A thorium-229 optical nuclear clock with feedback loop

This paper reports the implementation of a room-temperature thorium-229 optical nuclear clock embedded in a calcium fluoride crystal that achieves high stability through rapid laser feedback, enabling competitive constraints on ultralight dark matter models by surpassing previous limits on strong force and quark couplings.

The Gist

The Big Idea: A Clock Inside a Rock

Imagine you have a grandfather clock. Inside, a pendulum swings back and forth to keep time. The more perfectly that pendulum swings, the more accurate the clock is.

For the last 70 years, the world's most accurate clocks have used tiny atoms (like atoms of strontium or ytterbium) as their "pendulums." Scientists shine lasers on these atoms to make them vibrate, and they count those vibrations to tell time.

This paper describes a major breakthrough: the team has built a clock that uses the nucleus of an atom (the very heavy center) instead of the whole atom. Specifically, they are using the Thorium-229 isotope.

Think of it this way: If an atom is a solar system, the electrons are the planets orbiting the sun, and the nucleus is the sun itself. Previous clocks listened to the planets (electrons). This new clock listens to the sun (the nucleus). Because the sun is so heavy and isolated, it is much harder to bump into or disturb. This makes the "nuclear pendulum" incredibly stable and resistant to outside noise like temperature changes or magnetic fields.

How They Built It: The "Crystal Sandwich"

The team didn't trap single atoms in a vacuum (which is hard and expensive). Instead, they took a tiny, millimeter-sized crystal of calcium fluoride (the same stuff used in some high-end lenses) and "doped" it with a tiny amount of Thorium-229.

• The Analogy: Imagine a block of Jell-O. If you drop a few specks of glitter into it, the glitter is trapped inside but can still wiggle. The Thorium atoms are the glitter, trapped inside the crystal "Jell-O."
• The Challenge: To make this clock tick, they need to hit the Thorium nuclei with a very specific color of light (ultraviolet light with a wavelength of 148 nanometers). This is a very difficult color of light to produce and control.

The "Feedback Loop": Teaching the Laser to Listen

The core achievement of this paper is that they created a self-correcting system.

1. The Laser: They have a laser that tries to shine on the Thorium nuclei.
2. The Mistake: Lasers naturally drift over time, like a runner who starts to slow down or speed up without realizing it.
3. The Correction: The team set up a "feedback loop." They constantly check if the Thorium nuclei are absorbing the light.
   • If the laser is slightly off-key, the nuclei won't absorb the light.
   • A detector (a photomultiplier tube) sees this and sends a signal back to the laser: "Hey, you're too high! Slow down!" or "You're too low! Speed up!"
   • The laser instantly adjusts itself to match the exact frequency of the Thorium nuclei.

This is the first time a nuclear clock has operated as a stand-alone device that corrects its own errors in real-time, rather than just being a passive experiment.

How Accurate Is It?

The paper reports that this clock is incredibly stable.

• The Metric: They measure "fractional frequency instability." In simple terms, this is how much the clock "jitters."
• The Result: Over a single day of running, the clock's error is so small it approaches 1 part in 1,000,000,000,000,000 (10⁻¹⁵).
• The Catch: Right now, the clock is limited by "shot noise." Imagine trying to hear a whisper in a noisy room. If you only have a few people whispering (photons), it's hard to hear clearly. As they improve the laser power and the crystal, they expect the clock to become even more precise, potentially beating the best atomic clocks in the world.

Why Does This Matter? Hunting for "Dark Matter"

The paper doesn't just talk about telling time; it talks about using the clock as a detector for Dark Matter.

• The Theory: Scientists think the universe is filled with invisible, ultra-light particles called "scalar bosons" (a type of dark matter). These particles might be wiggling through the universe like waves in the ocean.
• The Effect: If these waves pass through our clock, they might slightly change the "weight" of the fundamental forces that hold the Thorium nucleus together. This would make the clock tick slightly faster or slower in a rhythmic pattern.
• The Result: Because the Thorium nucleus is so sensitive to these forces (much more so than regular atoms), this clock is a super-sensitive seismograph for dark matter.
  • The team looked at their data for 23 hours.
  • They found no evidence of these dark matter waves yet.
  • However, by not finding them, they were able to rule out certain theories about how heavy these particles could be and how strongly they interact with light. They set new, stricter "borders" for where scientists should look next.

Summary

The team successfully built a working clock based on the nucleus of a Thorium atom, trapped inside a crystal. They created a system where the clock laser constantly listens to the nucleus and fixes its own drift. While it is currently limited by the amount of light they can use, it is already so sensitive that it can be used to hunt for invisible dark matter particles, proving that "nuclear clocks" are a viable and powerful new tool for physics.

Technical Summary

Technical Summary: A Thorium-229 Optical Nuclear Clock with Feedback Loop

Problem and Context
For the past 70 years, atomic clocks based on microwave and optical electronic transitions have served as the primary frequency standards due to their high stability and accuracy. However, the search for next-generation timekeeping devices has turned toward the thorium-229 (Th-229) isotope. The Th-229 nucleus possesses an extraordinarily low-energy isomeric state at approximately 8.4 eV, offering a potential resonance quality factor on the order of $10^{19}$. Unlike electronic transitions, the nucleus couples weakly to external perturbative fields, allowing for the construction of a robust optical clock in a solid-state material at room temperature. Furthermore, the unique near-cancellation of MeV-scale Coulomb and nuclear contributions to the Th-229 binding energy makes this transition exceptionally sensitive to variations in fundamental constants, such as the fine-structure constant ($\alpha$), the QCD scale parameter ($\Lambda_{QCD}$), and quark masses ($m_q$). This sensitivity positions the Th-229 nuclear clock as a powerful tool for testing theories beyond the Standard Model, including the nature of ultralight dark matter.

