🔬 atomic physics

Atomic thermometry in optical lattice clocks: model discrepancies and their impact on lattice light shifts

This paper investigates atomic thermometry in the INRIM ytterbium optical lattice clock, revealing significant discrepancies (up to a factor of four) between temperature estimates derived from perturbed harmonic oscillator and Born-Oppenheimer models, which lead to lattice light shift frequency deviations of up to $8\times10^{-17}$ and underscore the need for refined theoretical modeling in state-of-the-art precision metrology.

Original authors: Irene Goti, Tommaso Petrucciani, Stefano Condio, Filippo Levi, Davide Calonico, Marco Pizzocaro

Published 2026-10-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Irene Goti, Tommaso Petrucciani, Stefano Condio, Filippo Levi, Davide Calonico, Marco Pizzocaro

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

To keep time with the utmost precision, scientists have turned to the atoms themselves. For decades, the most accurate clocks have relied on the steady vibration of atoms, but to read this vibration without error, the atoms must be held perfectly still. In modern experiments, researchers use a grid of laser light, known as an optical lattice, to trap thousands of atoms in place, much like eggs sitting in a carton. The challenge is that even in this trap, the atoms are not frozen; they jiggle with heat. This motion creates a subtle blur in the clock's reading, a problem that becomes critical when scientists aim for a level of accuracy where a clock would not lose a single second over the entire age of the universe. To fix this, they must measure exactly how hot the atoms are, but the tools used to take that measurement have been leading them in slightly different directions.

A team of researchers at the Istituto Nazionale di Ricerca Metrologica in Italy set out to resolve this confusion using a clock built from ytterbium atoms. They examined the atoms trapped in their laser grid under a wide variety of conditions, from deep, tight traps to very shallow ones, and they used a new cooling technique to make the atoms even colder than before. Their goal was to compare two different ways of calculating the atoms' temperature based on how the atoms respond to laser light. One method, the traditional approach, treats the atoms as if they are bouncing in a perfectly smooth, bowl-shaped trap. The newer method, which the team applied more rigorously, accounts for the fact that the laser trap is not a perfect bowl but has a more complex shape that changes as the atoms move further from the center.

The researchers found that these two methods do not agree. When they used the traditional model to analyze their data, it suggested the atoms were significantly hotter than the newer model indicated. In some cases, the difference was so large that the traditional method estimated the atoms to be four times hotter than the newer method did. This discrepancy was not just a minor mathematical quirk; it had a direct impact on how the team calculated the clock's accuracy. Because the motion of the atoms shifts the clock's frequency, using the wrong temperature estimate led to errors in the final timekeeping. The team calculated that relying on the older model could introduce a frequency error as large as 8 times 10 to the power of minus 17. While this number seems small, it is a massive gap for the most precise instruments on Earth, representing a difference that would be significant for the next generation of timekeeping standards.

To get a clearer picture, the team employed a technique called Sisyphus cooling, named after the Greek figure condemned to an endless, uphill struggle. In this process, the atoms are repeatedly pushed up a small energy hill by laser light and then allowed to slide back down, losing a bit of energy with each cycle. By applying this cooling method, the researchers were able to lower the temperature of the atoms significantly, allowing them to operate the clock in very shallow traps where the atoms are less tightly confined. When the atoms were this cold, the gap between the two temperature models shrank, but it did not disappear completely. Even with the coldest atoms, the two models still disagreed, suggesting that the traditional way of looking at the data is fundamentally missing something about how the atoms behave in the trap.

The study also compared these results with a third method called Doppler spectroscopy, which measures temperature by looking at how the atoms move sideways, similar to how a siren's pitch changes as an ambulance drives past. This method provided a direct check that aligned more closely with the newer, more complex model than with the traditional one. The researchers concluded that while the traditional model works well enough for less precise work, it is not sufficient for the most advanced clocks. The disagreement between the models means that scientists must be very careful about which mathematical description they choose when they try to correct for the effects of heat on their clocks.

Ultimately, the work highlights that achieving the highest levels of precision requires more than just better lasers or colder atoms; it requires a deeper understanding of the physics used to interpret the data. The team showed that by using a more complete description of the atoms' motion, they could reduce the uncertainty in their measurements. However, even with these improvements, a small amount of uncertainty remains, driven by the choice of model itself. This suggests that the field must continue to refine its theoretical tools and perhaps develop new ways to measure temperature that do not rely so heavily on specific assumptions about how the atoms move. For the scientists building the clocks of the future, the path to perfect timekeeping involves not just cooling the atoms, but also cooling down the assumptions used to describe them.

