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]

Coherence order classifies an operator component of NMR coherence by its response to rotation about the chosen total $z$ angular momentum. With dimensionless $F_z$ and rotation convention $U(\varphi )=e^{-i\varphi F_z}$, order $p$ meansThe order describes a phase transformation, rather than the amount of magnetization or the duration for which coherence persists.^[ref]

An original two-spin example shows why order zero is not synonymous with a diagonal population. Let $F_z=I_z+S_z$ for two spin one-half nuclei. For raising and lowering components with $[I_z,I_{+}]=I_{+}$ and $[S_z,S_{\pm }]=\pm S_{\pm }$,The first is zero-quantum coherence; the second is a double-quantum coherence component. In a product basis, the first connects the states with projections $(+1/2,-1/2)$ and $(-1/2,+1/2)$: different states with the same total projection. The second connects total projections−1 and+1. These operator components are not individually normalized density matrices.

At $\varphi =\pi /2$, the order-zero term remains unchanged while the order-two term changes sign. At $\varphi =\pi$, both remain unchanged. A phase comparison must therefore be chosen to discriminate the orders of interest; one phase change cannot distinguish every possible component.

Order zero also does not require zero time evolution. Under the Quantum Hamiltonian $H/\hslash ={\Omega }_I{I}_z+{\Omega }_S{S}_z$, the first component acquires $e^{-i({\Omega }_I-{\Omega }_S)t}$ when the two offsets differ. Total order and the separate offsets answer different questions.^[ref]

NMR phase cycling uses such phase responses to select contributions to an experiment. Selecting an order does not ensure that the spin system or pulse sequence actually generated it, nor that it is directly observable before further conversion.^[ref]

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 $U(\varphi )=e^{-i\varphi F_z}$, an order-$p$ component acquires $e^{-ip\varphi }$.^[ref]

For one spin one-half nucleus, $I_{+}$ and $I_{-}$ are the positive- and negative-order components. Transverse magnetization is represented by their combinations,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 usesIts eigenvalues are0.8 and0.2 for every $\varphi$, and its populations remain one half. Taking the traces against $I_x,I_y$ givesAt $\varphi =\pi /2$, 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.^[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.

Double-quantum coherence is a component of NMR coherence with coherence order $p=+2$ or $p=-2$. Under a common $z$ rotation $U(\varphi )=e^{-i\varphi F_z}$, an order-$p$ component transforms as $U{A}_p{U}^{†}=e^{-ip\varphi }{A}_p$. The term describes a phase response of the spin system.^[ref]

For two spin one-half nuclei with $F_z=I_z+S_z$, the component $I_{+}{S}_{+}$ has order+2: it connects the state with total projection−1 to that with total projection+1. Its adjoint $I_{-}{S}_{-}$ has order−2. A Hermitian density matrix can contain both components with conjugate coefficients.

An original constructed state makes the difference from populations explicit:Its density matrix has populations one half in each displayed basis state, plus the off-diagonal terms $|\uparrow \uparrow ⟩⟨\downarrow \downarrow |/2$ and its adjoint. A mixture of the two basis states with the same populations lacks those terms. This is an assumed mathematical preparation, rather than a claim about what a single pulse produces in a thermal sample.

Under the Quantum Hamiltonian $H/\hslash ={\Omega }_I{I}_z+{\Omega }_S{S}_z$, the positive-order off-diagonal component acquires phase $e^{-i({\Omega }_I+{\Omega }_S)t}$. A common phase rotation of $\pi /2$ changes its sign; a rotation of $\pi$ leaves it unchanged. The summed offsets and doubled phase response are consequences of the two projection changes.^[ref]

Conventional transverse detection reads appropriate single-quantum coherence. Experiments that select double-quantum coherence must generate it and subsequently convert it into an observable contribution. A phase-selection rule alone does not demonstrate that the necessary transfer occurred.^[ref]

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]