= Single-quantum coherence
<Single-quantum coherence> is a component of <NMR coherence> with <coherence order> $p=+1$ or $p=-1$. It connects basis states whose total spin projections differ by one unit. Under the convention $U(\phi)=e^{-i\phi F_z}$, an order-$p$ component acquires $e^{-ip\phi}$.<Understanding NMR Spectroscopy by James Keeler (Irvine course notes)>[Chapter9, section9.3.1, pp.9-9–9-10: single-quantum orders and z rotations]{ref}
For one <spin one-half> nucleus, $I_+$ and $I_-$ are the positive- and negative-order components. Transverse <magnetization> is represented by their combinations,
$$
I_x=\frac{I_++I_-}{2},\qquad I_y=\frac{I_+-I_-}{2i}.
$$
A Hermitian <density matrix> contains conjugate off-diagonal entries, so it can encode two real transverse components while containing both coherence-order signs.
An original state example uses
$$
\rho(\phi)=\begin{pmatrix}1/2&0.3e^{-i\phi}\\0.3e^{i\phi}&1/2\end{pmatrix}.
$$
Its eigenvalues are0.8 and0.2 for every $\phi$, and its populations remain one half. Taking the traces against $I_x,I_y$ gives
$$
\langle I_x\rangle=0.3\cos\phi,\qquad\langle I_y\rangle=0.3\sin\phi.
$$
At $\phi=\pi/2$, the transverse expectation has moved from $+x$ to $+y$ without changing the populations. This is a constructed quantum-state comparison, not a measured thermal polarization.
Conventional transverse <NMR> detection can access the appropriate single-quantum contribution. The complex receiver signal chooses a phase convention for combining its two measured components. This is why a statement about a detected phase sign should specify its convention, while the operator transformation above remains explicit.<Understanding NMR Spectroscopy by James Keeler (Irvine course notes)>[sections9.2.2–9.2.3 and9.3.4: quadrature detection, receiver phase and observable order]{ref}
Multispin single-quantum terms need not be merely isolated-spin magnetization: correlations can accompany the projection change. A <coherence order> identifies the transformation under a chosen common rotation, rather than uniquely specifying every spin correlation or the pulse sequence that generated it.
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