Dark state of a driven Hamiltonian ($H_1|D⟩=0,H_0|D⟩=E_D|D⟩$)

Such a quantum state remains an eigenstate of $H_0+f(t)H_1$ for every control input. Its projector commutes with all available generators, so its population is conserved. This obstructs full-space pure-state controllability even when the orthogonal bright subspace can be controlled. An instantaneous dark state of a slowly varying Quantum Hamiltonian instead supports adiabatic quantum control if a spectral gap and slowness condition are maintained.