Technical Summary: Atomic Thermometry in Optical Lattice Clocks: Model Discrepancies and Their Impact on Lattice Light Shifts

Problem Statement
Accurate measurement of atomic temperature is a critical diagnostic tool for optical lattice clocks, which currently aim for total uncertainties at the $10^{-18}$ level. In these systems, the motional state of trapped atoms directly influences systematic uncertainties, particularly the residual lattice light shift and collisional shifts. While sideband spectroscopy is the standard technique for thermometry in one-dimensional optical lattices, traditional analysis relies on the perturbed harmonic oscillator (PHO) approximation. This model assumes atoms occupy a small region near the trap minimum and treats anharmonicities as perturbations. However, recent theoretical developments, specifically the Born-Oppenheimer (BO) approximation applied to optical lattices, suggest that the PHO model may yield inaccurate temperature estimates, particularly as lattice depths decrease or atomic temperatures increase. The discrepancy between these modeling approaches and their subsequent impact on the evaluation of lattice light shifts at the $10^{-18}$ level had not been fully quantified experimentally.

Methodology
The authors investigated these discrepancies using the ytterbium optical lattice clock IT-Yb1 at INRIM. The study employed a wide range of trapping conditions, varying lattice depths ( $D$ ) from approximately 50 $E_R$ to 700 $E_R$ (where $E_R$ is the photon recoil energy). To achieve lower temperatures and access shallower lattice depths, the team implemented clock-line-mediated Sisyphus cooling, a technique utilizing two perpendicular 1D lattices at 1389 nm and a resonant 578 nm pulse to cool atoms longitudinally and radially.

Temperature measurements were conducted via two primary spectroscopic methods:

Sideband Spectroscopy: Performed along the strongly confined lattice axis. The resulting spectra were fitted using two distinct theoretical models:
- The Perturbed Harmonic Oscillator (PHO) model, which uses harmonic oscillator selection rules and treats the potential as a power series expansion.
- The Born-Oppenheimer (BO) model, which separates fast longitudinal motion from slow radial motion. This model utilizes the Franck-Condon principle and a turning-point approximation to map laser frequency to atomic energy, accounting for the full lattice potential (calculated via Mathieu functions) rather than a truncated expansion.
Transverse Doppler Spectroscopy: Performed along the weakly confined radial axis to provide an independent estimate of radial temperature ( $T_r$ ) based on Gaussian Doppler broadening.

The extracted temperatures from these models were then fed into lattice light shift calculations using the formalism developed by Beloy et al. and Ushijima et al. to assess the systematic impact of model choice on frequency shift evaluation.

Key Contributions and Results

Model Discrepancies: The study reveals significant discrepancies between temperature estimates derived from the PHO and BO models. In the absence of post-cooling, the PHO model overestimates the radial temperature by up to a factor of four compared to the BO model. Even with Sisyphus cooling (reducing temperatures to $T_z \approx 2$ $\mu$ K and $T_r \approx 0.8$ $\mu$ K), a factor-of-two difference in radial temperature estimates persists between the two models.
Physical Origin: The divergence arises from the different assumptions regarding the relationship between sideband frequency and atomic energy. The PHO model predicts a sideband frequency near the center of the allowed transition band, whereas the BO turning-point approximation associates the frequency with the inner edge of the band. This results in different slopes in the frequency-versus-energy relation, leading to different fitted temperatures for the same spectral data.
Impact on Lattice Light Shifts: The choice of thermometry model significantly affects the calculated lattice light shift. When comparing the BO model (Case 1) against the PHO model (Case 2), the authors observed relative frequency deviations of up to $8 \times 10^{-17}$ at higher lattice depths. Even at shallow depths ( $D=100 E_R$ ), discrepancies of $\sim 4 \times 10^{-18}$ were observed with Sisyphus cooling applied.
Cooling Effects: The implementation of clock-line-mediated Sisyphus cooling successfully reduced atomic temperatures by a factor of two and enabled stable clock operation at lattice depths as low as 50 $E_R$ . Increasing the cooling duration ( $T_{cool}$ ) progressively reduced the discrepancy between models, lowering the frequency deviation to the $1 \times 10^{-18}$ level for very cold atoms ( $T_{cool} > 100$ ms).
Rabi Time Dependence: The study noted that the PHO model yields inconsistent radial temperature estimates as the Rabi pulse duration increases, interpreting additional excitation on the shallow sideband slope as higher temperature. In contrast, the BO model provides consistent estimates across varying Rabi times.

Significance
The paper demonstrates that the theoretical model used to analyze sideband spectra is a non-negligible source of systematic uncertainty in state-of-the-art optical lattice clocks. While sufficiently low atomic temperatures and the consistent use of a single theoretical framework can reduce model-dependent discrepancies to the low $10^{-18}$ level, residual differences remain significant for clocks targeting uncertainties at this frontier. The authors conclude that these results motivate further refinement of sideband models beyond current approximations and the development of alternative thermometry techniques, such as carrier-based thermometry, that may be less sensitive to model-specific assumptions. The findings are presented as a cautionary note for high-accuracy frequency measurements and precision clock comparisons, emphasizing that model consistency is as crucial as experimental precision in the $10^{-18}$ regime.

More like this