  We investigate a valleytronic device based on graphene with charge separation
at different sublattices and correspondingly at nonequivalent valleys. We
characterize the maximality condition of valley polarization and investigate
the parameters and conditions upon which we can coherently control the carriers
and store data via valley degree of freedom. The valley polarization is
controlled by the amplitude as well as the carrier envelope phase of the pulse
and the curvature of the electron trajectory in the reciprocal space. When
strong field excitation is taken into account, the optical selection rule in
perturbative optics is replaced by the nonadiabatic geometric effects. As a
result, a nonperturbative valley polarization in two dimensional Dirac
materials is induced regardless of having an intrinsic bandgap.
Microscopically, such a nonreciprocal response of graphene in the chiral
electric field is encoded by the quantum Berry phase, as a (pseudo)
magnetoelectric monopole.
