Single-Ion Atomic Clock with $3times10^{-18}$ Systematic Uncertainty

We experimentally investigate an optical frequency standard based on the $^2S_{1/2} (F=0)to {}^2F_{7/2} (F=3)$ electric octupole (textit{E}3) transition of a single trapped $^{171}$Yb$^+$ ion. For the spectroscopy of this strongly forbidden transition, we utilize a Ramsey-type excitation scheme that provides immunity to probe-induced frequency shifts. The cancellation of these shifts is controlled by interleaved single-pulse Rabi spectroscopy which reduces the related relative frequency uncertainty to $1.1times 10^{-18}$. To determine the frequency shift due to thermal radiation emitted by the ions environment, we measure the static scalar differential polarizability of the textit{E}3 transition as $0.888(16)times 10^{-40}$ J m$^2$/V$^2$ and a dynamic correction $eta(300~text{K})=-0.0015(7)$. This reduces the uncertainty due to thermal radiation to $1.8times 10^{-18}$. The residual motion of the ion yields the largest contribution $(2.1times 10^{-18})$ to the total systematic relative uncertainty of the clock of $3.2times 10^{-18}$.