Combined experimental and theoretical probe of the lifetime of the $3d{}^2{D}_{5/2}$ state in ${}^{40}{\mathrm{Ca}}^{+}$

In light of the diverse range of reported values for the lifetimes of metastable states of ${}^{40}{\mathrm{Ca}}^{+}$, we have carried out afresh both measurements and theoretical investigation to confirm the lifetime of its $3d{}^2{D}_{5/2}$ state $\left({\tau }_{3d5/2}\right)$. A high-efficiency quantum state detection method by monitoring the quantum jumps of a laser-cooled single ${\mathrm{Ca}}^{+}$ ion in a miniature ring Paul trap was employed in the measurement. Also, sophisticated calculations were performed considering higher order nonlinear terms in the relativistic coupled-cluster (RCC) method with all possible single and double excitations, but accounting only for the important triple excitations from both the core and the valence orbitals. Systematic factors affecting measurement, such as collision with background gases, heating effects, the power of the 866-nm laser, and state detection errors were carefully analyzed. Our observational and theoretical values for ${\tau }_{3d5/2}$ are 1174(10) ms and 1172(3) ms, respectively, which agree well with the experimental results reported by P. A. Barton et al. [Phys. Rev. A 62, 032503 (2000)] and A. Kreuter et al. [Phys. Rev. A 71, 032504 (2005)]. The present theoretical analysis demonstrates that the contributions from the core triples and Breit interaction are notable, as they improve the theoretical results obtained in the previous RCC calculations [B. K. Sahoo, Phys. Rev. A 74, 062504 (2006)].

Figure 1

Partial energy-level diagram of ${}^{40}{\mathrm{Ca}}^{+}$ showing the principal transition, $4s{}^2{S}_{1/2}\to 3d{}^2{D}_{5/2}$, used in cooling, repumping, and probing of the dipole-forbidden transition.

Figure 2

Experimental apparatus for measurement of the $3d{}^2{D}_{5/2}$ lifetime in ${}^{40}{\mathrm{Ca}}^{+}$. A brief description is given in the text. M, mirror; DM, dichroic mirror; AOM, acoustic optical modulator.

Figure 3

Simplified pulse sequence for lifetime measurement of the $3d{}^2{D}_{5/2}$ state in ${}^{40}{\mathrm{Ca}}^{+}$. The preparation consisted of Doppler cooling with the 397-nm and 866-nm lasers (10 ms), repumping using the 854-nm laser (5 ms), optical pumping with the 729-nm laser (3 ms), and two PMT detection periods (3 ms). $\Delta t$ varies from 50 to 5000 ms.

Figure 4

Measured data points for the lifetime of the $3d{}^2{D}_{5/2}$ state in ${\mathrm{Ca}}^{+}$. The solid line is the fitting curve determined by applying the linear regression fitting and the least-squares method to experimental data from 50 to 5000 ms. This fit yields a lifetime ${\tau }_{3d5/2}$ = 1171(9) ms. The lower diagram shows the residuals, which are the differences between the data points and the fitting curve.

Figure 5

Variation in the measured lifetimes with the power of the 866-nm laser. The solid (red) line is the linear fit of five lifetime points of the 866-nm laser at different power values, while error bars are the statistical uncertainties from the fitting value of each run. The result shows that in the short detection period (3 ms) the lifetime is not affected by the increase in laser power.