 Double-quantum coherence

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]