A multi-ion optical clock with $\mathbf{5 \times 10^{-19}}$ uncertainty

This paper reports the development of a robust multi-ion optical clock using up to 10 strontium ions that achieves a record-breaking fractional frequency uncertainty of $5.3\times10^{-19}$ while reducing measurement time by a factor of 4.8 compared to single-ion systems.

The Gist

Imagine you are trying to measure the exact length of a single grain of sand. If you use a ruler, you might be off by a millimeter. If you use a laser micrometer, you might be off by a hair's width. But what if you needed to measure that grain of sand with such precision that you could detect if it had gained or lost a single atom?

That is the level of precision scientists at the Physikalisch-Technische Bundesanstalt (PTB) in Germany are working with. They have built a new kind of atomic clock so accurate that it would not lose or gain a single second over the entire age of the universe.

Here is the story of how they did it, explained without the heavy math.

The Problem: The "One-Person Band" vs. The "Choir"

For years, the most accurate clocks in the world have been single-ion clocks. Think of these as a solo singer holding a perfect note. They are incredibly precise, but they have a problem: noise.

In the quantum world, measuring a single particle is like trying to hear a whisper in a hurricane. There is a natural "fuzziness" (called quantum projection noise) that makes the measurement jittery. To get a clear picture, the solo singer has to hold that note for a very, very long time to average out the noise. This makes the clock slow to react and take a long time to get a result.

The scientists asked: What if we didn't use a solo singer, but a choir?

If you have 8 to 10 singers (ions) all singing the same note at the same time, the "fuzziness" cancels out much faster. You get a clear signal 4.8 times faster than with just one singer. This is the concept of a multi-ion clock.

The Challenge: The "Crowded Room" Effect

However, putting a choir together is tricky. In a single-ion clock, the singer is alone in a quiet room. In a multi-ion clock, the singers are packed into a tiny, invisible cage (a trap) next to each other.

When you crowd them together, two new problems arise:

1. The "Shoulder Push": The singers push against each other (electric repulsion). This changes the pitch of their voices slightly depending on where they stand in the line.
2. The "Magnetic Wind": The magnetic field holding them in place isn't perfectly uniform. It's like a wind that blows harder on the left side of the room than the right. This makes the singers on the left sing a slightly different note than the ones on the right.

If you don't fix this, your "choir" will sound out of tune, and your clock will be inaccurate. Previous attempts at multi-ion clocks struggled with this, often resulting in a "chorus" that was less accurate than the solo singer.

The Solution: The "Magic Angle" and the "Smart Camera"

The team solved these problems with two clever tricks:

1. The Magic Angle (The Tilted Compass)
The scientists realized that the "magnetic wind" and the "shoulder pushing" (quadrupole shift) could cancel each other out if they tilted the magnetic field just right.

• Analogy: Imagine trying to balance a stack of books on a wobbly table. If you tilt the table at a very specific angle (54.7 degrees), the books stop sliding off.
• By setting their magnetic field to this "magic angle," they neutralized the distortion caused by the ions pushing against each other. The residual error was so small it was practically invisible (below the level of $10^{-20}$).

2. The "Smart Camera" (Seeing the Individuals)
Usually, when you have a choir, you just listen to the whole group. But if one person is off-key, you might not know who it is.

• The team built a special camera (an EMCCD) that can see each individual ion in the chain.
• Analogy: It's like having a spotlight that can instantly zoom in on any single singer in the choir to check if they are singing the right note.
• This allowed them to detect and correct for tiny differences in position. If one ion was slightly out of place, they knew exactly how to adjust the math to fix it.

The Bonus: Catching the "Ghost"

Here is a surprising twist. Usually, having more particles makes a system more prone to errors (like more people in a room making more noise). But in this case, having more ions actually made the clock more accurate in one specific way.

The clock operates in a vacuum, but sometimes a stray gas molecule (a "ghost") bumps into an ion.

