Semiclassical wave packet dynamics in nonuniform electric fields

We study the semiclassical theory of wave packet dynamics in crystalline solids extended to include the effects of a nonuniform electric field. In particular, we derive a correction to the semiclassical 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 semiclassical 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 nonuniform 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 nonuniform electric fields in solids.