  We investigate the simple resonances of a 2 by 2 matrix of n-dimensional
semiclassical Shr\"odinger operators that interact through a first order
differential operator. We assume that one of the two (analytic) potentials
admits a well with non empty interior, while the other one is non trapping and
creates a barrier between the well and infinity. Under a condition on the
resonant state inside the well, we find an optimal lower bound on the width of
the resonance. The method of proof relies on Carleman estimates, microlocal
propagation of the microsupport, and a refined study of a non involutive double
characteristic problem in the framework of Sj\"ostrand's analytic microlocal
theory.
