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arXiv:1902.08701 · Physical Review A

Systematic uncertainty due to background-gas collisions in trapped-ion optical clocks

Aaron M. Hankin, Ethan Clements, Yao Huang, Samuel M. Brewer, Jwo-Sy Chen, C. W. Chou, David B. Hume, David R. Leibrandt

Published Sep 30, 2019• 30 citations
arXivHTMLPDF

Abstract

We describe a framework for calculating the frequency shift and uncertainty of trapped-ion optical atomic clocks caused by background-gas collisions, and apply this framework to an $^{27}\mathrm{Al}^{+}$ clock to enable a total fractional systematic uncertainty below ${10}^{\ensuremath{-}18}$. For this clock, with 38(19) nPa of room-temperature ${\mathrm{H}}_{2}$ background gas, we find that collisional heating generates a non-thermal distribution of motional states with a mean time-dilation shift of order ${10}^{\ensuremath{-}16}$ at the end of a 150 ms probe, which is not detected by sideband thermometry energy measurements. However, the contribution of collisional heating to the spectroscopy signal is highly suppressed and we calculate the BGC shift to be $\ensuremath{-}0.6(2.4)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}19}$, where the shift is due to collisional heating time dilation and the uncertainty is dominated by the worst case $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$ bound used for collisional phase shift of the $^{27}\mathrm{Al}^{+}$ superposition state. We experimentally validate the framework and determine the background-gas pressure in situ using measurements of the rate of collisions that cause reordering of mixed-species ion pairs.

