorescence emitted when the excited state radiatively decays. It is convenient to take the ratio of two time-integrated fluorescent signals measured for orthogonal linear polarizations, since this is independent of the excited-state number density. The plot of this ratio versus magnetic field is very well approximated by a Lorentzian curve whose half-width is proportional to the magnetic dipole hyperfine coupling constant. This method was used to determine the magnitude of the magnetic dipole constant of the cesium 8D3/2{\mathit{D}}_{3/2}D3/2​ state to be 3.92\ifmmode\pm\else\textpm\fi{}0.10 MHz, which is in excellent agreement with previous results.
We report the first experimental determination of the hyperfine quenching rate of the $6{s}^{2}\text{ }^{1}{S}_{0}(F=1/2)\ensuremath{-}6s6p\text{ }^{3}{P}_{0}(F=1/2)$ transition in 171Yb^{171}\mathrm{Yb}171Yb with nuclear spin I=1/2I=1/2I=1/2. This rate determines the natural linewidth and the Rabi frequency of the clock transition of a Yb optical frequency standard. Our technique involves spectrally resolved fluorescence decay measurements of the lowest lying 3P0,1^{3}{P}_{0,1}3P0,1​ levels of neutral Yb atoms embedded in a solid Ne matrix. The solid Ne provides a simple way to trap a large number of atoms as well as an efficient mechanism for populating 3P0^{3}{P}_{0}3P0​. The decay rates in solid Ne are modified by medium effects including the index-of-refraction dependence. We find the 3P0^{3}{P}_{0}3P0​ hyperfine quenching rate to be $(4.42\ifmmode\pm\else\textpm\fi{}0.35)\ifmmode\times\else\texttimes\fi{}1{0}^{\ensuremath{-}2}\text{ }\text{ }{\mathrm{s}}^{\ensuremath{-}1}$ for free 171Yb^{171}\mathrm{Yb}171Yb, which agrees with recent ab initio calculations.
We present an experimental determination on the Landé g-factors for the 5s 2 1 S 0 and 5 s 5 p 3 P 0 states in ultra-cold atomic systems, which is important for evaluating the Zeeman shift of the clock transition in the 87 Sr optical lattice clock. The Zeeman shift of the 5s5p 3 P 0 –5s 2 1 S 0 forbidden transition is measured with the π-polarized and σ ± -polarized interrogations at different magnetic field strengths. Moreover, in the g-factor measurement with the σ ± -transition spectra, it is unnecessary to calibrate the external magnetic field. By this means, the ground state 5s 2 1 S 0 g-factor for the 87 Sr atom is −1.306(52)×10 −4 , which is the first experimental determination to the best of our knowledge, and the result matches very well with the theoretical estimation. The differential g-factor δg between the 5s5p 3 P 0 state and the 5s 2 1 S 0 state of the 87 Sr atoms is measured in the experiment as well, which are −7.67(36) × 10 −5 with π-transition spectra and −7.72(43) × 10 −5 with σ ± -transition spectra, in good agreement with the previous report [Phys. Rev. A 76 (2007) 022510]. This work can also be used for determining the differential g-factor of the clock states for the optical clocks based on other atoms .