# Non-equilibrium nature of non-linear optical response: Application to the bulk photo voltaic effect

## TL;DR

This paper investigates the non-linear optical bulk photovoltaic effect, emphasizing its non-equilibrium nature and the role of electron-phonon interactions, revealing non-local responses and the limitations of equilibrium theories.

## Contribution

It introduces a semiclassical approach and quantum master equation analysis to understand the non-linear response, highlighting the importance of out-of-time-ordered correlators.

## Key findings

- Non-linear optical response involves out-of-time-ordered correlators.
- Scattering rate significantly influences non-linear DC response.
- Non-local effects are crucial for spatially inhomogeneous excitation profiles.

## Abstract

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.