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arXiv:1907.10805 · Physical Review A

Frequency shifts due to Stark effects on a rubidium two-photon transition

Kyle W. Martin, Benjamin Stuhl, Jon Eugenio, Marianna S. Safronova, Gretchen R. Phelps, John H. Burke, N. Lemke

Published Aug 28, 2019• 37 citations
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Abstract

The $5{S}_{1/2}\ensuremath{\rightarrow}5{D}_{5/2}$ two-photon transition in Rb is of interest for the development of a compact optical atomic clock. Here we present a rigorous calculation of the 778.1-nm ac Stark shift $[2.30(4)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}{(\mathrm{mW}/{\mathrm{mm}}^{2})}^{\ensuremath{-}1}]$ that is in good agreement with our measured value of $2.5(2)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}{(\mathrm{mW}/{\mathrm{mm}}^{2})}^{\ensuremath{-}1}$. We include a calculation of the temperature-dependent blackbody radiation (BBR) shift, and we predict that the clock could be operated either with zero net BBR shift $[T=495.9(27)$ K] or with zero first-order sensitivity $[T=368.1(14)$ K]. Also described is the calculation of the dc Stark shift of $5.5(1)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}15}/(\mathrm{V}/{\mathrm{cm}}^{2}$) as well as clock sensitivities to optical alignment variations in both a cat's eye and a flat mirror retroreflector. Finally, we characterize these Stark effects, discussing mitigation techniques necessary to reduce final clock instabilities.

DOI
10.1103/physreva.100.023417

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Frequency standards based on optical transitions between electronic states of trapped ions have reached uncertainties below 10−18 [1]. The Stark shift caused by room-temperature blackbody radiation (BBR) causes the largest correction for most high-performance optical clock. The systematic uncertainty of 171Yb+ clocks based on the 2S1/2 (F = 0) → 2D3/2 (F = 2) electric quadrupole (E2) and 2S1/2 (F = 0) → 2F7/2 (F = 3) electric octupole (E3) transitions is currently limited to 28×10−18 and 1.5×10−18, respectively, by the fractional accuracy of the differential polarizability Δαdc of approximately 2% [2]. In the scope of this thesis, a dual species clock based on 171Yb+ and 88Sr+ ions was set up, operated and evaluated, to overcome this barrier. Since Δαdc is known with a smaller fractional uncertainty of 0.04% for the 2S1/2 → 2D5/2 E2 transition of 88Sr+ [3], this allows for a transfer of Δαdc from 88Sr+ to 171Yb+ following a method proposed in [4]. The optical intensity of a laser is calibrated with the Stark shift it causes on the 88Sr+ clock transition, allowing for the determination of Δαdc of the 171Yb+ clock transitions. This allows for a reduction of the fractional uncertainty of Δαdc for both 171Yb+ clock transitions below 0.4% and clock operation on the E2 and E3 transition with a fractional BBR shift uncertainty of 0.84×10−18 and 0.24×10−18 is enabled, respectively. However the value of Δαdc determined in this thesis does not agree with [2], a fractional offset of approximately 14% is observed. To investigate this discrepancy further, Δαdc is measured in the same manner as in [2] for the 88Sr+ clock transition. The same offset of approximately 14% is found, calling the accuracy of at least one of these methods into question. Finally, the first optical frequency ratio ℛ between the 88Sr+ clock transition and the 171Yb+ E3 transition is measured. The fractional uncertainty of ℛ is 23×10−18. ℛ in combination with the absolute frequency of the E3 transition allows for the determination of the 88Sr+ absolute frequency νSr+ with an uncertainty limited by the caesium references. The value of νSr+ falls outside of the recommended range, but is corroborated by measurements of the same quantity by other institutes [5, 6]. [1] M. Marshall et al., Phys. Rev. Lett. 135, 033201 (2025) [2] N. Huntemann et al., Phys. Rev. Lett. 116, 063001 (2016) [3] T. Lindvall et al., Phys. Rev. Lett. 135, 043402 (2025) [4] M. D. Barrett et al., Phys. Rev. A 100, 043418 (2019) [5] C. Marceau et al., Metrologia 62.4, 045001 (2025) [6] T. Lindvall et al., Phys. Rev. Appl. 24, 044082 (2025)

