Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 61 / 1 / f / Solution

Write $\cos {\omega }_{1}t=(e^{i{\omega }_{1}t}+e^{-i{\omega }_{1}t})/2$ and $\Delta \omega ={\omega }_2-{\omega }_1$. Multiplication of the matrix from part (e) givesThese are exactly the constant, difference-frequency, and sum-frequency terms in the PDF, with each Hermitian pair visible. For a weak, slowly varying envelope, the terms at $2{\omega }_1$ and ${\omega }_1+{\omega }_2$ average out by the rotating-wave approximation. To discard the other transition as well, its detuning must be nonzero and large compared with the drive coupling and envelope bandwidth. Sufficient scale conditions arewhere $B$ is a characteristic envelope bandwidth; turn-on transients must satisfy the same spectral restrictions. ThenSmall coupling compared with the optical frequencies alone is not enough for this selective approximation: if $|\Delta \omega |$ is comparable with the pulse bandwidth or Rabi scale, the spectator transition remains driven. A long selective gate also accumulates a small off-resonant Stark phase of order $(A_1{d}_2{)}^2/\Delta \omega$, so that error must be acceptable or compensated. These assumptions distinguish resonant quantum control from an exact decoupling.