  Mobile quantum impurities interacting with a fermionic bath form
quasiparticles known as Fermi polarons. We demonstrate that a force applied to
the bath particles can generate a drag force of similar magnitude acting on the
impurities, realizing a novel, nonperturbative Coulomb drag effect. To prove
this, we calculate the fully self-consistent, frequency-dependent
transconductivity at zero temperature in the Baym-Kadanoff conserving
approximation. We apply our theory to excitons and exciton polaritons
interacting with a bath of charge carriers in a doped semiconductor embedded in
a microcavity. In external electric and magnetic fields, the drag effect
enables electrical control of excitons and may pave the way for the
implementation of gauge fields for excitons and polaritons. Moreover, a
reciprocal effect may facilitate optical manipulation of electron transport.
Our findings establish transport measurements as a novel, powerful tool for
probing the many-body physics of mobile quantum impurities.
