Nonequilibrium nature of nonlinear optical response: Application to the bulk photovoltaic effect

The bulk photovoltaic effect is an example of a nonlinear optical response that leads to a DC current that is relevant for photovoltaic applications. In this paper, we theoretically study this effect in the presence of electron-phonon interactions. Using the response function formalism, we find that the nonlinear 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 noninteracting result) for nonlinear 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 nonlinear DC response. Most interestingly, the semiclassical treatment shows that the nonlinear DC response for spatially inhomogeneous excitation profiles is strongly nonlocal and must involve the aforementioned out-of-time-ordered correlators that cannot be computed by equilibrium field theory.