One of Maxwell's other equations links the E-field to charge density. You can't talk about the time derivative of the E-field without also talking about the time derivative of charge density.
It's true that interactions between charged particles invoke photons, but you can have charged particles (and their associated E-fields) travelling at constant velocity in a vaccum and still define an associated current density without considering photons. I'm not sure your interpretation produces a useful intuition for this situation.
One of the underlying ideas of the paper is that, in conventional metals where conduction band electrons are much more wave than particle, you can still measure when they 'enter' and 'exit' the material through perturbations in the field, that is bursts of photons that occur when electron waves interact with an obstacle.
It's true that interactions between charged particles invoke photons, but you can have charged particles (and their associated E-fields) travelling at constant velocity in a vaccum and still define an associated current density without considering photons. I'm not sure your interpretation produces a useful intuition for this situation.
One of the underlying ideas of the paper is that, in conventional metals where conduction band electrons are much more wave than particle, you can still measure when they 'enter' and 'exit' the material through perturbations in the field, that is bursts of photons that occur when electron waves interact with an obstacle.