  We consider the flow of polarization current J(t)=dP/dt produced by a
homogeneous electric field E(t) or by rapidly varying some other parameter in
the Hamiltonian of a solid. For an initially insulating system and a
collisionless time evolution, the dynamic polarization P(t) is given by a
nonadiabatic version of the King-Smith--Vanderbilt geometric-phase formula.
This leads to a computationally convenient form for the Schroedinger equation
where the electric field is described by a linear scalar potential handled on a
discrete mesh in reciprocal space. Stationary solutions in sufficiently weak
static fields are local minima of the energy functional of Nunes and Gonze.
Such solutions only exist below a critical field that depends inversely on the
density of k points. For higher fields they become long-lived resonances, which
can be accessed dynamically by gradually increasing E. As an illustration the
dielectric function in the presence of a dc bias field is computed for a
tight-binding model from the polarization response to a step-function
discontinuity in E(t), displaying the Franz-Keldysh effect.
