Precision measurement and calculation of the 3d $^2$D-level lifetimes of $^{40}$Ca$^+$

We report measurements of the lifetimes of the 3d $^2$D$_{5/2}$ and 3d $^2$D$_{3/2}$ metastable states of a single laser-cooled $^{40} $Ca$^+$ ion in a linear Paul trap. The result for the natural lifetime of the D$_{5/2}$ state of 1168(9)~ms agrees excellently with the most precise published value. The lifetime of the D$_{3/2}$ state is measured with a single ion for the first time and yields 1176(11)~ms which improves the statistical uncertainty of previous results by a factor of four. We perform high-precision {\it ab initio} all-order calculation of these lifetimes and conduct a detailed study of their accuracy. The resulting theoretical values (D$_{3/2}$ state: 1196(11)~ms, D$_{5/2}$ state: 1165(11)~ms) are in very good agreement with experimental values. These calculations represent an excellent test of high-precision atomic theory and will serve as a benchmark for the study of parity nonconservation in Ba$^+$ which has similar atomic structure.