Precision frequency measurements with polarized ${}_{}{}^3{}_{}{}^{}\mathrm{He}$, ${}_{}{}^{21}{}_{}{}^{}\mathrm{Ne}$, and ${}_{}{}^{129}{}_{}{}^{}\mathrm{Xe}$ atoms

We present the principles, applications, and performance of new techniques developed to measure the frequency spectrum of freely precessing polarized noble-gas atoms. The noble-gas polarization technique of spin exchange with laser optically pumped Rb enables us to simultaneously polarize several species ${(}^3$He, ${}_{}{}^{21}{}_{}{}^{}\mathrm{Ne}$, and ${}_{}{}^{129}{}_{}{}^{}\mathrm{Xe}$). With signal-to-noise ratios of ${10}^4$, the free precession frequencies are determined with a precision of less than ${10}^{\mathrm{-}6}$ Hz out of ${10}^3$ Hz in 1024 s. The technique will be applied to determine frequency shifts due to dipole couplings to applied fields and quadrupole splittings due to local Lorentz-invariance-violating spatial anisotropy. In all cases, the frequencies of two or more species are measured simultaneously in order to provide magnetometry for increased precision and to provide measurement and cancellation of systematic effects.