While previous studies successfully demonstrated the excitation of Th-229 nuclei using vacuum-ultraviolet (VUV) lasers stabilized to external frequency standards, a system where the laser interrogating the nuclei is steered by the nuclear transition itself had not been realized. This paper addresses the challenge of constructing a stand-alone nuclear clock that operates with a feedback loop based on continuous absorption spectroscopy.

Methodology
The authors implemented a Th-229 nuclear clock by stabilizing a continuous-wave (CW) laser to the 148 nm nuclear transition. The experimental setup consists of two main subsystems:

1. Clock Laser and Frequency Generation: A high-finesse cavity-stabilized external-cavity diode laser (ECDL) operating at 1187 nm serves as the clock laser for short-term stability. This laser is frequency-quadrupled via a commercial laser system and a single-pass second harmonic generation (SHG) stage using a randomly quasi-phase-matched (RQPM) strontium tetraborate (SBO) crystal to generate the required 148 nm VUV radiation.
2. Nuclear Interrogation and Feedback: The 148 nm radiation is directed into a vacuum chamber containing a millimeter-sized calcium fluoride (CaF$_2$) crystal doped with Th-229 (specifically the X2 crystal segment). The nuclei are embedded in the crystal at room temperature. Absorption spectroscopy is performed using a photomultiplier tube (PMT) with a CsI photocathode.
   • Feedback Loop: To generate an error signal, the interrogation frequency is modulated between two frequencies separated by $f_{FWHM}/\sqrt{3}$ (where $f_{FWHM}$ is the full width at half maximum of the absorption peak). The difference in detected photon counts provides a signal with a zero crossing at the absorption peak. This error signal is used as feedback to an electro-optical modulator (EOM) to compensate for the long-term drift of the cavity, effectively locking the laser to the nuclear transition.
   • Comparison: The clock laser's frequency is compared to a Yb$^+$ single-ion clock located at the Bundesamt für Eich- und Vermessungswesen (BEV) in Vienna. This comparison is performed via a frequency comb stabilized by a separate high-finesse cavity and linked via a Doppler-compensated fiber link.

Key Contributions and Results

• First Stand-Alone Nuclear Clock: The paper reports the first implementation of a nuclear clock where the laser is steered by the nuclear transition itself, operating as a stand-alone device rather than relying solely on external references.
• Stability Performance: The nuclear clock demonstrates a fractional frequency instability that follows a simple shot-noise limited scaling of $3 \cdot 10^{-12} / \sqrt{\tau/s}$. Over 1 day of continuous operation, the instability approaches $10^{-15}$.
• Operational Modes: The system was tested in two modes: a fast feedback mode with an integration time ($T$) of 20 seconds and a slower mode with $T = 30$ minutes. The 20-second mode allows for faster comparison with the Yb$^+$ clock, extending the accessible mass range for dark matter searches, while the 30-minute mode minimizes the impact of feedback noise on short-term stability.
• Reproducibility and Systematics: The authors investigated the reproducibility of the clock by measuring line centers at different positions within the crystal. They observed frequency shifts of up to 1.7 kHz between different crystal locations, attributed to local strains induced by structural inhomogeneity. However, measurements at the same position were reproducible.
• Dark Matter Constraints: Using the clock data (specifically the $T=20$ s mode over $\approx 23$ hours), the authors searched for periodic fluctuations and slow drifts in the nuclear transition energy. No oscillations were detected above the 5% detection threshold. Consequently, they established upper limits on the coupling of ultralight scalar dark matter to photons, the strong force, and quarks.

Significance and Claims
The paper claims that the Th-229 nuclear clock, despite being in a development phase, offers enhanced sensitivity for fundamental physics tests compared to current atomic clocks. Specifically:

• Dark Matter Coupling: The constraints on dark matter coupling to photons are competitive with the best atomic clock comparisons. More significantly, the constraints regarding coupling to the strong force and quarks go beyond previous measurements, reaching a parameter space 100 to 1000 times deeper than earlier atomic clock experiments.
• Solid-State Advantage: The ability to operate in a solid-state crystal at room temperature provides a robust platform with a large signal-to-noise ratio compared to single-nucleus approaches.
• Future Potential: The authors project that improvements in VUV laser power, crystal doping homogeneity, and the use of host crystals with reduced linewidths (such as stoichiometric ThF$_4$ or spinless solids) could lead to fractional frequency instabilities of $\sim 10^{-16}/\sqrt{\tau/s}$. This would bring the nuclear clock to the same stability level as state-of-the-art optical atomic clocks while maintaining the simplicity of a solid-state device and offering four orders of magnitude increased sensitivity to variations in fundamental constants.

The work represents a critical step in realizing the potential of nuclear clocks, moving from proof-of-principle excitation to a functional, feedback-stabilized timekeeping device capable of probing new physics.