Analytic estimation of transition between instantaneous eigenstates of quantum two-level system

Transition amplitudes between instantaneous eigenstates of quantum two-level system are evaluated analytically on the basis of a new parametrization of its evolution operator, which has recently been proposed to construct exact solutions. In particular, they are estimated when the Hamiltonian varies infinitesimally slowly. The results, not only confirm the adiabatic theorem in the adiabatic limit, but also bring us with an analytic estimation of the adiabatic approximation. The condition under which no transition between different instantaneous eigenstates is allowed is also clarified.