We report precision measurement of the unperturbed ${ ^{ 3}}D_1$ microwave transition frequencies in $^{ 176}\mathrm{ Lu}^+$ to a fractional uncertainty of $4\times10^{ -14}$. We find the $|F,m_F\rangle=|8,0\rangle$ to $|7,0\rangle$ hyperfine transition frequency to be $10\,491\,519\,945.228\,82(38)\,$Hz and the $|7,0\rangle$ to $|6,0\rangle$ transition frequency to be $11\,290\,004\,289.881\,61(36)\,$ Hz. At this precision we are able to observe the hyperfine-mediated effects in the ratio of the quadrupole shifts, from which we can directly infer the residual quadrupole moment after $^3D_1$ hyperfine averaging. We find a residual quadrupole moment of ${ -2.48(23)\times10^{ -4}}\,e a_0^2$, consistent with a previous assessment using a different and less direct method. With the unperturbed microwave frequencies accurately known, the residual quadrupole shift for a $^{ 176}\mathrm{ Lu}^+$ ($^3D_1$) optical frequency standard can henceforth be readily evaluated to $<10^{ -20}$ uncertainty by routine ${ ^{ 3}}{ D}_1$ microwave spectroscopy.
Categories: physics.atom-ph