  The bulk photovoltaic effect is an example of a non-linear optical response
that leads to a DC current that is relevant for photo-voltaic applications. In
this work, we theoretically study this effect in the presence of
electron-phonon interactions. Using the response function formalism we find
that the non-linear optical response, in general, contains three operator
correlation functions, one of which is not ordered in time. This latter
correlator cannot be computed from equilibrium field theory. Using a
semiclassical approach instead, we show that the bulk photovoltaic effect can
be attributed to the dipole moment of the generated excitons. We then confirm
the validity of the semiclassical result (which agrees with the non-interacting
result) for non-linear DC response from a quantum master equation approach.
From this formalism we find that, in contrast to usual linear response, the
scattering rate has a strong implicit effect on the non-linear DC response.
Most interestingly, the semiclassical treatment shows that the non-linear DC
response for spatially inhomogeneous excitation profiles is strongly non-local
and must involve the aforementioned out-of-time-ordered correlators that cannot
be computed by equilibrium field theory.
