Measurement of Dipole Matrix Elements with a Single Trapped Ion

We demonstrate a method to determine dipole matrix elements by comparing measurements of dispersive and absorptive light ion interactions. We measure the matrix element pertaining to the Ca II H line, i.e., the $4{}^2{S}_{1/2}\leftrightarrow 4{}^2{P}_{1/2}$ transition of ${}^{40}{\mathrm{Ca}}^{+}$, for which we find the value 2.8928(43) $e{a}_0$. Moreover, the method allows us to deduce the lifetime of the $4{}^2{P}_{1/2}$ state to be 6.904(26) ns, which is in agreement with predictions from recent theoretical calculations and resolves a long-standing discrepancy between calculated values and experimental results.

Figure 1

Measurement scheme: (a) a laser off-resonantly couples two Zeeman ground state levels to an excited state with decay rate $\gamma$. (b) The coupling gives rise to a differential ac-Stark shift ${\mathrm{\Delta }}_S$ between the Zeeman levels and incoherent redistribution of population at rates $R_{\pm }$. (c) Relevant energy levels in ${}^{40}{\mathrm{Ca}}^{+}$. The shelving to the metastable $3{}^2{D}_{5/2}$ state as well to the $3{}^2{D}_{3/2}$ state acting as a population sink are indicated. State populations, which result in a bright (dark) detection event are marked by an open (closed) circle.

Figure 2

Dark event fraction versus time of exposure to an off-resonant laser pulse for initializations in $|\uparrow ⟩$ and $|\downarrow ⟩$. Data points indicated by blue squares (red circles) are measured with the laser detuned by 12.03 GHz (13.94 GHz) from resonance. Solid lines show the fits to the model, Eq. (7). The inset shows the dark event fraction versus pulse duration for a spin-echo experiment. The blue line connects the data points, the red line is the resulting fit. The oscillation frequency directly corresponds to the differential ac-Stark shift ${\mathrm{\Delta }}_S$.

Figure 3

Determination of the four detunings chosen for our measurements. For each of the 65 acquired data sets, we plot the quantity $\mathrm{sign}(\mathrm{\Delta }){\mathrm{\Delta }}_S/\delta R\propto \mathrm{\Delta }$ versus the optical frequency as measured by a wavelength meter. The zero crossing of a linear fit, depicted by the solid red line, reveals the resonance frequency with a standard measurement uncertainty of $2\pi \times 21\mathrm{MHz}$. The insets show details of the data sets measured at the different detunings with the abscissa magnified 166 times. The shaded backgrounds indicate their uncertainties along the frequency axis. The arrows on the top frequency scale indicate the values of the four different detunings at $-13.94$, $-12.03$, 11.52, and 13.42 GHz.

Figure 4

Comparison of the measured lifetime with theoretical and experimental results for $P_{1/2}$-level lifetimes from Refs. [8, 16, 27, 28, 29, 30, 31, 32, 33, 34, 35].