Systematic uncertainty due to background-gas collisions in trapped-ion optical 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 $^{27}$Al$^+$ clock to enable a total fractional systematic uncertainty below $10^{-18}$. For this clock, with 38(19) nPa of room temperature 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 $10^{-16}$ 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 $-0.6(2.4)\times 10^{-19}$, where the shift is due to collisional heating time-dilation and the uncertainty is dominated by the worst case $\pm \pi/2$ bound used for collisional phase shift of the $^{27}$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.