🔬 atomic physics

Reassessment of line profile asymmetry in measurements of the 1s-2s energy interval in hydrogen

This paper revises the theoretical analysis of line profile asymmetry in hydrogen's 1s-2s transition, demonstrating that this asymmetry causes a frequency shift consistent with modern experimental error budgets and necessitating adjustments to the standard measurement models.

Original authors: D. Solovyev, A. Anikin, T. Zalialiutdinov, L. Labzowsky

Published 2026-07-24

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CC BY 4.0

Original authors: D. Solovyev, A. Anikin, T. Zalialiutdinov, L. Labzowsky

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

Imagine you are a detective trying to measure the exact height of a mountain, but the mountain is an atom, and the "height" is the energy it takes to jump from one level to another. In the world of physics, hydrogen is the simplest mountain of all, just one proton and one electron. Because it's so simple, scientists have been measuring its "steps" (energy transitions) with incredible precision for decades. They are so good at it that they can measure these steps to within a few parts in 10^15. This isn't just about being precise for the sake of it; these measurements are the rulers we use to define the fundamental constants of the universe, like the size of the proton itself. If our ruler is even a tiny bit crooked, our understanding of the universe's building blocks gets a little wobbly.

For a long time, scientists thought they had the perfect ruler. They measured the jump from the ground floor (1s) to the second floor (2s) of the hydrogen atom using two photons at once. The measurements were so consistent that the number didn't change even after repeating the experiment years later. However, there was a nagging suspicion: the "shape" of the signal they were measuring might be slightly lopsided. Imagine trying to find the exact center of a bell curve, but the bell is slightly leaning to one side. If you assume it's perfectly symmetrical, your center point will be off, even if just by a hair. This paper dives into that specific "lean," asking if the way the signal is distorted could be hiding a tiny but important error in our most precise measurements.

The authors of this paper, a team of physicists from Russia, decided to take a fresh look at the theoretical "blueprint" of how this hydrogen jump happens. They focused on a specific quirk in the experiment: the setup where the atom is excited in one place, flies through the air, and then is hit by an electric field in a different place to make it glow. They realized that this electric field mixes two different states of the atom (the 2s and 2p states) together, like mixing two colors of paint. Because one of these "colors" (the 2p state) is naturally very unstable and fades away quickly, this mixing creates a unique interference pattern.

Think of it like two musicians playing the same note. If they are perfectly in sync, you hear a clear tone. But if one musician is slightly out of tune and their sound fades away at a different rate, the resulting sound isn't a perfect, symmetrical bell shape; it gets a little "bent" or asymmetrical. The paper shows that this bending is real and is caused by a quantum effect called "quantum interference," where the paths the atom takes to get excited and then decay overlap in a way that distorts the final signal.

The researchers built a new mathematical model to describe this bent shape, which they call an "asymmetric line profile." When they compared this new, bent model to the old, perfectly symmetrical model (which everyone had been using), they found a difference. The old model was effectively "looking" at the peak of the curve in the wrong spot. Their calculations suggest that this mistake shifts the measured frequency of the jump by a tiny amount—somewhere between 0.9 and 14.8 Hertz, depending on the strength of the electric field used in the experiment.

To make sure this wasn't just a theory, they also accounted for the messy reality of the experiment: the atoms are flying at different speeds, and the electric field doesn't turn on instantly but switches on over a short time. They simulated how these factors change the signal. They found that while cooling the atoms and waiting for a specific time delay (about 1210 microseconds) helps narrow the signal down to a very sharp line (about 200 Hertz wide), the underlying "lean" or asymmetry doesn't disappear. It's still there, hiding in the data.

The paper concludes that this asymmetry is significant enough to matter. The shift they found is right at the level of the current experimental error margins. This means that the tiny disagreements scientists have seen between different measurements might not be random noise, but rather a systematic error caused by using a symmetrical model for an asymmetrical reality. By adopting this new, bent model to analyze their data, scientists can potentially correct the frequency value, tightening the uncertainty and giving us a more accurate ruler for the universe. The authors aren't claiming they have solved the mystery of the proton's size, but they are pointing out a very specific, previously overlooked tilt in the data that needs to be straightened out to get the next level of precision.

Technical Summary: Reassessment of Line Profile Asymmetry in 1s-2s Hydrogen Measurements

Problem Statement
The determination of the 1s–2s two-photon transition frequency in hydrogen represents one of the most precise spectroscopic measurements available, with experimental precision reaching parts in $10^{15}$ . Despite high precision, the measured frequency value has remained stable over two decades, yet theoretical efforts to fully explain the line profile have not revealed significant contributions that would alter this value. A persistent discrepancy exists in the determination of fundamental constants, such as the proton charge radius and the Rydberg constant. While previous analyses (e.g., Refs. [19, 20]) suggested negligible effects on the transition frequency, they were evaluative in nature. The current work addresses the potential influence of line profile asymmetry, driven by quantum interference effects (QIE) and the mixing of states in external fields, which may introduce a frequency shift comparable to current experimental uncertainties.

