 NMR coherence

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.^[ref]

For an original spin one-half example, use dimensionless $I_z=\mathrm{diag}(1/2,-1/2)$ and $I_x={\sigma }_x/2$. A valid density matrix isIts populations in this basis are both one half, but its off-diagonal entries are $\epsilon /2$. The eigenvalues $(1+\epsilon )/2$ and $(1-\epsilon )/2$ are nonnegative, and the trace is one. Comparing $\epsilon =0$ with $\epsilon =1$ 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 $U=\exp (-i\varphi I_z)$, the upper-right entry of $U\rho U^{†}$ becomes $(\epsilon /2)e^{-i\varphi }$, while the populations remain unchanged. More generally, a component $A_p$ of coherence order $p$ obeysSingle-quantum orders are $p=\pm 1$; a Hermitian transverse operator contains both signs. Longitudinal magnetization has order zero, but in a multispin system order zero also includes terms beyond populations.^[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.^[ref]