Precision Characterization of the $^2$D$_{5/2}$ State and Quadratic Zeeman Coefficient in $^{171}$Yb$^+$

We report measurements of the branching fraction, hyperfine constant, and second-order Zeeman coefficient of the D$_{5/2}$ level in $^{171}$Yb$^+$ with up to two orders-of-magnitude improvement in precision compared to previously reported values. We estimate the electric quadrupole reduced matrix element of the S$_{1/2}$ $\leftrightarrow$ D$_{5/2}$ transition to be 12.5(4) $e a_0^2$. Furthermore, we determine the transition frequency of the F$_{7/2}$ $\leftrightarrow$ $^{1}$D$[3/2]_{3/2}$ at 760 nm with a $\sim$25-fold improvement in precision. These measurements provide benchmarks for quantum-many-body atomic-physics calculations and provide valuable data for efforts to improve quantum information processors based on Yb$^+$.