Optimal Calibration-Free Observable for the Nucleon-Coupling Ratio in a Dual-Alkali Comagnetometer for Dark Matter Searches
A dual-alkali single-cell $^{87}$Rb-$^{39}$K-$^{3}$He comagnetometer can read an axion-like dark matter signal through two optical rotation channels, encoding the ratio $\mathcal{R}=ξ_n/ξ_p$ of the field's neutron and proton spin couplings in their relative response. The inter-species phase difference $Δφ$ 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 estimate of $\mathcal{R}$ depends on the two channels only through their complex ratio, which splits into $Δφ$ and an amplitude ratio, of which only $Δφ$ 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 $Δφ$ is no longer near-sufficient. Below $\sim\!40$ Hz, the amplitude ratio would improve the precision on $\mathcal{R}$ by a factor of $\gtrsim2$. Recovering that information, however, requires the relative gain to be known sufficiently accurately, so $Δφ$ remains the robust observable even where it is not the most informative one.