Precision Spectroscopy of 2S-nS Transitions in Atomic Hydrogen: A Determination of the Proton Charge Radius

This paper reports high-precision absolute frequency measurements of 2S-nS (n=8, 9, 10) two-photon transitions in cryogenic atomic hydrogen, yielding a proton charge radius of 0.8433(31) fm and a Rydberg frequency that align well with CODATA 2022 recommendations.

The Gist

Imagine the hydrogen atom as the "perfectly tuned guitar string" of the universe. Because it's so simple (just one proton and one electron), physicists can calculate exactly how it should vibrate. If the real-world guitar sounds even slightly different from the math, it means either our math is wrong, or there's a hidden variable we haven't accounted for yet.

This paper is about a team of scientists who decided to tune that guitar string with extreme precision to measure the size of the proton (the nucleus of the atom) and to check if our fundamental laws of physics are holding up.

Here is a breakdown of what they did, using everyday analogies:

1. The Goal: Measuring the "Proton's Belly Button"

For a long time, scientists have been trying to measure the size of the proton. It's like trying to measure the exact diameter of a tiny marble inside a spinning top. Recently, there was a "proton radius puzzle": measurements using regular hydrogen disagreed with measurements using "muonic hydrogen" (a heavier, exotic version of hydrogen).

This team wanted to settle the score by measuring specific jumps the electron makes inside a regular hydrogen atom. They focused on the electron jumping from a low-energy orbit (2S) to higher-energy orbits (8S, 9S, and 10S).

2. The Setup: A Super-Cold, Super-Slow Train

To measure these jumps accurately, the atoms can't be zooming around like race cars; they need to be moving slowly so the scientists can "listen" to them.

• The Cryogenic Beam: They created a beam of hydrogen atoms that were super cold (cryogenic). Think of this as a train of atoms moving very slowly and smoothly, rather than a chaotic crowd of people running in a stadium.
• The Laser "Tuning Fork": They used lasers to hit the atoms. If the laser frequency matches the exact energy the atom needs to jump, the atom absorbs the energy.
• The "Depletion" Trick: They didn't measure the atoms that jumped; they measured the ones that didn't jump. Imagine a crowd of people (atoms) in a dark room. If you shine a specific light, the people who jump up disappear from the floor. By counting how many people are left on the floor, they can tell exactly what color of light caused the jump.

3. The Big Problem: The "Static Electricity" of Light

When you shine a bright light on an atom, it doesn't just sit there; the light pushes on the atom, slightly changing its energy levels. This is called the AC Stark shift.

• The Analogy: Imagine trying to weigh a feather on a scale, but a strong fan (the laser) is blowing on it, making the scale read heavier or lighter than it really is.
• The Solution: In previous experiments, this "fan" effect was huge and messy. In this experiment, the team used a clever trick: they used a second laser to actively "cancel out" the push of the first laser. It's like having a second fan blowing in the exact opposite direction to create a perfectly still air pocket. This allowed them to see the atom's true frequency without the laser pushing it around.

4. The Results: A New, Precise Measurement

After running hundreds of measurements over seven months, they found:

• The Proton Radius: They calculated the proton's size to be 0.8433 femtometers (a femtometer is one-quadrillionth of a meter).
• The Rydberg Constant: They also refined a fundamental number in physics that describes how atoms emit light.

Why does this matter?
Their result agrees very well with the "official" recommended values (CODATA 2022). This suggests that the "proton radius puzzle" might be getting resolved, or at least that regular hydrogen measurements are consistent with the latest theoretical calculations.

5. What They Didn't Find (and Why That's Important)

The paper notes a small tension: their result for the proton size differs slightly (by about 2.5 "sigma") from a previous measurement they did using a different type of jump (2S to 8D).

• The Analogy: It's like measuring a room with a tape measure and getting 10 feet, but measuring it with a laser ruler and getting 10.05 feet.
• The Conclusion: They couldn't find a specific error in their math or equipment to explain this difference. However, they argue that their new method (measuring S-to-S jumps) is likely more reliable because it avoids certain "distortions" that happen in the other method (like the atom getting confused by nearby energy levels).

Summary

Think of this paper as a high-stakes calibration of the universe's most basic ruler. By cooling down hydrogen atoms, silencing the "noise" of the lasers, and counting the survivors, the team measured the size of the proton with a precision of about 1 part in 400 billion. Their findings support current theories but leave a tiny mystery open for future detectives to solve.

Technical Summary

1. Problem Statement

The hydrogen atom serves as a fundamental testbed for Quantum Electrodynamics (QED) and searches for Beyond Standard Model (BSM) physics. However, a significant discrepancy known as the "proton radius puzzle" has persisted, where proton charge radius ($r_p$) values extracted from muonic hydrogen spectroscopy differ from those derived from electronic hydrogen spectroscopy. While recent electronic measurements have shown some agreement with muonic results, tensions remain.

Specifically, previous high-precision measurements of the $2S_{1/2} - 8D_{5/2}$ transition in hydrogen (by the same group) showed a $\approx 2.5\sigma$ discrepancy with other determinations. The authors hypothesize that systematic effects inherent to $D$-state measurements (such as unresolved hyperfine structure and linear Stark shifts due to near-degenerate states) may be the cause. To resolve this, they present new, ultra-precise measurements of $2S_{1/2} - nS_{1/2}$ transitions ($n=8, 9, 10$), which offer superior systematic control compared to $D$-state transitions.

