  We study the semi-classical theory of wave packet dynamics in crystalline
solids extended to include the effects of a non-uniform electric field. In
particular, we derive a correction to the semi-classical equations of motion
(EOMs) for the dynamics of the wave packet center that depends on the gradient
of the electric field and on the quantum metric (also called the Fubini-Study,
Bures, or Bloch metric) on the Brillouin zone. We show that the physical origin
of this term is a contribution to the total energy of the wave packet that
depends on its electric quadrupole moment and on the electric field gradient.
We also derive an equation relating the electric quadrupole moment of a sharply
peaked wave packet to the quantum metric evaluated at the wave packet center in
reciprocal space. Finally, we explore the physical consequences of this
correction to the semi-classical EOMs. We show that in a metal with broken
time-reversal and inversion symmetry, an electric field gradient can generate a
longitudinal current which is linear in the electric field gradient, and which
depends on the quantum metric at the Fermi surface. We then give two examples
of concrete lattice models in which this effect occurs. Our results show that
non-uniform electric fields can be used to probe the quantum geometry of the
electronic bands in metals and open the door to further studies of the effects
of non-uniform electric fields in solids.
