Quantitative sufficient conditions for adiabatic approximation

Two linear In this letter, we prove the following conclusions by introducing a function Fn(t): (1) If a quantum system S with a time-dependent non-degenerate Hamiltonian H(t) is initially in the n-th eigenstate of H(0), then the state of the system at time t will remain in the n-th eigenstate of H(t) up to a multiplicative phase factor if and only if the values Fn(t) for all t are always on the circle centered at 1 with radius 1; (2) If a quantum system S with a time-dependent Hamiltonian H(t) is initially in the n-th eigenstate of H(0), then the state of the system at time t will remain ɛ-uniformly approximately in the n-th eigenstate of H(t) up to a multiplicative phase factor if and only if the values Fn(t) for all t are always outside of the circle centered at 1 with radius 1−ɛ. Moreover, some quantitative sufficient conditions for the state of the system at time t to remain ɛ-uniformly approximately in the n-th eigenstate of H(t) up to a multiplicative phase factor are established. Lastly, our results are illustrated by a spin-half particle in a rotating magnetic field.