= 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 $U(\phi)=e^{-i\phi F_z}$, an order-$p$ component transforms as $UA_pU^\dagger=e^{-ip\phi}A_p$. The term describes a phase response of the spin system.<Understanding NMR Spectroscopy by James Keeler (Irvine course notes)>[Chapter9, section9.3.1, pp.9-9–9-10: coherence orders and multispin products]{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:
$$
|\psi\rangle=\frac{|\uparrow\uparrow\rangle+|\downarrow\downarrow\rangle}{\sqrt2}.
$$
Its <density matrix> has populations one half in each displayed basis state, plus the off-diagonal terms $|\uparrow\uparrow\rangle\langle\downarrow\downarrow|/2$ 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> $H/\hbar=\Omega_I I_z+\Omega_S S_z$, 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.<Understanding NMR Spectroscopy by James Keeler (Irvine course notes)>[section9.3.2, pp.9-10–9-11: offset evolution of product operators]{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.<Understanding NMR Spectroscopy by James Keeler (Irvine course notes)>[section9.3.4, p.9-13: pathway generation and observable orders]{ref}
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