= Zero-quantum coherence
<Zero-quantum coherence> is a component of <NMR coherence> with total <coherence order> $p=0$, invariant under a common $z$ rotation. In a multispin system it can connect distinct basis states having the same total spin projection. This makes it different from merely specifying diagonal populations.<Understanding NMR Spectroscopy by James Keeler (Irvine course notes)>[Chapter9, section9.3.1, pp.9-9–9-10: zero-quantum terms and coherence order]{ref}
For two <spin one-half> nuclei, consider $A=I_+S_-$. With $F_z=I_z+S_z$,
$$
[F_z,A]=(1-1)A=0.
$$
It transfers one projection upward and the other downward, keeping their total fixed. The operator is proportional to $|\uparrow\downarrow\rangle\langle\downarrow\uparrow|$, an off-diagonal matrix element in the product basis.
An original state comparison uses
$$
|\psi\rangle=\frac{|\uparrow\downarrow\rangle+|\downarrow\uparrow\rangle}{\sqrt2}.
$$
The pure-state <density matrix> has a zero-quantum off-diagonal pair. An equal mixture of the two basis states has the same populations and no such pair. Both have total projection zero, but their coherent properties differ.
Invariance under a common rotation does not imply that the component never evolves. For $H/\hbar=\Omega_I I_z+\Omega_S S_z$, its evolution is
$$
A(t)=e^{-i(\Omega_I-\Omega_S)t}A(0).
$$
For invented offsets $\Omega_I/(2\pi)=100\,\mathrm{Hz}$ and $\Omega_S/(2\pi)=70\,\mathrm{Hz}$, its coefficient completes a phase cycle in $1/30\,\mathrm{s}$. Adding the same offset to both nuclei leaves that difference unchanged. Unequal perturbations can still change it; the example is not a universal protection against noise.<Understanding NMR Spectroscopy by James Keeler (Irvine course notes)>[section9.3.2, pp.9-10–9-11: separate offsets and summed operator phases]{ref}
The zero order helps distinguish contributions in <NMR phase cycling> and gradient-based coherence selection. Longitudinal <magnetization> also has order zero, so an order label alone does not identify a unique physical component or guarantee direct detection.<Understanding NMR Spectroscopy by James Keeler (Irvine course notes)>[sections9.3.1 and9.3.4: order-zero classification and observable pathways]{ref}
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