Coherence order
= Coherence order
{title2=$p$}
<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 $U(\phi)=e^{-i\phi F_z}$, order $p$ means
$$
[F_z,A_p]=pA_p,\qquad U(\phi)A_pU(\phi)^\dagger=e^{-ip\phi}A_p.
$$
The order describes a phase transformation, rather than the amount of <magnetization> or the duration for which coherence persists.<Understanding NMR Spectroscopy by James Keeler (Irvine course notes)>[Chapter9, section9.3.1, pp.9-9–9-10: classification under z rotations]{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 $[S_z,S_\pm]=\pm S_\pm$,
$$
[F_z,I_+S_-]=0,\qquad [F_z,I_+S_+]=2I_+S_+.
$$
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 $\phi=\pi/2$, the order-zero term remains unchanged while the order-two term changes sign. At $\phi=\pi$, 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> $H/\hbar=\Omega_I I_z+\Omega_S S_z$, 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.<Understanding NMR Spectroscopy by James Keeler (Irvine course notes)>[section9.3.2, pp.9-10–9-11: evolution under separate offsets]{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.<Understanding NMR Spectroscopy by James Keeler (Irvine course notes)>[section9.3.4, p.9-13: generated versus desired pathways and detection]{ref}