2. Methodology

Experimental Apparatus:

• Source: A highly collimated, cryogenic beam of atomic hydrogen.
• Excitation: Atoms are excited to the metastable $2S_{1/2}(F=1)$ state using cavity-enhanced 243 nm radiation.
• Spectroscopy: The $2S_{1/2} \to nS_{1/2}$ two-photon transitions are driven by cavity-enhanced radiation in the 759–778 nm range.
• Detection: Atoms that undergo the transition decay via a $P$-state to the ground state. The signal is the depletion of the remaining metastable $2S$ atoms, detected by a Channel Electron Multiplier (CEM).
• Frequency Reference: Absolute frequency is determined using an ensemble of Cesium beam clocks at NIST, linked via a stabilized fiber optic network.

Systematic Error Mitigation:

• AC Stark Shift: This is the dominant systematic error. The experiment employs an active mitigation technique using auxiliary radiation at 651.3 nm to cancel the AC Stark shift induced by the spectroscopy laser. This reduces the shift by over an order of magnitude and restores Lorentzian lineshapes.
• DC Stark Shift: Stray electric fields are passively mitigated by coating surfaces with colloidal graphite. The residual field is characterized using the highly sensitive $2S_{1/2} - 16S_{1/2}$ and $10S_{1/2} - 16S_{1/2}$ transitions, yielding a field magnitude of $\approx 0.415$ V/m.
• Other Shifts: Corrections are applied for the second-order Doppler shift (measured via time-of-flight), blackbody radiation shifts, and Zeeman shifts (negligible for $S$-states).

Data Analysis:

• 48 blinded measurements were performed over seven months.
• Resonance scans are fitted with a Lorentzian function to determine the line center.
• Extrapolation to zero laser power is performed to eliminate residual AC Stark shifts. The authors utilize a linear model for the shift vs. power, validated by a numerical model of atom-laser dynamics.
• Theoretical inputs include newly calculated relativistic Bethe logarithms ($A_{60}$) and QED remainder functions for $n=9$ and $10$.

3. Key Contributions

1. First Precision Measurements: This work presents the first precision measurements of the $2S_{1/2} - 9S_{1/2}$ and $2S_{1/2} - 10S_{1/2}$ transitions.
2. Improved $2S-8S$ Precision: The uncertainty of the $2S_{1/2} - 8S_{1/2}$ transition was reduced by a factor of 4.4 compared to previous work.
3. Advanced Systematic Control: The demonstration of active AC Stark shift mitigation allows for the recovery of ideal Lorentzian lineshapes and significantly reduces non-linearities in the extrapolation to zero power.
4. Theoretical Advances: The authors provide new theoretical calculations for the relativistic Bethe logarithms and QED remainder functions for $n=9$ and $10$ states, essential for extracting fundamental constants from these transitions.
5. Resolution of Discrepancy: By comparing these $S$-state results with previous $D$-state results, the authors isolate potential systematic errors in the $D$-state measurements (specifically regarding hyperfine structure and Stark distortions).

4. Results

• Transition Frequencies: The authors measured the $2S_{1/2} - nS_{1/2}$ transitions ($n=8, 9, 10$) with a fractional uncertainty of $\approx 2.6 \times 10^{-12}$.
• Proton Charge Radius ($r_p$): Combining these results with the $1S_{1/2} - 2S_{1/2}$ transition frequency, they extracted:
  $$r_p = 0.8433(31) \text{ fm}$$
  This value is in excellent agreement with the CODATA 2022 recommended value and recent muonic hydrogen results.
• Rydberg Constant ($R_\infty$): The extracted Rydberg frequency is:
  $$cR_\infty = 3\,289\,841\,960\,252.9(9.7) \text{ kHz}$$
  This also agrees well with CODATA 2022.
• Consistency Check: The results from this work are consistent with the CODATA 2022 values but disagree with the authors' previous $2S_{1/2} - 8D_{5/2}$ measurement by $\approx 2.5\sigma$. The authors attribute this to systematic effects specific to $D$-state measurements (unresolved hyperfine structure and linear Stark shifts) rather than new physics or underestimated errors in the $S$-state measurements.

5. Significance

• Resolution of the Proton Radius Puzzle: The results strongly support the smaller proton radius value derived from muonic hydrogen, confirming that the "puzzle" is likely resolved by refining electronic hydrogen measurements and correcting for systematic errors in specific transition types.
• Validation of QED: The agreement between experiment and theory (including high-order QED corrections) reinforces the validity of the Standard Model in the atomic sector.
• BSM Constraints: The high precision and the use of multiple transitions with different principal quantum numbers ($n$) provide stringent constraints on BSM theories involving light bosons that would modify the Coulomb potential.
• Future Directions: The paper highlights the potential for extending these $2S-nS$ measurements to Deuterium, which would offer even more sensitive tests of isotope shifts and BSM physics.

In conclusion, this work represents a milestone in atomic spectroscopy, utilizing advanced systematic mitigation techniques to achieve record-breaking precision in hydrogen $S$-state transitions, thereby providing a definitive determination of the proton charge radius that aligns with the muonic hydrogen results.