Narrow linewidth spectroscopy in quantum degenerate metastable helium

Combined with high-precision spectroscopy, QED theory allows extraction of the nuclear charge radius from spectroscopy in simple atomic systems. This recently lead to a significant discrepancy in the proton charge radius determined from hydrogen and muonic hydrogen spectroscopy, now known as the `proton size puzzle'. Spectroscopy in helium can provide additional insight in this conundrum. Our group previously measured the very weak $2\ ^3S \to 2\ ^1S$ transition ($\lambda=1557$ nm, $\Gamma=2\pi \times 8$ Hz) to $10^{-11}$ relative accuracy in quantum degenerate ($T=0.2\ \mu$K) metastable $^4$He and $^3$He, allowing a 1\% accurate determination of the charge radius difference of the $\alpha$ particle and the helion. Recent measurements in muonic He$^+$ aim for a precision of $3\times 10^{-4}$. In order to provide a similar precision, we aim to remeasure the transition to sub-kHz precision by reducing the linewidth of the spectroscopy laser by over an order of magnitude to the kHz level and by implementing a magic wavelength ($\lambda=320$ nm) dipole trap operating at 2 W CW power. First measurements in a helium BEC have shown a 10 kHz asymmetric line profile due to mean-field effects. This allows for the first determination of the unknown $2\ ^3S-2\ ^1S$ scattering length.