Precise determination of the quadrupole transition matrix element of ${}^{40}\mathrm{Ca}^{+}$ via branching-fraction and lifetime measurements

We report the experimental determination of the $4s{}^2{S}_{1/2}$ $\leftrightarrow$ $3d{}^2{D}_{5/2}$ quadrupole transition matrix element in ${}^{40}\mathrm{Ca}^{+}$ by measuring the branching fraction of the $3d{}^2{D}_{5/2}$ state decaying into the ground state $4s{}^2{S}_{1/2}$ and the lifetime of the $3d{}^2{D}_{5/2}$ state, using a technique of a highly synchronized measurement sequence for laser control and highly efficient quantum state detection for quantum jumps. The measured branching fraction and improved lifetime are, respectively, 0.998(12) and 1.1649(44) s, which yield the value of the quadrupole transition matrix element (in absolute value) of 9.733(52) $e{a}_0^2$ with an uncertainty of 0.54%. The measured quadrupole transition matrix element is in good agreement with the most precise many-body atomic structure calculations. Our method can be universally applied to measurements of transition matrix elements in single ions and atoms of similar structure.

Figure 1

(a) Partial energy-level diagram of ${}^{40}\mathrm{Ca}^{+}$. The 397- and 866-nm lasers are for cooling and detection, the 854-nm laser is for quenching, and the 729-nm laser is for shelving. (b) A simplified measurement sequence.

Figure 2

Measured data points for the branching-fraction measurements for decay from $3d{}^2{D}_{5/2}$ into $4s{}^2{S}_{1/2}$. The final value is determined using $1.5\times {10}^6$ measurement cycles.

Figure 3

Two fitting methods to the data points for the $3d{}^2{D}_{5/2}$ state lifetime measurement in ${}^{40}\mathrm{Ca}^{+}$. (a) The fitting curve determined by the method of linear regression and least-squares fit yields the lifetime ${\tau }_{5/2}=1.1650\left(43\right)$ s with 95% confidence ($2\sigma$). The lower left quarter shows the verified result of the normal distribution. (b) The fitting curve determined by the maximum likelihood fit yields the lifetime ${\tau }_{5/2}=1.1649\left(34\right)$ s with 95% confidence. Each point represents 30 000 measurements per $\mathrm{\Delta }t$ from 10 to 6000 ms. The lower diagram shows the residuals between the data points and the fitting curve.

Figure 4

Comparison of recently measured and calculated quadrupole transition matrix elements.