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Motivated by recent direct measurement of the forbidden $2\phantom{\rule{0.16em}{0ex}}{}^{3}{S}_{1}\ensuremath{\rightarrow}3\phantom{\rule{0.16em}{0ex}}{}^{3}{S}_{1}$ transition in helium [Thomas et al., Phys. Rev. Lett. 125, 013002 (2020)], where the ac Stark shift is one of the main systematic uncertainties, we propose a dichroic two-photon transition measurement for $2\phantom{\rule{0.16em}{0ex}}{}^{3}{S}_{1}\ensuremath{\rightarrow}3\phantom{\rule{0.16em}{0ex}}{}^{3}{S}_{1}$, which could effectively suppress the ac Stark shift by utilizing magic wavelengths: one magic wavelength is used to realize state-insensitive optical trapping; the other magic wavelength is used as one of the two lasers driving the two-photon transition. Carrying out calculations based on the no-pair Dirac-Coulomb-Breit Hamiltonian with mass shift operator included, we report the magic wavelength of 1265.615 9(4) nm for $^{4}\mathrm{He}$ [or 1265.683 9(2) nm for $^{3}\mathrm{He}]$ can be used to design an optical dipole trap; the magic wavelength of 934.234 5(2) nm for $^{4}\mathrm{He}$ [or 934.255 4(4) nm for $^{3}\mathrm{He}]$ can be as one excitation laser in the two-photon process and the ac Stark shift can be reduced to less than 100 kHz, as long as the intensity of the other excitation laser does not exceed $1\ifmmode\times\else\texttimes\fi{}{10}^{4}\phantom{\rule{3.33333pt}{0ex}}{\text{W/cm}}^{2}$. Alternatively, by selecting detuning frequencies relative to the $2\phantom{\rule{0.16em}{0ex}}^{3}P$ state in the region of 82--103 THz, as well as adjusting the intensity ratios of the two lasers, the ac Stark shift in the $2\phantom{\rule{0.16em}{0ex}}{}^{3}{S}_{1}\ensuremath{\rightarrow}3\phantom{\rule{0.16em}{0ex}}{}^{3}{S}_{1}$ two-photon transition can be canceled.

