Adiabatic quantum control

Design a slowly varying Quantum Hamiltonian path whose relevant initial eigenstate connects to a desired final eigenstate. Under the gap and slowness hypotheses of the quantum adiabatic theorem, the state follows that instantaneous eigenstate up to phase. The path, spectral gap and duration all matter; a level crossing or excessive speed can spoil the intended state transfer.

For a resonant three-level chain with couplings ${\Omega }_1,{\Omega }_2$, the zero-energy dark state of a driven Hamiltonian is $|D⟩=({\Omega }_2|1⟩-{\Omega }_1|3⟩)/\sqrt{{\Omega }_1^2+{\Omega }_2^2}$. Turning on the target-to-intermediate coupling first and sweeping ${\Omega }_1/{\Omega }_2$ from zero to infinity rotates this state from $|1⟩$ to $-|3⟩$. The other eigenvalues are $\pm \sqrt{{\Omega }_1^2+{\Omega }_2^2}$, so $\hslash |\dot{\theta }|\ll \sqrt{{\Omega }_1^2+{\Omega }_2^2}$ controls nonadiabatic leakage. Ideally the intermediate state remains unoccupied, suppressing spontaneous-emission loss. The gap must not close while the dark state is rotating.