Spectroscopy of the $^{85}$Rb 4$D_{3/2}$ state for hyperfine-structure determination

We report a measurement of the hyperfine-structure constants of the $^{85}$Rb 4$D_{3/2}$ state using a two-photon 5$S_{1/2}\rightarrow$4$D_{3/2}$ transition. The hyperfine transitions are probed by measuring the transmission of the low-power 795-nm lower-stage laser beam through a cold-atom sample as a function of 795-nm laser frequency, with the frequency of the upper-stage 1476-nm laser fixed. All 4 hyperfine components are well-resolved in the recorded transmission spectra. AC shifts are carefully considered. The field-free hyperfine line positions are obtained by extrapolating measured line positions to zero laser power. The magnetic-dipole and electric-quadrupole constants, $A$ and $B$, are determined from the hyperfine intervals to be 7.419(35)~MHz and 4.19(19)~MHz, respectively. The results are evaluated in context with previous works. Possible uses of the Rb 4$D_J$ states in Rydberg-atom-physics, precision-metrology and quantum-technology applications are discussed.