arXiv:2106.15149•Physical Review A•Oct 25, 2021•4 citations
The 5S1/2\ensuremath→5D5/25{S}_{1/2}\ensuremath{\rightarrow}5{D}_{5/2}5S1/2​\ensuremath→5D5/2​ two-photon transition in Rb is of interest for the development of a compact optical atomic clock. Here we present a rigorous calculation of the 778.1-nm ac Stark shift [2.30(4)\ifmmode×\else\texttimes\fi10\ensuremath−13(mW/mm2)\ensuremath−1][2.30(4)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}{(\mathrm{mW}/{\mathrm{mm}}^{2})}^{\ensuremath{-}1}][2.30(4)\ifmmode×\else\texttimes\fi10\ensuremath−13(mW/mm2)\ensuremath−1] that is in good agreement with our measured value of 2.5(2)\ifmmode×\else\texttimes\fi10\ensuremath−13(mW/mm2)\ensuremath−12.5(2)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}{(\mathrm{mW}/{\mathrm{mm}}^{2})}^{\ensuremath{-}1}2.5(2)\ifmmode×\else\texttimes\fi10\ensuremath−13(mW/mm2)\ensuremath−1. We include a calculation of the temperature-dependent blackbody radiation (BBR) shift, and we predict that the clock could be operated either with zero net BBR shift [T=495.9(27)[T=495.9(27)[T=495.9(27) K] or with zero first-order sensitivity [T=368.1(14)[T=368.1(14)[T=368.1(14) K]. Also described is the calculation of the dc Stark shift of 5.5(1)\ifmmode×\else\texttimes\fi10\ensuremath−15/(V/cm25.5(1)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}15}/(\mathrm{V}/{\mathrm{cm}}^{2}5.5(1)\ifmmode×\else\texttimes\fi10\ensuremath−15/(V/cm2) as well as clock sensitivities to optical alignment variations in both a cat's eye and a flat mirror retroreflector. Finally, we characterize these Stark effects, discussing mitigation techniques necessary to reduce final clock instabilities.
An atomic clock designed around the 778 nm two-photon transition in rubidium vapor has become an appealing option for future portable optical clocks. However, two-photon transitions are known to suffer from large AC Stark shifts. We present an analysis of the AC Stark shift in the5S1/2→5D5/25\mathrm{S}_{1/2}\rightarrow 5\mathrm{D}_{5/2}5S1/2​→5D5/2​two photon transition of Rb, along with an improved determination of the blackbody radiation Stark shift.
We evaluated the static and dynamic polarizabilities of the 5s21S05{s}^{2}\phantom{\rule{0.16em}{0ex}}{}^{1}{S}_{0}5s21S0​ and 5s5p3P0o5s5p\phantom{\rule{0.16em}{0ex}}{}^{3}{P}_{0}^{o}5s5p3P0o​ states of Sr using the high-precision relativistic configuration interaction combined with the all-order method. Our calculation explains the discrepancy between the recent experimental $5{s}^{2}\phantom{\rule{0.16em}{0ex}}{}^{1}{S}_{0}\ensuremath{-}5s5p\phantom{\rule{0.16em}{0ex}}{}^{3}{P}_{0}^{o}$ dc Stark shift measurement $\ensuremath{\Delta}\ensuremath{\alpha}=247.379(7)$ [Middelmann et al., Phys. Rev. Lett. 109, 263004 (2012)] and the earlier theoretical result of 261(4) a.u. [Porsev and Derevianko, Phys. Rev. A 74, 020502(R) (2006)]. Our present value of 247.5 a.u. is in excellent agreement with the experimental result. We also evaluated the dynamic correction to the BBR shift with 1% uncertainty; $\ensuremath{-}0.1492(16)Hz.ThedynamiccorrectiontotheBBRshiftisunusuallylargeinthecaseofSr(70.1492(16) Hz. The dynamic correction to the BBR shift is unusually large in the case of Sr (70.1492(16)Hz.ThedynamiccorrectiontotheBBRshiftisunusuallylargeinthecaseofSr(7%$) and it enters significantly into the uncertainty budget of the Sr optical lattice clock. We suggest future experiments that could further reduce the present uncertainties.
Two-photon vapor cell-based optical clocks are strong candidates for next-generation portable atomic standards, offering simplicity, compactness, and high performance. Their narrow clock transitions with counter-propagating beams enable first-order Doppler-free operation. However, systematic perturbations such as the AC Stark shift, temperature-induced shift, and drifts resulting from the laser system pose challenges cause instabilities to medium- to long-term performance. This paper provides a comprehensive overview of Rb two-photon vapor cell optical standards, focusing on the long-term performance-limiting effects and potential mitigation strategies, aiming for clock stabilities better than 1 × 10−15 over the averaging time of a day and beyond.
