  Nuclear magnetic resonance is a promising experimental approach to search for
ultra-light axion-like dark matter. Searches such as the cosmic axion
spin-precession experiments (CASPEr) are ultimately limited by
quantum-mechanical noise sources, in particular, spin-projection noise. We
discuss how such fundamental limits can potentially be reached. We consider a
circuit model of a magnetic resonance experiment and quantify three noise
sources: spin-projection noise, thermal noise, and amplifier noise. Calculation
of the total noise spectrum takes into account the modification of the circuit
impedance by the presence of nuclear spins, as well as the circuit back-action
on the spin ensemble. Suppression of the circuit back-action is especially
important in order for the spin-projection noise limits of searches for
axion-like dark matter to reach the quantum chromodynamic axion sensitivity.
