  Berry phases have long been known to significantly alter the properties of
periodic systems, giving rise to anomalous terms in the semiclassical equations
of motion describing wave-packet dynamics. In non-Hermitian systems,
generalizations of the Berry connection have been proposed and shown to have
novel effects on dynamics and transport. In this work, we expand upon these
results by deriving the full set of semiclassical equations of motion for
wave-packet dynamics in a non-Hermitian system subject to complex external
electric fields, which are realizable as gain gradients. We show that the
non-Hermiticities of both the band Hamiltonian and the external potential give
rise to anomalous weight rate and velocity terms which depend on the geometric
properties of the eigenfunctions, including the quantum metric tensor. These
analytical results are compared with numerical lattice simulations which reveal
these anomalous terms even in one-dimension. Our work expands the range of
phenomena expected to be detectable in experimental setups, which should be
realizable in currently available metamaterials and classical wave systems,
including mechanical, acoustic, and optical.
