  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.
