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

This paper reports the precision measurement of unperturbed $^{176}\mathrm{Lu}^+$ $^3D_1$ microwave clock transition frequencies with a fractional uncertainty of $4\times10^{-14}$ , enabling the direct inference of the residual quadrupole moment and facilitating routine evaluation of quadrupole shifts for optical frequency standards at the $10^{-20}$ level.

Original authors: M. D. K. Lee, Qi Zhao, Qin Qichen, Zhao Zhang, N. Jayjong, K. J. Arnold, M. D. Barrett

Published 2026-09-29

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CC BY 4.0

Original authors: M. D. K. Lee, Qi Zhao, Qin Qichen, Zhao Zhang, N. Jayjong, K. J. Arnold, M. D. Barrett

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Time is not a constant river flowing at the same speed for everyone; it is a local phenomenon that can be stretched or squeezed by gravity, motion, and even the invisible electric fields surrounding an atom. To measure time with the greatest possible precision, scientists build clocks that count the vibrations of atoms. The most accurate clocks in the world today are optical clocks, which use light waves vibrating at incredibly high frequencies to mark the passage of time. However, to reach the next level of accuracy, researchers must account for tiny, subtle forces that nudge the atoms out of their perfect rhythm. One such force is the electric field gradient, a variation in electric force across the tiny space where an atom sits. This gradient interacts with the atom's internal shape, causing a slight shift in its ticking rate. For decades, scientists have known how to cancel out the largest of these shifts, but a faint, lingering effect has remained hidden, too small to see with previous tools.

In a recent study, a team of researchers at the National University of Singapore has finally brought this hidden effect into clear view. They worked with a single ion of the element lutetium, specifically the isotope lutetium-176, trapped in a vacuum chamber. This ion acts as a tiny, perfect clock, but to read its time, the scientists had to listen to it in two different ways. First, they used lasers to measure the ion's optical transitions, which are the high-frequency jumps that define the clock's main tick. Second, and crucially for this new work, they used microwaves to probe the ion's internal structure at a much lower frequency. By combining these two methods, the team was able to measure the ion's microwave transitions with a precision so high that they could detect the tiny, residual shift caused by the electric field gradient. Their measurements revealed the exact frequency of these transitions to a degree of uncertainty so small it is equivalent to losing less than one second over the entire age of the universe.

The researchers did not simply measure the frequency; they mapped out how the ion responded to the electric fields in its environment. They carefully oriented the magnetic field around the trapped ion in three different directions, effectively taking a three-dimensional snapshot of the forces at play. By averaging the results from these three directions, they were able to cancel out the dominant shifts and isolate the tiny remainder. This remainder, which they call the residual quadrupole moment, is a subtle distortion in the atom's shape that persists even after the main effects are averaged out. The team found this value to be a specific, negative number, confirming a prediction made by earlier theoretical calculations. This confirmation is significant because it validates a complex method used to understand the behavior of atoms in high-precision clocks, proving that the theoretical models used to design these instruments are accurate.

The implications of this work extend beyond a single measurement. By knowing the exact frequency of the microwave transitions, scientists can now calculate the tiny shifts caused by electric fields in the optical clock with extreme confidence. This means that future optical clocks based on lutetium can be corrected for these subtle effects with an uncertainty smaller than one part in ten quintillion. The researchers achieved this by first calibrating their equipment with extreme care, ensuring that the magnetic fields they applied were perfectly aligned with the lasers used to prepare the atom. They then measured the microwave frequencies over many hours, averaging out random noise to reveal the true signal. The result is a new standard of precision that allows scientists to see the invisible forces acting on the atom, turning a theoretical prediction into an observed reality. This level of control over atomic behavior is a testament to the power of modern experimental physics, where the smallest details of nature are brought into sharp focus through patience, ingenuity, and the relentless pursuit of accuracy.

Technical Summary: Precision Measurement of the 176Lu+ 3D1 Microwave Clock Transitions

Problem Statement
Optical and microwave frequency standards utilizing states with angular momentum $J > 1/2$ are susceptible to quadrupole shifts caused by the coupling between the electronic quadrupole moment and electric field gradients. While hyperfine averaging (HA) in $^{176}\text{Lu}^+$ effectively mitigates the leading quadrupole shift by realizing an effective $J=0$ state, a residual quadrupole moment (RQM) persists due to hyperfine-mediated mixing between fine-structure manifolds. Although this RQM induces a small shift ( $< 10^{-18}$ fractional) on the optical clock frequency, it must be assessed at the $10^{-19}$ level of systematic uncertainty to meet current precision standards. Previous assessments of the $^{176}\text{Lu}^+$ (3D $_1$ ) RQM relied on a combination of $g$ -factor measurements and theoretical matrix element calculations, lacking a direct, in situ experimental measurement of the electric field gradient effects. Furthermore, the unperturbed microwave clock transition frequencies, necessary for evaluating these shifts, required higher precision than previously available.

