Resonant quantum control

A near-resonant drive selects a transition and, in an appropriate rotating frame and rotating-wave approximation, controls a two-level effective Quantum Hamiltonian. If $H_{\mathrm{eff}}=\Omega (t){\sigma }_x/2$ with fixed axis, then $U(T)=\exp [-i{\sigma }_x{\int }_0^T\Omega (t)dt/2]$. Pulse area fixes the rotation angle; drive phase chooses a transverse axis. Selectivity and neglect of rapidly oscillating terms require the appropriate detuning and drive-strength regime.

For a fixed dimensionless Hermitian operator $A$ driven by $H(t)=\Omega (t)A$, all instantaneous generators commute. The exact unitary operator is $U(T)=\exp (-i\theta A)$, so the area, rather than the detailed envelope, fixes the rotation. For $A^2=I$, it is $\cos \theta I-i\sin \theta A$. This convention uses the Quantum Hamiltonian coefficient directly; a convention with $\hslash {\Omega }_{R}A/2$ instead has $\theta =\frac{1}{2}\int {\Omega }_{R}dt$.

For a Hermitian operator satisfying $A^3=A$, its squared operator $P=A^2$ is the projector onto the active eigenvalue-$\pm 1$ subspace. All remaining states have eigenvalue zero. Separate even and odd powers in the matrix exponential to obtain the displayed rotation on the active subspace and the identity on its spectator subspace. Thus an embedded two-level coupling need not square to the full Hilbert-space identity.