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arXiv:0807.1381 · Physical Review A

Determination of Sr properties for a high-accuracy optical clock

S. G. Porsev, Andrew D. Ludlow, Martin M. Boyd, Jun Ye

Published Sep 5, 2008• 79 citations
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Abstract

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 $5{s}^{2}$ $^{1}S_{0}$ and $5s5p$ $^{3}P_{0}^{o}$ 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 $^{3}P_{0}^{o}$ 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 $^{3}P_{0}^{o}$ 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.

DOI
10.1103/physreva.78.032508

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We have carried out calculations towards the goal of reducing the inaccuracy of the Sr optical atomic clock to 1\ifmmode×\else\texttimes\fi10\ensuremath−171\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}17}1\ifmmode×\else\texttimes\fi10\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 1S0\ensuremath-3P0o^{1}S_{0}\text{\ensuremath{-}}^{3}P_{0}^{o}1S0​\ensuremath-3P0o​ 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×\else\texttimes\fi10\ensuremath−171\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}17}1\ifmmode×\else\texttimes\fi10\ensuremath−17. The calculations are confirmed by a number of experimental measurements on various Sr properties.
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.
Optical clocks have the potential to be 100 times more accurate than current best cesium atomic clocks within a fraction of the averaging time.This corresponds to a fractional uncertainty of the clock frequency on the level of 10 -18 and requires highaccuracy knowledge of systematic frequency shifts, such that they can be avoided or corrected for.In strontium optical lattice clocks an ensemble of ultracold strontium atoms is conned in an optical lattice, to allow for spectroscopy of the reference transition 5s 2 1 S 0 5s5p 3 P 0 in the Lamb-Dicke regime.The by far largest systematic frequency shift of the strontium clock transition is caused by its high sensitivity to blackbody radiation (BBR).The knowledge of the resulting frequency shift limited the achievable clock uncertainty to about 1 × 10 -16 .In this thesis for the rst time an experimental approach was followed, to determine the sensitivity of the strontium clock transition to blackbody radiation.At an environmental temperature of 300 K the resulting frequency shift corresponds to 2.277 8(23) Hz.The achieved uncertainty contributes with 5 × 10 -18 to the fractional systematic uncertainty of the clock frequency.The determination is based on a precision measurement of the dierence of static polarizabilities of the two clock states ∆α dc = α(5s5p 3 P 0 ) -α(5s 2 1 S 0 ) = 4.078 73(11) × 10 -39 Cm 2 /V.For this the dc Stark shift of the clock transition has been measured in the accurately known electric eld of a precision plate capacitor, which has been developed in this work.The attained static polarizability dierence ∆α dc corresponds to the rst term of a power series of the sensitivity to BBR.Higher orders are accumulated as dynamic part of the BBR shift.Which has been modelled using ∆α dc and experimental data for other atomic properties.To interrogate the ultracold atoms in the electric eld a novel transport technique has been developed, which uses the magic wavelength (813 nm) optical lattice to convey the atoms.For this, the interference pattern is translated together with the foci of the far-detuned lattice beams, by simultaneous translation of the lattice optics.In this way ultracold atoms have been transported within 250 ms over 5 cm, or several Rayleigh ranges respectively without signicant atom loss or heating.In the course of this work the rst frequency measurement with the strontium lattice clock of Physikalisch-Technischen Bundesanstalt (PTB) has been performed.The frequency of the 87 Sr clock transition was measured to be 429 228 004 229 872.9(5)Hz.The strontium lattice clock itself contributed with 1.5 × 10 -16 to the overall fractional uncertainty.Furthermore the thermal radiation eld in a cryogenic blackbody environment was modelled and a draft of a corresponding environment of the atoms was designed, which is suitable to be implemented in the strontium lattice clock of PTB.
We report on an evaluation of an optical clock that uses the ^{2}S_{1/2}→^{2}D_{5/2} transition of a single ^{88}Sr^{+} ion as the reference. In contrast to previous work, we estimate the effective temperature of the blackbody radiation that shifts the reference transition directly during operation from the corresponding frequency shift and the well-characterized sensitivity to thermal radiation. We measure the clock output frequency against an independent ^{171}Yb^{+} ion clock, based on the ^{2}S_{1/2}(F=0)→^{2}F_{7/2}(F=3) electric octupole (E3) transition, and determine the frequency ratio with a total fractional uncertainty of 2.3×10^{-17}. Relying on a previous measurement of the ^{171}Yb^{+} (E3) clock frequency, we find the absolute frequency of the ^{88}Sr^{+} clock transition to be 444 779 044 095 485.277(59) Hz. Our result reduces the uncertainty by a factor of 3 compared with the previously most accurate measurement and may help to resolve so far inconsistent determinations of this value. We also show that for three simultaneously interrogated ^{88}Sr^{+} ions, the increased number causes the expected improvement of the short-term frequency instability of the optical clock without degrading its systematic uncertainty.
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)
In the recent work by Yamaguchi et al. [Phys. Rev. Lett. 123, 113201 (2019)] Cd was identified as an excellent candidate for a lattice clock. Here, we carried out computations needed for further clock development and made an assessment of the higher-order corrections to the light shift of the 5s25{s}^{2}5s2 $^{1}{S}_{0}\ensuremath{-}5s5p$ 3P0o^{3}{P}_{0}^{o}3P0o​ clock transition. We carried out calculations of the magnetic dipole and electric quadrupole polarizabilities and linear and circular hyperpolarizabilities of the 5s25{s}^{2}5s2 1S0^{1}{S}_{0}1S0​ and 5s5p5s5p5s5p 3P0o^{3}{P}_{0}^{o}3P0o​ clock states at the magic wavelength and estimated uncertainties of these quantities. We also evaluated the second-order Zeeman clock transition frequency shift.
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 $5{s}^{2}{\phantom{\rule{0.16em}{0ex}}}^{1}{S}_{0}\text{\ensuremath{-}}5s5p{\phantom{\rule{0.16em}{0ex}}}^{3}{P}_{0}$ Sr clock transition between 350 and 500 nm which can also serve as precision benchmarks.