Quantum brachistochrone problem for a spin-1 system in a magnetic field

We study the quantum brachistochrone problem for a spin-1 system in a magnetic field of constant absolute value. Such a system gives us the possibility to examine in detail the statement that the state vectors realizing evolution with the minimal time of passage evolve along the subspace spanned by the initial and final state vectors [Carlini , Phys. Rev. Lett. 96, 060503 (2006); Brody and Hook, J. Phys. A 39, L167 (2006)]. Considering an explicit example, we show the existence of a quantum brachistochrone with the minimal possible time; however, the state vector we study leaves the subspace spanned by the initial and final state vectors during evolution. This is the result of our choice of a more constrained Hamiltonian than the one assumed in the general quantum brachistochrone problem. It is worth noting that such an evolution, being more complicated, is time optimal but with larger time than in the general case. This might be important for experiments, where a general Hamiltonian with all the allowed parameters is difficult to implement, but a constrained one, depending on the magnetic field, can be realized. However, for the preconstrained Hamiltonian not all final states are accessible. The present result does not contradict the general statement of the quantum brachistochrone problem, but gives additional insight into possible realization of the time-optimal passage.