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

A dual-alkali single-cell 87Rb–39K–3He comagnetometer can read an axion-like dark matter signal through two optical rotation channels, encoding the ratio R=ξn/ξp 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 R. Treating the extraction of R as a statistical estimation problem, we show that the estimate of 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 ∼100 Hz the phase difference alone captures most of the coupling-ratio information. At lower frequencies Δφ is no longer near-sufficient. Below ∼40 Hz, the amplitude ratio would improve the precision on R by a factor of ≳2. 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.