The effective Hamiltonian of the Pound-Overhauser controlled-NOT gate

In NMR-based quantum computing, it is known that the controlled-NOT gate can be implemented by applying a low-power, monochromatic radio-frequency field to one peak of a doublet in a weakly-coupled two-spin system. This is known in NMR spectroscopy as Pound-Overhauser double resonance. The ``transition'' Hamiltonian that has been associated with this procedure is however only an approximation, which ignores off-resonance effects and does not correctly predict the associated phase factors. In this paper, the exact effective Hamiltonian for evolution of the spins' state in a rotating frame is derived, both under irradiation of a single peak (on-transition) as well as between the peaks of the doublet (on-resonance). The accuracy of these effective Hamiltonians is validated by comparing the observable product operator components of the density matrix obtained by simulation to those obtained by fitting the corresponding experiments. It is further shown how both the on-transition and on-resonance fields can be used to implement the controlled-NOT gate exactly up to conditional phases, and analytic expressions for these phases are derived. In Appendices, the on-resonance Hamiltonian is analytically diagonalized, and proofs are given that, in the weak-coupling approximation, off-resonance effects can be neglected whenever the radio-frequency field power is small compared to the difference in resonance frequencies of the two spins.