Measurements of ${}^{27}\mathrm{Al}^{+}$ and ${}^{25}\mathrm{Mg}^{+}$ magnetic constants for improved ion-clock accuracy

We have measured the quadratic Zeeman coefficient for the ${}^{1}S_0\leftrightarrow {}^{3}P_0$ optical clock transition in ${}^{27}\mathrm{Al}^{+}$, $C_2=-71.944\left(24\right)\mathrm{MHz}/{\mathrm{T}}^2$, and the unperturbed hyperfine splitting of the ${}^{25}\mathrm{Mg}^{+}{}^{2}S_{1/2}$ ground electronic state, $\mathrm{\Delta }W/h=1788762752.85\left(13\right)\mathrm{Hz}$, with improved uncertainties. Both constants are relevant to the evaluation of the ${}^{27}\mathrm{Al}^{+}$ quantum-logic clock systematic uncertainty. The measurement of $C_2$ is in agreement with a previous measurement and a recent calculation at the $1\sigma$ level. The measurement of $\mathrm{\Delta }W$ is in good agreement with a recent measurement and differs from a previously published result by approximately $2\sigma$. With the improved value for $\mathrm{\Delta }W$, we deduce an improved value for the nuclear-to-electronic $g$-factor ratio $g_I/g_J=9.299308313\left(60\right)\times {10}^{-5}$ and the nuclear $g$-factor for the ${}^{25}\mathrm{Mg}$ nucleus $g_I=1.86195782\left(28\right)\times {10}^{-4}$. Using the values of $C_2$ and $\mathrm{\Delta }W$ presented here, we derive a quadratic Zeeman shift of the ${}^{27}\mathrm{Al}^{+}$ quantum-logic clock of $\mathrm{\Delta }\nu /\nu =-(9241.8\pm 3.7)\times {10}^{-19}$, for a bias magnetic field of $B\approx 0.12\mathrm{mT}$.

Figure 1

(a) Energy levels and relevant transitions measured for the determination of the ${}^{27}\mathrm{Al}^{+}$ quadratic Zeeman coefficient, $C_2$. (b) Energy levels and relevant transitions measured for the determination of the unperturbed ${}^{25}\mathrm{Mg}^{+}$ hyperfine splitting, $\mathrm{\Delta }W$.

Figure 2

Measurement of the dc magnetic field at the location of the ${}^{27}\mathrm{Al}^{+}$ ion. (a) The magnetic field $B_{\text{dc}}$ [see Eq. (2)] is shown as a function of time. Sections in gray indicate times when the ${}^{27}\mathrm{Al}^{+}$ clock was not operating due to ion loss. (b) Allan deviation of the ${\nu }_{{\mathrm{Al}}^{+}}/{\nu }_{Yb}$ frequency ratio (see text) at $\approx 1\mathrm{mT}$ bias magnetic field in the ${}^{27}\mathrm{Al}^{+}$ clock. The asymptote of the ratio is fit (red line) to a stability of $\sigma \left(\tau \right)=2.1\times {10}^{-15}/\sqrt{\tau }$, where $\tau$ is the averaging time in seconds.

Figure 3

Measurement of the ${}^{27}\mathrm{Al}^{+}$ frequency shift as a function of the bias magnetic field. The frequency shift is fit to a quadratic function to extract the quadratic Zeeman coefficient $C_2$. The fit residuals are shown below. A comparison to the previous $C_2$ measurement ($C_{2,0}$) [4], as well as a theoretical calculation presented in Sec. 3 (red dashed line) are shown in the right panel.

Figure 4

Measurement of the $B_{\text{dc}}$ corrected ${\nu }_{0,0}^{\prime }$ frequency as a function of the trap drive power. The error bars are statistical and the fit is weighted by those uncertainties. The data taken at ${\nu }_{\text{rf}}=76\mathrm{MHz}$ was used for the measurement of $\mathrm{\Delta }W$ and the data taken at ${\nu }_{\text{rf}}=40.72\mathrm{MHz}$ is used for the evaluation of the ac Zeeman shift in the ${}^{27}\mathrm{Al}^{+}$ clock.

Figure 5

Lower panel: Comparison of the uncorrected ${}^{25}\mathrm{Mg}^{+}$ hyperfine constant, $A_{\text{hfs}}^{\prime }$, which ignores higher order effects in the hyperfine intraction, from separate experiments. The $A_{\text{hfs}}^{\prime }$ values are shown relative to the NIST Penning trap measurement, $A_{\text{hfs},0}^{\prime }$ [21]. The measurements [20, 22] were performed at low magnetic field in an rf trap. Upper panel: Low magnetic-field-based measurements, including the work presented here.