Optimal Calibration-Free Observable for the Nucleon-Coupling Ratio in a Dual-Alkali Comagnetometer for Dark Matter Searches

Abstract

A dual-alkali single-cell $^{87}$Rb-$^{39}$K-$^{3}$He comagnetometer can read an axionlike dark matter signal through two optical-rotation channels, encoding the ratio $\mathcal{R}=\xi_n/\xi_p$ of the field's neutron and proton spin couplings in their relative response. The inter-species phase difference $\Delta\varphi$ has been proposed as a calibration-free readout that is sensitive to $\mathcal{R}$. Treating the extraction of $\mathcal{R}$ as a statistical estimation problem, we show that the optimal observable is the complex inter-channel ratio, which splits into $\Delta\varphi$ and an amplitude ratio, of which only $\Delta\varphi$ is insensitive to the relative gain and hence calibration-free. For our choice of comagnetometer parameters, above $\sim\!100$ Hz the phase difference alone captures most of the coupling-ratio information. At lower frequencies $\Delta\varphi$ is not near-sufficient: there the amplitude ratio would improve the precision on $\mathcal{R}$ by a factor of $\gtrsim2$ below $\sim\!40$ Hz. Recovering that information, however, requires the relative gain to be known sufficiently accurately, so $\Delta\varphi$ stays the robust observable even where it is not the optimal one.