Penning-trap eigenfrequency measurements with optical radiofrequency detectors

We use an electric-dipole laser-driven transition to precisely measure the cyclotron-frequency ratios of the pairs ${}^{42}$Ca${}^{+}$-${}^{40}$Ca${}^{+}$, ${}^{44}$Ca${}^{+}$-${}^{40}$Ca${}^{+}$ and ${}^{48}$Ca${}^{+}$-${}^{40}$Ca${}^{+}$ in a 7-tesla Penning trap. A single laser-cooled ($T\approx 1$~mK) ion serves, together with photon-counting and/or photon-imaging units, as a radiofrequency detector covering a broad-band frequency spectrum, in the present case from kHz to a few MHz. Such detectors (${}^{40,42,44,48}$Ca${}^{+}$) allow measuring extremely small forces, with measured normalized sensitivities down to $7.4(3.5)$ yN$\mathrm{/}\sqrt{\text{Hz}}$ and $24.9(9.9)$ yN$\mathrm{/}\sqrt{\text{Hz}}$ in the MHz and kHz regime, respectively. The direct determination of the ions’ amplitudes makes a cyclotron-frequency measurement process more robust against inhomogeneities of the magnetic field and/or deviations of the electric quadrupole field due to mechanical imperfections of the trap.