Laser Cooling and Hyperfine Measurements of Radium-225 Ions
$^{225}$Ra$^+$ ions (nuclear spin $I=1/2$) have transitions that are first-order insensitive to magnetic field noise, which is advantageous for optical clocks and quantum information science. We report on laser cooling and trapping of $^{225}$Ra$^+$ ions and hyperfine splitting measurements of the ion's $\mathrm{7s}$ $^2\mathrm{S}_{1/2}$, $\mathrm{7p}$ $^2\mathrm{P}_{1/2}$, and $\mathrm{6d}$ $^2\mathrm{D}_{3/2}$ states. We measured the ground state hyperfine constant, $A(\mathrm{S}_{1/2}) =$ $-27.684511052(5)$ $\mathrm{GHz}$, and the quadratic Zeeman coefficient, $C_2 =$ $142.3(1.0)$ $\mathrm{Hz\ G}^{-2}$, of the $^2\mathrm{S}_{1/2} (F=0, m_F = 0) \leftrightarrow~^2\mathrm{S}_{1/2} (F=1, m_{F} = 0)$ transition. Our result addresses a discrepancy in the literature for the ground state hyperfine splitting. We measured the hyperfine constants of the $^2\mathrm{P}_{1/2}$ state, $A(\mathrm{P}_{1/2}) =$ $-5.447(4)$ $\mathrm{GHz}$, and the $^2\mathrm{D}_{3/2}$ state, $A(\mathrm{D}_{3/2}) =$ $-619.7(1.1)$ $\mathrm{MHz}$. We also performed state preparation and measurement using the ground state hyperfine levels and realized a fidelity of $0.9951(9)$.