Further perspective on the theory of heteronuclear decoupling

An exact general theory of heteronuclear decoupling is presented for spin-1/2 IS systems. RF irradiation applied to the I spins decouples the S spins by reversing the sign of antiphase magnetization. A one-to-one correspondence is demonstrated between the sign of $2S_yI_z$ and the sense of the S-spin coupling evolution, with magnetization $S_x$ refocused when $\int 2S_yI_z\,dt = 0$. The exact instantaneous coupling at any time during the decoupling sequence is readily obtained in terms of the system states, showing that the creation of two-spin coherence is crucial for reducing the coupling, as required during the refocusing process. Representative examples from standard cyclic, periodic composite-pulse and adiabatic decoupling sequences illustrate the decoupling mechanism. More general aperiodic sequences, obtained using optimal control, realize the potential inherent in the theory for significantly improved decoupling. The recently derived equivalence between the dynamics of any N-level quantum system and a system of classical coupled harmonic oscillators provides additional insight into the mechanics of decoupling.