Methodology
The authors employ finite-time quantum electrodynamics (QED) theory to construct a detailed theoretical line profile for the 1s–2s two-photon absorption process followed by a time-delayed one-photon decay. The methodology involves:

Electron Propagator in External Fields: The analysis utilizes an electron propagator operating at finite times, accounting for the external electric field used to mix the 2s and 2p states (specifically $2s_{1/2}$ and $2p_{1/2}$ ) to enable Lyman- $\alpha$ detection. The wave functions are treated as solutions to the Dirac equation in the presence of the nucleus and the external field.
S-Matrix Formalism: The transition amplitude is evaluated using the S-matrix formalism, incorporating both resonant and non-resonant terms. The authors explicitly include terms previously omitted in other studies, such as the mixing of $2s$ and $2p$ states via the external field and the interference between different photon absorption pathways (e.g., $E1E1$ vs. $E1E2/E1M1$ ).
Asymmetric Line Profile Construction: By considering the quantum interference between the $2s$ and $2p$ states, the authors derive an asymmetric line profile (Fano–Voigt type) rather than a symmetric Lorentzian. The profile is characterized by coefficients $a$ , $b$ , and $C$ , which depend on transition amplitudes, the Lamb shift ( $\Delta E_L$ ), and the natural widths of the states.
Non-Adiabatic Field Switching and Time Delay: The model accounts for the spatial separation between the excitation and de-excitation regions in experiments [1–3]. The external electric field is treated as switching on non-adiabatically at a time delay $\tau$ . This introduces time-dependence into the mixed state width ( $\Gamma_{2\tilde{s}}$ ) and the line profile numerator.
Numerical Integration: The final line profile is obtained by integrating over the Maxwellian velocity distribution of the atoms, incorporating the second-order Doppler effect and the specific time delay $\tau$ used to select slower atoms.

Key Contributions and Results

Derivation of Asymmetric Profile: The paper derives a line profile equation (Eq. 8) that explicitly includes asymmetry parameters. This profile is shown to differ from the symmetric Lorentzian contour used in standard data processing.
Quantification of Frequency Shift: The authors calculate the frequency shift ( $\Delta(x)$ ) induced by the asymmetry. For a full-width at half-maximum (FWHM) analysis, the shift is estimated to be in the range of $[0.9; 14.8]$ Hz for electric field strengths between 10 and 20 V/cm. The shift at the line maximum is approximately half this value.
Impact of Time Delay and Velocity Selection: The study demonstrates that while time delays (e.g., $\tau = 1210$ $\mu$ s) and atomic cooling (to 5 K) significantly narrow the line profile (reducing FWHM from ~150 kHz to ~200 Hz), they do not eliminate the asymmetry shift. The shift persists across the ensemble of detected emissions, reaching values up to ~7 Hz for specific atomic velocities in the 5 K case.
Resolution of Linewidth Discrepancy: The inclusion of the mixed state width and non-adiabatic field switching in the theoretical model helps explain the ~200 Hz discrepancy previously observed between theoretical predictions (Ref. [32]) and experimental linewidths for cooled atoms.
Independence from Orientation: The calculated frequency shift is shown to be independent of the electric field orientation, laser direction, and polarization, distinguishing it from other angular-dependent effects.

Significance and Claims
The paper asserts that the observed line profile asymmetry can lead to a frequency shift in the 1s–2s transition at the level of modern experimental accuracy. The authors claim that:

The asymmetry shift is consistent with the line shape model contribution that forms part of the error budget in experimental measurements.
The use of an asymmetric profile (Eq. 8 or Eq. 15) for processing experimental data, rather than a symmetric Lorentzian, is theoretically justified and necessary for further improving the accuracy of the transition frequency determination.
The findings provide a theoretical basis for the "line shape uncertainty" acknowledged in previous high-precision experiments.
The results suggest that the frequency shift derived from the asymmetric profile can be used to adjust experimental values, potentially refining the determination of physical constants like the Rydberg constant and the proton charge radius.

The authors maintain a modest tone, noting that their analysis relies on specific approximations (e.g., neglecting Stark shifts and higher-order state mixing) but argue that the derived shifts are significant enough to warrant inclusion in the error analysis of future high-precision spectroscopy.

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