Y. Huang

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 $\pm π/2$ 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.

We report a frequency measurement of the clock transition of a single ^40Ca^+ ion trapped and laser cooled in a miniature ring Paul trap with 10^-15 level uncertainty. In the measurement, we used an optical frequency comb referenced to a Hydrogen maser, which was calibrated to the SI second through the Global Positioning System (GPS). Two rounds of measurements were taken in May and June 2011, respectively. The frequency was measured to be 411 042 129 776 393.0(1.6) Hz with a fractional uncertainty of 3.9{\times}10^-15 in a total averaging time of > 2{\times}10^6 s within 32 days.