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 $\Delta \phi$ 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 $\Delta \phi$ and an amplitude ratio, of which only $\Delta \phi$ 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 $\Delta \phi$ is not near-sufficient: there the amplitude ratio would improve the precision on $\mathcal{R}$ by a factor of $\gtrsim 2$ below $\sim 40$ Hz. Recovering that information, however, requires the relative gain to be known sufficiently accurately, so $\Delta \phi$ stays the robust observable even where it is not the optimal one.

Comments: 13 pages, 3 figures