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Coherence order classifies an operator component of NMR coherence by its response to rotation about the chosen total $z$ angular momentum. With dimensionless $F_z$ and rotation convention $U(\varphi )=e^{-i\varphi F_z}$, order $p$ meansThe order describes a phase transformation, rather than the amount of magnetization or the duration for which coherence persists.^[ref]

An original two-spin example shows why order zero is not synonymous with a diagonal population. Let $F_z=I_z+S_z$ for two spin one-half nuclei. For raising and lowering components with $[I_z,I_{+}]=I_{+}$ and $[S_z,S_{\pm }]=\pm S_{\pm }$,The first is zero-quantum coherence; the second is a double-quantum coherence component. In a product basis, the first connects the states with projections $(+1/2,-1/2)$ and $(-1/2,+1/2)$: different states with the same total projection. The second connects total projections−1 and+1. These operator components are not individually normalized density matrices.

At $\varphi =\pi /2$, the order-zero term remains unchanged while the order-two term changes sign. At $\varphi =\pi$, both remain unchanged. A phase comparison must therefore be chosen to discriminate the orders of interest; one phase change cannot distinguish every possible component.

Order zero also does not require zero time evolution. Under the Quantum Hamiltonian $H/\hslash ={\Omega }_I{I}_z+{\Omega }_S{S}_z$, the first component acquires $e^{-i({\Omega }_I-{\Omega }_S)t}$ when the two offsets differ. Total order and the separate offsets answer different questions.^[ref]

NMR phase cycling uses such phase responses to select contributions to an experiment. Selecting an order does not ensure that the spin system or pulse sequence actually generated it, nor that it is directly observable before further conversion.^[ref]

Double-quantum coherence is a component of NMR coherence with coherence order $p=+2$ or $p=-2$. Under a common $z$ rotation $U(\varphi )=e^{-i\varphi F_z}$, an order-$p$ component transforms as $U{A}_p{U}^{†}=e^{-ip\varphi }{A}_p$. The term describes a phase response of the spin system.^[ref]

For two spin one-half nuclei with $F_z=I_z+S_z$, the component $I_{+}{S}_{+}$ has order+2: it connects the state with total projection−1 to that with total projection+1. Its adjoint $I_{-}{S}_{-}$ has order−2. A Hermitian density matrix can contain both components with conjugate coefficients.

An original constructed state makes the difference from populations explicit:Its density matrix has populations one half in each displayed basis state, plus the off-diagonal terms $|\uparrow \uparrow ⟩⟨\downarrow \downarrow |/2$ and its adjoint. A mixture of the two basis states with the same populations lacks those terms. This is an assumed mathematical preparation, rather than a claim about what a single pulse produces in a thermal sample.

Under the Quantum Hamiltonian $H/\hslash ={\Omega }_I{I}_z+{\Omega }_S{S}_z$, the positive-order off-diagonal component acquires phase $e^{-i({\Omega }_I+{\Omega }_S)t}$. A common phase rotation of $\pi /2$ changes its sign; a rotation of $\pi$ leaves it unchanged. The summed offsets and doubled phase response are consequences of the two projection changes.^[ref]

Conventional transverse detection reads appropriate single-quantum coherence. Experiments that select double-quantum coherence must generate it and subsequently convert it into an observable contribution. A phase-selection rule alone does not demonstrate that the necessary transfer occurred.^[ref]