Methodology
The authors performed precision measurements on a single trapped $^{176}\text{Lu}^+$ ion using the "Lu-1" apparatus. The experimental approach involved three primary components:

Magnetic Field Alignment and Calibration: To precisely control the orientation of the magnetic field relative to the ion trap's electric field gradient, the authors developed a spectroscopic alignment scheme. They utilized a nominally $\pi$ -polarized 646-nm optical pumping (OP) laser as a fixed reference in the space-fixed frame. By minimizing the depumping rate from the $|3\text{D}_1, 7, 0\rangle$ state, they aligned the magnetic field to the laser polarization. A second "side" OP laser and 804-nm electric quadrupole (E2) transition selection rules were used to establish a third orthogonal axis. This allowed the magnetic field to be set with a systematic uncertainty of $0.5^\circ$ in orthogonality.
Quadratic Zeeman Coefficient Determination: The authors remeasured the quadratic Zeeman coefficients ( $\alpha_1$ and $\alpha_2$ ) for the $f_1$ ( $|7,0\rangle \to |8,0\rangle$ ) and $f_2$ ( $|7,0\rangle \to |6,0\rangle$ ) transitions. Ramsey spectroscopy was conducted over magnetic field amplitudes ranging from 0.13 mT to 2.6 mT. The coefficients were extracted by fitting the frequency shifts, subtracting a quartic component derived from perturbation theory.
Unperturbed Frequency and RQM Measurement: The unperturbed microwave frequencies were determined by averaging measurements taken along three mutually orthogonal magnetic field directions ( $\hat{x}, \hat{y}, \hat{z}$ ). This averaging cancels the first-order quadrupole shift. To measure the RQM, the authors analyzed the ratio of the quadrupole shifts on the $f_1$ and $f_2$ transitions. This was performed under two conditions: typical operating voltages and a high-bias configuration (38 V on endcaps) to amplify the quadrupole shift and resolve hyperfine-mediated effects.

Key Contributions and Results

Precision Frequency Measurement: The authors report the unperturbed 3D $_1$ $_{1}$ microwave transition frequencies with a fractional uncertainty of $4 \times 10^{-14}$ $4 \times 1 0^{- 14}$ .
- $f_1$ ( $F=7 \to 8$ ): $10\,491\,519\,945.228\,82 \pm (32)_{\text{stat}} \pm (23)_{\text{sys}}$ Hz
- $f_2$ ( $F=7 \to 6$ ): $11\,290\,004\,289.881\,61 \pm (32)_{\text{stat}} \pm (20)_{\text{sys}}$ Hz
Improved Zeeman Coefficients: The quadratic Zeeman coefficients were determined as $\alpha_1 = 2\,038.589\,1(31)$ Hz mT $^{-2}$ and $\alpha_2 = 2\,428.947\,0(26)$ Hz mT $^{-2}$ . These values represent a reduction in uncertainty by approximately one order of magnitude compared to previous reports.
Direct RQM Determination: By measuring the ratio of quadrupole shifts on the two microwave transitions, the authors directly inferred the residual quadrupole moment arising from hyperfine mixing. The measured value is $\langle \delta\Theta \rangle_F \approx -2.48(23) \times 10^{-4} ea_0^2$ .
Systematic Characterization: The study provided a detailed characterization of systematic shifts, including excess micromotion (EMM), microwave and rf ac-Zeeman shifts, and phase chirp in the synthesis chain. The dominant systematic uncertainties were identified as EMM and the incomplete cancellation of the quadrupole shift due to magnetic field alignment orthogonality.

Significance
The paper claims that these results provide an independent experimental validation of the previously reported RQM value, which was derived indirectly from $g$ -factor measurements and theory. The consistency between the direct spectroscopic method and the previous assessment increases confidence in the RQM evaluation for $^{176}\text{Lu}^+$ optical frequency standards.

With the unperturbed microwave frequencies now accurately known, the residual quadrupole shift for a $^{176}\text{Lu}^+$ (3D $_1$ ) optical frequency standard can be readily evaluated to an uncertainty of $< 10^{-20}$ through routine 3D $_1$ microwave spectroscopy. This capability removes a significant obstacle in the pursuit of ultra-precise ion-based optical clocks. The authors note that while the work was not intended to develop a microwave frequency standard, the achieved fractional accuracy is competitive with, though slightly lower than, the best existing microwave standards based on trapped ions (Cd $^+$ , Hg $^+$ , Yb $^+$ ).

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