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Physical Review A

Collisional frequency shifts in Sr + optical clocks

Iosif V. Leibin, Inna S. Kalinina, Amar C. Vutha, Alexei A. Buchachenko

Published Sep 17, 2025• 1 citation
Publisher

Abstract

Collisions of atomic ions in optical clocks with background gas molecules result in systematic frequency shifts. We evaluate the frequency shift in ${\mathrm{Sr}}^{+}$ 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 ${\mathrm{Sr}}^{+}$ optical clocks.

DOI
10.1103/knhc-k4p8

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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.
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 with background gas particles can shift the resonance frequencies of atoms in atomic clocks. The internal quantum states of atoms can also become entangled with their motional states due to the recoil imparted by a collision, which leads to a further shift of the clock frequency through the relativistic Doppler shift. It can be complicated to evaluate the Doppler and collisional frequency shifts for clock atoms in such entangled states, but estimates of these shifts are essential in order to improve the accuracy of optical atomic clocks. We present a formalism that describes collisions and relativistic Doppler shifts in a unified manner, and can therefore be used to accurately estimate collisional frequency shifts in trapped-atom clocks.
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 ${10}^{\ensuremath{-}18}$. 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 ${10}^{\ensuremath{-}16}$ 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 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.