• Solo Clock: If a ghost bumps the single singer, the singer falls off the stage, and the clock stops. The scientists might not even know it happened until the clock restarts.
• Choir Clock: If a ghost bumps one singer in a choir of 10, the camera sees that one singer stop singing while the other 9 keep going. The system instantly knows, "Oh, we had a collision," and fixes it immediately.
• Result: Because they can spot these collisions so easily, the "ghost" errors are reduced by a factor of four compared to the single-ion clock.

The Result: A New Standard

The team compared their new "Choir Clock" (using Strontium ions) against a world-famous "Solo Clock" (using Ytterbium ions).

• They found that the two clocks agreed with each other to an incredible degree of precision.
• The uncertainty was so low that it is now one of the few measurements in the world precise enough to help redefine the second itself.

Why Does This Matter?

You might ask, "Who cares if a clock is this accurate?"

1. GPS and Navigation: Current GPS relies on clocks that are good, but not this good. Future GPS could pinpoint your location to within a few millimeters, not just a few meters.
2. Finding New Physics: If time flows differently in different places (due to gravity or new forces), this clock is sensitive enough to detect it. It's a microscope for time itself.
3. The Future of Time: We are moving toward an "optical second" that is defined by light waves rather than the vibration of atoms in a microwave oven. This clock proves that we can build these new clocks using groups of atoms, making them faster and more robust.

In summary: The scientists took a solo act, turned it into a choir, taught the choir to stand in a perfect formation, gave them a camera to watch every member, and ended up with a timekeeper so precise it could measure the age of the universe without losing a second.

Technical Summary

1. Problem Statement

Optical atomic clocks based on single trapped ions currently represent the most precise timekeeping instruments, achieving fractional frequency uncertainties below $1 \times 10^{-18}$. However, their frequency instability is fundamentally limited by quantum projection noise, which scales as $1/\sqrt{N}$ (where $N=1$ for a single ion). This necessitates long averaging times to achieve high statistical precision.

Scaling to multi-ion systems (Coulomb crystals) offers a pathway to reduce measurement time by a factor of $\sqrt{N}$. However, this introduces significant challenges:

• Inhomogeneous Frequency Shifts: In a chain of ions, external perturbations (magnetic fields, electric field gradients) vary across the ion positions, causing inhomogeneous broadening and systematic frequency shifts.
• Quadrupole Shifts: The excited state of the clock transition has a large electric quadrupole moment. Inhomogeneous electric field gradients along the ion chain couple to this moment, creating position-dependent shifts that are difficult to suppress.
• Background Gas Collisions: While multi-ion operation increases the signal, it was unclear if systematic uncertainties (specifically from collisions) could be maintained at the $10^{-19}$ level compared to single-ion systems.

The central challenge addressed is achieving state-of-the-art systematic uncertainty (comparable to single-ion clocks) while utilizing multiple ions to reduce measurement time.

2. Methodology

The researchers developed a multi-ion optical clock using $^{88}\text{Sr}^+$ ions confined in a linear radiofrequency (RF) Paul trap.

• Ion Configuration: The system operates with 8 to 10 ions simultaneously interrogated in a linear Coulomb crystal.
• Clock Transition: The $^2S_{1/2} \leftrightarrow {}^2D_{5/2}$ electric quadrupole transition at 674 nm.
• Key Technical Innovations:
  • Ion-Resolved State Detection: An Electron-Multiplying Charge-Coupled Device (EMCCD) camera is used to detect the fluorescence of individual ions. This allows for the measurement of excitation imbalances across the chain, enabling the characterization and minimization of position-dependent shifts.
  • Static Suppression of Quadrupole Shifts: Instead of using complex dynamical decoupling sequences, the team utilized static suppression. By orienting the quantization axis (magnetic field) at the "magic angle" of $\theta \approx 54.7^\circ$ relative to the trap axis, the tensor component of the quadrupole shift is nulled.
  • Magnetic Field Gradient Compensation: A dedicated set of coils in an anti-Helmholtz configuration compensates for magnetic field gradients along the chain, reducing residual gradients to $<0.1$ nT/mm.
  • Sideband Cooling: To mitigate thermal dephasing caused by low-frequency motional modes, the axial center-of-mass (COM) mode is continuously cooled via resolved sideband cooling before interrogation.
  • Collision Detection: The system monitors for "crystal melting" events (caused by background gas collisions) which are more frequent and detectable in multi-ion chains than in single-ion chains.

