Transitionless quantum driving

For a general quantum system driven by a slowly time-dependent Hamiltonian, transitions between instantaneous eigenstates are exponentially weak. But a nearby Hamiltonian exists for which the transition amplitudes between any eigenstates of the original Hamiltonian are exactly zero for all values of slowness. The general theory is illustrated by spins driven by changing magnetic fields, and implies that any spin expectation history, including those where the spin never precesses, can be generated by infinitely many driving fields, here displayed explicitly. Asymptotically, the absence of transitions is explained by continuation to complex time, where the complex degeneracies in the transitionless driving fields have a nongeneric structure for which there is no Stokes phenomenon; this is analogous to the explanation of reflectionless potentials.