Combined microwave and optical spectroscopy for hyperfine structure analysis in thulium atoms

We present refined values of the hyperfine splitting frequencies of the ground $|g\ensuremath{\rangle}=|4{f}^{13}{(}^{2}{F}^{o})6{s}^{2},J=7/2\ensuremath{\rangle}$ and clock $|c\ensuremath{\rangle}=|4{f}^{13}{(}^{2}{F}^{o})6{s}^{2},J=5/2\ensuremath{\rangle}$ states of thulium atoms, ${f}_{g}^{\text{HFS}}=1\phantom{\rule{0.16em}{0ex}}496\phantom{\rule{0.16em}{0ex}}550\phantom{\rule{0.16em}{0ex}}658.23(3)$ Hz and ${f}_{c}^{\text{HFS}}=2\phantom{\rule{0.16em}{0ex}}113\phantom{\rule{0.16em}{0ex}}946\phantom{\rule{0.16em}{0ex}}873.08(9)$ Hz, respectively. The measurements are performed on an ultracold atomic ensemble in an optical lattice using combined microwave and optical transition spectroscopy. The achieved measurement accuracy of less than 0.1 ppb represents an improvement of 2 and 7 orders of magnitude for ${f}_{g}^{\text{HFS}}$ and ${f}_{c}^{\text{HFS}}$, respectively, compared to the previously published values. We also refine the value of the Land\'e $g$-factor of the clock level to ${g}_{c}=0.854\phantom{\rule{0.16em}{0ex}}786(12)$. The results of this work can be used in the frequency standards and quantum simulators based on thulium atoms being developed today, as well as in a number of fundamental studies that require precise knowledge of the hyperfine structure characteristics of various elements.