Frequency standards based on optical transitions between electronic states of trapped ions have reached uncertainties below 10−18 [1]. The Stark shift caused by room-temperature blackbody radiation (BBR) causes the largest correction for most high-performance optical clock. The systematic uncertainty of 171Yb+ clocks based on the 2S1/2 (F = 0) → 2D3/2 (F = 2) electric quadrupole (E2) and 2S1/2 (F = 0) → 2F7/2 (F = 3) electric octupole (E3) transitions is currently limited to 28×10−18 and 1.5×10−18, respectively, by the fractional accuracy of the differential polarizability Δαdc of approximately 2% [2]. In the scope of this thesis, a dual species clock based on 171Yb+ and 88Sr+ ions was set up, operated and evaluated, to overcome this barrier. Since Δαdc is known with a smaller fractional uncertainty of 0.04% for the 2S1/2 → 2D5/2 E2 transition of 88Sr+ [3], this allows for a transfer of Δαdc from 88Sr+ to 171Yb+ following a method proposed in [4]. The optical intensity of a laser is calibrated with the Stark shift it causes on the 88Sr+ clock transition, allowing for the determination of Δαdc of the 171Yb+ clock transitions. This allows for a reduction of the fractional uncertainty of Δαdc for both 171Yb+ clock transitions below 0.4% and clock operation on the E2 and E3 transition with a fractional BBR shift uncertainty of 0.84×10−18 and 0.24×10−18 is enabled, respectively. However the value of Δαdc determined in this thesis does not agree with [2], a fractional offset of approximately 14% is observed. To investigate this discrepancy further, Δαdc is measured in the same manner as in [2] for the 88Sr+ clock transition. The same offset of approximately 14% is found, calling the accuracy of at least one of these methods into question. Finally, the first optical frequency ratio ℛ between the 88Sr+ clock transition and the 171Yb+ E3 transition is measured. The fractional uncertainty of ℛ is 23×10−18. ℛ in combination with the absolute frequency of the E3 transition allows for the determination of the 88Sr+ absolute frequency νSr+ with an uncertainty limited by the caesium references. The value of νSr+ falls outside of the recommended range, but is corroborated by measurements of the same quantity by other institutes [5, 6]. [1] M. Marshall et al., Phys. Rev. Lett. 135, 033201 (2025) [2] N. Huntemann et al., Phys. Rev. Lett. 116, 063001 (2016) [3] T. Lindvall et al., Phys. Rev. Lett. 135, 043402 (2025) [4] M. D. Barrett et al., Phys. Rev. A 100, 043418 (2019) [5] C. Marceau et al., Metrologia 62.4, 045001 (2025) [6] T. Lindvall et al., Phys. Rev. Appl. 24, 044082 (2025)
Motivated by recent direct measurement of the forbidden $2\phantom{\rule{0.16em}{0ex}}{}^{3}{S}_{1}\ensuremath{\rightarrow}3\phantom{\rule{0.16em}{0ex}}{}^{3}{S}_{1}$ transition in helium [Thomas et al., Phys. Rev. Lett. 125, 013002 (2020)], where the ac Stark shift is one of the main systematic uncertainties, we propose a dichroic two-photon transition measurement for $2\phantom{\rule{0.16em}{0ex}}{}^{3}{S}_{1}\ensuremath{\rightarrow}3\phantom{\rule{0.16em}{0ex}}{}^{3}{S}_{1}$, which could effectively suppress the ac Stark shift by utilizing magic wavelengths: one magic wavelength is used to realize state-insensitive optical trapping; the other magic wavelength is used as one of the two lasers driving the two-photon transition. Carrying out calculations based on the no-pair Dirac-Coulomb-Breit Hamiltonian with mass shift operator included, we report the magic wavelength of 1265.615 9(4) nm for 4He^{4}\mathrm{He}4He [or 1265.683 9(2) nm for 3He]^{3}\mathrm{He}]3He] can be used to design an optical dipole trap; the magic wavelength of 934.234 5(2) nm for 4He^{4}\mathrm{He}4He [or 934.255 4(4) nm for 3He]^{3}\mathrm{He}]3He] can be as one excitation laser in the two-photon process and the ac Stark shift can be reduced to less than 100 kHz, as long as the intensity of the other excitation laser does not exceed $1\ifmmode\times\else\texttimes\fi{}{10}^{4}\phantom{\rule{3.33333pt}{0ex}}{\text{W/cm}}^{2}$. Alternatively, by selecting detuning frequencies relative to the 23P2\phantom{\rule{0.16em}{0ex}}^{3}P23P state in the region of 82--103 THz, as well as adjusting the intensity ratios of the two lasers, the ac Stark shift in the $2\phantom{\rule{0.16em}{0ex}}{}^{3}{S}_{1}\ensuremath{\rightarrow}3\phantom{\rule{0.16em}{0ex}}{}^{3}{S}_{1}$ two-photon transition can be canceled.