Hyperfine and fine structure measurements of the 2 $^{\mathrm{3}}$S and 2 $^{\mathrm{3}}$P states of $^{\mathrm{7}}$Li$^{\mathrm{+}}$

Precision spectroscopy of Li$^{\mathrm{+}}$ is a promising tool to test QED and measure fundamental constants. Here, we investigate the hyperfine and fine structures of 2 $^{\mathrm{3}}$S and 2 $^{\mathrm{3}}$P in $^{\mathrm{7}}$Li$^{\mathrm{+}}$ using saturated fluorescence spectroscopy based on a \textasciitilde 500 eV metastable ion beam. We measure the 2 $^{\mathrm{3}}$S$_{\mathrm{1}}\leftrightarrow $2 $^{\mathrm{3}}$P$_{\mathrm{0,1,2}}$ transitions in $^{\mathrm{7}}$Li$^{\mathrm{+}}$. The widths of \textasciitilde 50 MHz in FWHM are determined by Lamb dips, which are generated by two counter-propagating lasers perpendicular to the Li$^{\mathrm{+}}$ beam. With a triple nested loop scanning method, the long-term drift and systematic uncertainties are reduced or eliminated. The systematic uncertainties caused by the Doppler effect, line profile, laser power, frequency calibration and Zeeman effect are evaluated, giving a total uncertainty \textless 100 kHz. For the 2 $^{\mathrm{3}}$S hyperfine splittings, the accuracy is close to the previous works. For the 2 $^{\mathrm{3}}$P fine and hyperfine splittings, our values are one order of magnitude more accurate than the previous experiments and have similar accuracy to the theoretical values.