Precision measurement of the $^{176}\mathrm{Lu}^+$ $^3D_1$ microwave clock transitions

TL;DR

The paper presents precision measurements of the unperturbed $^{176}$Lu$^{+}$ ${^3D_1}$ microwave clock transitions $f_1$ and $f_2$ with a fractional uncertainty of $4\times10^{-14}$, achieved by Ramsey spectroscopy across multiple magnetic-field orientations and careful averaging to eliminate quadrupole shifts. By extracting accurate quadratic Zeeman coefficients and calibrating magnetic-field alignment, the authors determine the orientation and magnitude of the electric field gradient and measure the quadrupole shift experimentally. They then extract the ratio of quadrupole shifts for the two microwave transitions to quantify hyperfine-mediated effects, obtaining a residual quadrupole moment $\langle \delta \Theta \rangle_F = (-2.48\pm(17)_\mathrm{stat}\pm(16)_\mathrm{sys})\times10^{-4}\,e a_0^2$, in agreement with prior theory. These results enable direct, in-situ evaluation of the residual quadrupole shift for $^{176}$Lu$^{+}$ optical frequency standards at the $<10^{-20}$ level and provide an experimental test of hyperfine-mediated corrections in the system, with implications for error budgets in high-accuracy clocks.

Abstract

We report precision measurement of the unperturbed ${^{3}}D_1$ microwave transition frequencies in $^{176}\mathrm{Lu}^+$ to a fractional uncertainty of $4\times10^{-14}$. We find the $|F,m_F\rangle=|8,0\rangle$ to $|7,0\rangle$ hyperfine transition frequency to be $10\,491\,519\,945.228\,82(38)\,$Hz and the $|7,0\rangle$ to $|6,0\rangle$ transition frequency to be $11\,290\,004\,289.881\,61(36)\,$ Hz. At this precision we are able to observe the hyperfine-mediated effects in the ratio of the quadrupole shifts, from which we can directly infer the residual quadrupole moment after $^3D_1$ hyperfine averaging. We find a residual quadrupole moment of ${-2.48(23)\times10^{-4}}\,e a_0^2$, consistent with a previous assessment using a different and less direct method. With the unperturbed microwave frequencies accurately known, the residual quadrupole shift for a $^{176}\mathrm{Lu}^+$ ($^3D_1$) optical frequency standard can henceforth be readily evaluated to $<10^{-20}$ uncertainty by routine ${^{3}}{D}_1$ microwave spectroscopy.

Figures (5)

• Figure 1: (a) Atomic-level structure of $^{176}$Lu$^+$ showing the wavelengths of repump (dashed lines), cooling/detection (double line), and clock (solid lines) transitions used. (b) Microwave clock transitions measured in this work. (c) Ion trap geometry with the orientation of lasers and polarizers used to precisely set the applied magnetic field direction as described in the main text.
• Figure 2: (a) Example of alignment of the magnetic field to the linear OP polarization. (b) Example of alignment of $\hat{k}_{s}$ to the magnetic field via the 804-nm transition E2 coupling.
• Figure 3: Evaluation of microwave quadratic Zeeman coefficients. (a) Measurements of both ${^{3}}D_1$ microwave transitions measured as a function of magnetic field amplitude. Solid lines are a quadratic fit. (b) Residuals with respect to the fit. The larger error bars are the combined total uncertainty, and the black error bars are the statistical component only.
• Figure 4: Measurements of the ${^{3}}D_1$$\ket{7,0}$ to $\ket{8,0}$ transition for a range of magnetic field orientations $(\theta,\phi)$ for (a) the typical applied dc biases, and (b) no applied dc bias. Solid lines are the fit to the quadrupole shift model given Eq. (\ref{['eq:quad']}). The reduced $\chi^2$ is $0.93$ with 25 degrees of freedom and 1.5 with 8 degrees of freedom for (a) and (b) respectively. Fit residuals are shown in the bottom plots.
• Figure 5: (a,b) Quadrupole shifts measured with no bias applied (a) and with high bias applied (b). Error bars are smaller than the points. (c,d) measured ratio of quadruple shift compared to the expected value (green line) based on the evaluation hyperfine mediated effects in zhiqiang2020hyperfine (Eq. (\ref{['eq:ratio']})). (e) Allan deviation of ratio measurement at high bias.