Figure 1. Classical and quantum NMR detection regimes.

(a) Inductive NMR: Interaction between the sensor and individual nuclear spins (Γ_S) is much weaker than interaction between nuclei (Γ_N). For large spin ensembles (≳10¹² nuclear spins) the time-averaged signal, is proportional to the sample polarization (multiplied by the spin magnetic moment, μ). In the high temperature limit, sample magnetization is , where N is the number of spins, B is the external magnetic field, T is the temperature and k_B is the Boltzmann constant. (b) Nanoscale NMR: For detection of small ensembles (~10⁴ nuclear spins) with a weakly coupled sensor, is still proportional to the sample magnetization. Now statistical fluctuations may exceed thermal polarization, so sample magnetization is given by. (c) Quantum NMR: Detection of individual or few nuclei may occur when interaction of each nuclear spin with the sensor (Γ_S) is stronger than the coupling to the surrounding bath (Γ_N). In this strong coupling regime, the magnetic field of individual spins is measured rather than sample magnetization. The detected signal, , is then proportional to Nμ regardless of whether the nuclei have a net polarization.