Spectroscopy of $^4$He at 0.25 ppt Uncertainty and Improved Alpha-Helion Charge-Radius Difference Determination

Abstract

High-precision spectroscopy of simple atomic systems can be used to advance the theory of atomic energy levels but can also serve as a sensitive probe of nuclear charge radii. For this last purpose, we report an improved measurement of the $2\,^3{S}_1 \to 2\,^1{S}_0$ transition frequency in $^4$He with 48 Hz uncertainty (0.25 ppt), using a Bose-Einstein condensed sample confined in a magic-wavelength optical dipole trap. A systematic Doppler shift from condensate motion is suppressed by time-resolved ion detection, and the transition frequency is calibrated via a White Rabbit link to a remote active hydrogen maser clock. Combined with previous $^3$He measurements and improved theory, we obtain the most precise determination to date of the charge-radius difference between the alpha and helion particles of $r_{h}^2 -r_α^2$ of $1.0676(10)\text{fm}^2$. This is consistent with other recent determinations and confirms that the current discrepancy between QED theory and experimentally observed ionization energies of excited states in helium is not apparent in the isotope shift.

Figures (5)

• Figure 1: Overview of the optical dipole trap and frequency metrology. Using a phase-locked loop (PLL), the 1557 nm spectroscopy laser is locked to an ultrastable laser via an optical frequency comb, by mixing the beat note from a photodiode (PD). The OFC is referenced through a White Rabbit (WR) link to an active hydrogen maser at VSL. Part of the 1557 nm light is used together with a separate 1085 nm laser, locked to a wavemeter (WM), to generate the UV laser light. The UV laser light is sent through an AOM, and subsequently split into two beams of orthogonal linear polarization that are focused to form the two arms of the optical dipole trap. The BEC is trapped at the crossing point of the two beams. By using shutters, we excite the BEC alternately from opposite directions. The He$_{(2)}^+$ ions created by the excitation are detected using an MCP, and the arrival times are registered using a time-to-digital converter (TDC).
• Figure 2: a) shows an histogram of from the arrival times of the He$^+$ ions over the 100 ms spectroscopy time measured with the time to digital converter. The axial trap oscillation induces a time-dependent Doppler shift periodic with the inverse of the trap frequency $\nu_{\text{trap}}$. As a result the BEC is only periodically resonant with the spectroscopy signal. To prevent a systematic Doppler shift, we sum the ion counts over a time window equal to the trap oscillation period, as indicated by the shaded area. Note that for the example shown here we deliberately increased the oscillation amplitude to illustrate the effect, but for all regular measurements the oscillations were first minimized (see text). b) shows a typical $m_J=\pm1$ spectrum. The $m_J = \pm1$ lines are alternately excited. The black points represent the ion counts with the spectroscopy laser on, while the grey points are background counts measured immediately before the spectroscopy. The typical transition linewidth is 3 kHz.
• Figure 3: Overview of the multiple linear regression analysis and the resulting transition frequencies for all measurement sets. For measurement set 2, part a–c shows the partial residual plots as functions of the ODT power, probe power, and chemical potential, respectively. The slope of the ODT power regression yields the differential ac Stark shift, while the slope of the chemical potential regression is used to extract the triplet-singlet scattering length. The black circles in d) represent the measured frequencies for this data set, while the red points show the residuals after correcting for the measured systematic effects. Error bars correspond to the statistical uncertainty of the fit, with the residual errors bars adjusted to include the uncertainty from the fitted slopes. e) and f) show the measured transition frequencies from the various measurement sets with excitation from the left and right sides respectively. The solid lines represent the weighted mean, and the shaded areas indicate the standard error of the weighted mean. The result from the extrapolation shown in a–d corresponds to the transition frequency of set 3 in e). The square points represent measurements done with a local cesium clock.
• Figure 4: The measured (differential) ac Stark shift as a function of dipole trap wavelength between the $2\,^3{S}_1$ and $2\,^1{S}_0$ states. Each point represents the ac Stark shift determined from the multiple linear regression analysis for a specific measurement set. The blue curve is the fitted polarizability curve from Notermans2014. The magic wavelength, found at the zero crossing of the differential ac Stark shift, is determined to be 319.81602(4) nm. The grey shaded region indicates the 1$\sigma$ confidence interval on the fit.
• Figure 5: Overview of differential nuclear charge radius measurements between the alpha and helion particle, $r_{h}^2 -r_{\alpha}^2$. A comparison is made between measurements based on spectroscopy in electronic helium vanderWerf2025vanRooij2011Canciopastor2012Canciopastor2004Shiner1995Zheng2017Wen2025ClausenMerkt2025Huang2020, muonic helium Schuhmann2025, and electron scattering experiments Sick2014. Zheng Zheng2017 and Wen Wen2025 measured the $^{4}$He transition frequency and used the $^{3}$He result of Cancio Pastor Canciopastor2012 to determine $r_{h}^2 - r_{\alpha}^2$.