= NMR coherence
{c}
<NMR coherence> describes phase-bearing components of a nuclear-spin <density matrix>. Its description depends on a specified basis and rotation convention. Transverse <magnetization> is an observable example, while multispin coherences can require additional pulses before they contribute to the detected <nuclear magnetic resonance> <signal>.<Understanding NMR Spectroscopy by James Keeler (Irvine course notes)>[Chapter9, sections9.3.1–9.3.4, pp.9-9–9-13: coherence orders and observable pathways]{ref}
For an original <spin one-half> example, use dimensionless $I_z=\operatorname{diag}(1/2,-1/2)$ and $I_x=\sigma_x/2$. A valid <density matrix> is
$$
\rho=\frac{\mathbf1}{2}+\varepsilon I_x
=\frac12\begin{pmatrix}1&\varepsilon\\\varepsilon&1\end{pmatrix},\qquad |\varepsilon|\leq1.
$$
Its populations in this basis are both one half, but its off-diagonal entries are $\varepsilon/2$. The <eigenvalues> $(1+\varepsilon)/2$ and $(1-\varepsilon)/2$ are nonnegative, and the <trace> is one. Comparing $\varepsilon=0$ with $\varepsilon=1$ shows that identical populations do not require identical transverse coherence. This prepared-state example is not the thermal state of an actual bulk sample.
With rotation convention $U=\exp(-i\phi I_z)$, the upper-right entry of $U\rho U^\dagger$ becomes $(\varepsilon/2)e^{-i\phi}$, while the populations remain unchanged. More generally, a component $A_p$ of <coherence order> $p$ obeys
$$
[I_z,A_p]=pA_p,\qquad UA_pU^\dagger=e^{-ip\phi}A_p.
$$
Single-quantum orders are $p=\pm1$; a Hermitian transverse operator contains both signs. Longitudinal <magnetization> has order zero, but in a multispin system order zero also includes terms beyond populations.<Understanding NMR Spectroscopy by James Keeler (Irvine course notes)>[section9.3.1, printed pp.9-9–9-10: classification under z rotations]{ref}
This phase response makes <NMR phase cycling> useful: contributions can be selected by how they transform under controlled phase changes. Selection alone does not establish that an experiment generated the desired component. The spin system and pulse sequence must also support it.<Understanding NMR Spectroscopy by James Keeler (Irvine course notes)>[section9.3.4, p.9-13: desired versus actually generated coherence pathways]{ref}
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