  The optical conductivity encodes the current response to a time dependent
electric field. We develop a theory of the optical conductivity
$\sigma(\omega)$ in presence of a dc supercurrent. Current induced optical
response is prohibited by Galilean invariance from occurring in systems with a
single parabolic band. However, we show that lattice effects give rise to a
pronounced current dependent peak in $\sigma(\omega)$ at the gap edge $\omega =
2 \Delta$, which diverges in the clean limit. We demonstrate this in a model of
a multi-band superconductor. Our theory explains the recent observation of a
current induced peak in the optical conductivity in NbN by Nakamura et al.
(2019), and provides a new mechanism for direct activation of the Higgs mode
with light.
