Sub-Hertz Optical Frequency Comparisons between Two TrappedYb+171Ions

We compare the frequencies of the $6s{}^{2}S_{1/2}(F=0)\to 5d{}^{2}D_{3/2}(F=2)$ reference transition in ${}^{171}\mathrm{Yb}^{+}$ for two single ions stored in independent traps. The quadrupole moment of the $5d{}^{2}D_{3/2}$ state is measured to be $9.32(48)\times {10}^{-40}{\mathrm{C}\mathrm{m}}^2$ and from the quadratic Stark shift the relevant scalar and tensor polarizabilities are determined to be ${\alpha }_S(S_{1/2})-{\alpha }_S(D_{3/2})=-6.9(1.4)\times {10}^{-40}{\mathrm{J}\mathrm{m}}^2/{\mathrm{V}}^2$ and ${\alpha }_T(D_{3/2})=-13.6(2.2)\times {10}^{-40}{\mathrm{J}\mathrm{m}}^2/{\mathrm{V}}^2$, respectively. In the absence of external perturbations we find a mean frequency difference between the two trapped ions of 0.26(42) Hz, corresponding to a relative difference of $3.8(6.1)\times {10}^{-16}$. This is comparable to the agreement found in the most accurate comparisons between cesium fountain clocks.