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.^[ref]

For two spin one-half nuclei, consider $A=I_{+}{S}_{-}$. With $F_z=I_z+S_z$,It transfers one projection upward and the other downward, keeping their total fixed. The operator is proportional to $|\uparrow \downarrow ⟩⟨\downarrow \uparrow |$, an off-diagonal matrix element in the product basis.

An original state comparison usesThe 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 $H/\hslash ={\Omega }_I{I}_z+{\Omega }_S{S}_z$, its evolution isFor 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.^[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.^[ref]