High-speed driving of a two-level system

A simple result is found for the optimal protocol of drivings for a general two-level Hamiltonian, which transports a given initial state to a given final state in minimal time, under additional conditions on the drivings. If one of the three possible drivings is unconstrained in strength the problem is analytically completely solvable. A surprise arises for a class of states when one driving is bounded by a constant c and the other drivings are constant. Then, for large c, the optimal driving is of type bang-off-bang and for increasing c one recovers the unconstrained result. However, for smaller c the optimal driving can suddenly switch to bang-bang type. It is also shown that for general states one may have a multistep protocol. The present paper explicitly proves and considerably extends the author's results contained in Phys. Rev. Lett. 111, 260501 (2013).