3. Key Contributions

• Demonstration of High-Precision Multi-Ion Operation: The paper reports the first multi-ion optical clock with a fully evaluated systematic uncertainty in the $10^{-19}$ range.
• Suppression of Inhomogeneous Shifts: The authors demonstrated that residual frequency shift variations along a chain of up to 10 ions can be suppressed to below the $10^{-20}$ level through static geometric alignment and active gradient compensation.
• Systematic Uncertainty Reduction via Multi-Ion Operation: Counter-intuitively, the multi-ion system achieved a lower systematic uncertainty than its single-ion counterpart ($5.3 \times 10^{-19}$ vs. $6.5 \times 10^{-19}$). This is attributed to the improved detection of background gas collisions; in a multi-ion chain, collisions cause crystal melting (invalid cycles) that are easily detected, whereas they often go undetected in single-ion operation.
• Frequency Ratio Comparison: A rigorous comparison was performed against an established single-ion $^{171}\text{Yb}^+$ clock (based on the electric octupole transition), yielding a frequency ratio with unprecedented precision.

4. Results

• Systematic Uncertainty: The total fractional frequency uncertainty of the $^{88}\text{Sr}^+$ multi-ion clock is $5.3 \times 10^{-19}$.
  • The dominant uncertainty source is Blackbody Radiation (BBR) shift ($4.4 \times 10^{-19}$), followed by thermal motion ($2.6 \times 10^{-19}$).
  • The uncertainty contribution from background gas collisions was reduced from $3.9 \times 10^{-19}$ (single ion) to $0.9 \times 10^{-19}$ (multi-ion).
  • Residual quadrupole shift inhomogeneity was constrained to an uncertainty of $0.1 \times 10^{-19}$.
• Measurement Time Reduction: Operating with 8–10 ions reduced the measurement time required to reach a specific statistical uncertainty by a factor of 4.8 compared to single-ion operation.
• Frequency Ratio: The comparison with the $^{171}\text{Yb}^+$ clock yielded an unperturbed frequency ratio:
  $$\frac{\nu_{\text{Sr}^+}}{\nu_{\text{Yb}^+}} = 0.6926711632159660405(20)$$
  • Statistical Uncertainty: $0.9 \times 10^{-18}$
  • Combined Uncertainty: $2.9 \times 10^{-18}$
• Stability: The multi-ion clock demonstrated an instability of approximately $1.1 \times 10^{-15}/\sqrt{\tau}$, limited by dead time associated with re-cooling after crystal melting.

5. Significance

• Redefinition of the Second: The results represent a significant step toward the future optical redefinition of the SI second. The frequency ratio between two different atomic species ($^{88}\text{Sr}^+$ and $^{171}\text{Yb}^+$) now meets the criterion of an overall fractional uncertainty $\le 5 \times 10^{-18}$, a prerequisite for establishing a robust optical time scale.
• Scalability: This work proves that multi-ion systems are not just viable for reducing instability but can also achieve superior systematic performance compared to single-ion clocks. This opens the door for scaling to even larger ion numbers or other ion species (e.g., $^{115}\text{In}^+$, $^{176}\text{Lu}^+$) to further improve clock performance.
• Fundamental Physics: The high precision and stability of these clocks enhance their utility as tools for testing fundamental physics, such as Lorentz symmetry violations and variations of fundamental constants.

In conclusion, this paper establishes that multi-ion optical clocks can overcome the historical trade-off between speed (instability) and accuracy (systematic uncertainty), offering a robust path forward for next-generation timekeeping standards.