Precision measurement of the ground-state hyperfine constant of ${}_{}{}^{25}{}_{}{}^{}\mathrm{Mg}_{}^{+}{}_{}{}^{}$

The ground-state hyperfine constant $A$ and the nuclear-to-electronic $g$-factor ratio $\frac{g_I}{g_J}$ of ${}_{}{}^{25}{}_{}{}^{}\mathrm{Mg}_{}^{+}{}_{}{}^{}$ have been measured by a laser optical-pumping double-resonance technique. The ions were stored in a Penning trap at a magnetic field of about 1 T. The results are $A=-596.254\mathrm{} 376\left(54\right)\mathrm{}$ MHz and $\frac{g_I}{g_J}=9.299\mathrm{} 484\left(75\right)\mathrm{}\times {10}^{-5\mathrm{}}$. The magnetic field at the ions was stabilized by servoing it to an ($\Delta m_I=0\mathrm{}$, $\Delta m_J=\pm 1$) electronic Zeeman transition. Other hyperfine ($\Delta m_I=\pm 1$, $\Delta m_J=0\mathrm{}$) transitions were detected while the field was thus stabilized. The derivative of the $(m_I,m_J)=(\frac{-3\mathrm{}}{2}, \frac{1}{2}) \mathrm{to} (\frac{-1\mathrm{}}{2}, \frac{1}{2})$ transition frequency with respect to magnetic field $B_0$ goes to zero at $B_0\simeq 1.24$ T. The corresponding resonance was observed at this field with linewidths as small as 0.012 Hz ($Q=2.4\mathrm{}\times {10}^{10}$) by implementing the Ramsey interference method with two coherent rf pulses separated in time by up to 41.4 s.