DOI
10.1103/physreva.100.033419

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We describe a framework for calculating the frequency shift and uncertainty of trapped-ion optical atomic clocks caused by background-gas collisions, and apply this framework to an 27Al+^{27}\mathrm{Al}^{+}27Al+ clock to enable a total fractional systematic uncertainty below 10\ensuremath−18{10}^{\ensuremath{-}18}10\ensuremath−18. For this clock, with 38(19) nPa of room-temperature H2{\mathrm{H}}_{2}H2​ background gas, we find that collisional heating generates a non-thermal distribution of motional states with a mean time-dilation shift of order 10\ensuremath−16{10}^{\ensuremath{-}16}10\ensuremath−16 at the end of a 150 ms probe, which is not detected by sideband thermometry energy measurements. However, the contribution of collisional heating to the spectroscopy signal is highly suppressed and we calculate the BGC shift to be \ensuremath−0.6(2.4)\ifmmode×\else\texttimes\fi10\ensuremath−19\ensuremath{-}0.6(2.4)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}19}\ensuremath−0.6(2.4)\ifmmode×\else\texttimes\fi10\ensuremath−19, where the shift is due to collisional heating time dilation and the uncertainty is dominated by the worst case \ifmmode±\else\textpm\fi\ensuremathπ/2\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2\ifmmode±\else\textpm\fi\ensuremathπ/2 bound used for collisional phase shift of the 27Al+^{27}\mathrm{Al}^{+}27Al+ superposition state. We experimentally validate the framework and determine the background-gas pressure in situ using measurements of the rate of collisions that cause reordering of mixed-species ion pairs.
Summary form only given. As the performance of optical atomic clocks improves steadily, new sources of systematic errors need to be evaluated with increasingly higher accuracy. At present, the collisional frequency shift (CFS), due to interactions with background gas atoms in the environment of the clock ions, is in some cases the next-to-leading source of systematic uncertainty for trapped-ion optical clocks. Existing approaches towards evaluating the CFS have either assumed that only glancing-angle collisions are relevant, or assumed a worst-case scenario for the phase shift imparted to clock state superposition by a collision. The former approach, which was developed for laser-cooled cesium clocks, is not applicable to the deeply trapped atomic ions used in optical clocks, while the latter approach is likely to significantly overestimate the CFS. To address this problem we have developed a framework, based on a non-perturbative quantum channel description of the collision process. Our method leads to a convenient master equation and a compact expression for the CFS. In particular, this framework can be used to estimate the CFS in situations involving many partial waves and large collisional phase shifts, which is the case with thermal background gas atoms scattering off a clock ion. Our method lends itself to practical evaluations of the CFS, by using readily calculated molecular potential curves for the ion-background gas system to extract the collisional phase shifts which lead to the CFS. As an application of this method to a realistic and relevant situation, we have numerically estimated the frequency shift of the 674 nm clock transition in Sr+due to collisions with background helium gas. We find that the CFS is much smaller than the worst-case estimate. In this talk, we describe the theoretical background and motivate the steps that lead to our framework for calculating the CFS. Following this, we present our calculations for the Sr+-He system as an example of the general method, and discuss possible extensions and improvements that will allow us to evaluate the CFS for the Sr+clock at the National Research Council Canada.
Collisions with background gas can perturb the transition frequency of trapped ions in an optical atomic clock. We develop a nonperturbative framework based on a quantum channel description of the scattering process and use it to derive a master equation which leads to a simple analytic expression for the collisional frequency shift. As a demonstration of our method, we calculate the frequency shift of the Sr+{\mathrm{Sr}}^{+}Sr+ optical atomic clock transition due to elastic collisions with helium.
Collisions of atomic ions in optical clocks with background gas molecules result in systematic frequency shifts. We evaluate the frequency shift in Sr+{\mathrm{Sr}}^{+}Sr+ clocks due to collisions with helium atoms and hydrogen molecules, using ab initio spin-orbit-coupled potential energy surfaces together with a master equation for the collisional shift. Our results lead to improved uncertainty estimates for collisional frequency shifts in Sr+{\mathrm{Sr}}^{+}Sr+ optical clocks.
Ions possess both center-of-mass motion and internal states, rendering trapped-ion optical clocks susceptible to perturbations of both degrees of freedom, which results in systematic shifts and fluctuations with motional and relativistic origins. We develop a quantum theory that connects these effects to filter functions determined by the employed pulse sequences and show how the interrogation can be tailored to suppress or enhance specific perturbations. Our treatment includes laser frequency noise and motional effects such as recoil and Doppler shifts, micromotion and excess micromotion, as well as state-dependent trap imperfections and accelerations, e.g., caused by Stark shifts. We further account for relativistic effects including time dilation and the mass defect, and show how the same framework can be applied to hypothetical dark-matter signals. By considering ensemble-averaged phases and their variances, we characterize both systematic shifts and fluctuations, while asynchronous operation relative to the Paul-trap drive allows motional fluctuations to average down between runs. Our results provide a unified framework for characterizing and controlling the systematic and statistical contributions that determine the trueness and precision of trapped-ion optical clocks.
We present a liquid-nitrogen-cooled Ca\mathrm{Ca}Ca+{}^{+}+ optical clock with an overall systematic uncertainty of $3.0\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}18}$. In contrast to the room-temperature Ca\mathrm{Ca}Ca+{}^{+}+ optical clock that we have reported previously, the cryogenic black-body radiation (BBR) shield in vacuum is cooled to $82\ifmmode\pm\else\textpm\fi{}5$ K using liquid nitrogen. We also implement an ion trap with a reduced heating rate and improved laser cooling. This allows the ion temperature to fall to the Doppler-cooling limit during the clock operation and the systematic uncertainty associated with the secular (thermal) motion of the ion is reduced to $<1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}18}$. The uncertainty arising from the probe laser light shift and the servo error is also reduced to $<1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}19}$ and $4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}19}$ with the hyper-Ramsey method and the higher-order servo algorithm, respectively. By comparing the output frequency of the cryogenic clock to that of a room-temperature clock, the differential BBR shift between the two is determined with a fractional statistical uncertainty of $7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}18}$. The differential BBR shift is used to calculate the static differential polarizability and the result is found to be in excellent agreement with our previous measurement using a different method. This work suggests that the BBR shift of optical clocks can be suppressed well in a liquid-nitrogen environment. Systems similar to what is presented here can also be used to suppress the BBR shift significantly in other types of optical clocks, such as Yb\mathrm{Yb}Yb+{}^{+}+, Sr\mathrm{Sr}Sr+{}^{+}+, Yb\mathrm{Yb}Yb, Sr, etc.