  The understanding of out-of-equilibrium fluctuation relations in small open
quantum systems has been a focal point of research in recent years. In
particular, for systems with adiabatic time-dependent driving, it was shown
that the fluctuation relations known from stationary systems do no longer apply
due the geometric nature of the pumping current response. However, the precise
physical interpretation of the corrected pumping fluctuation relations as well
as the role of many-body interactions remained unexplored. Here, we study
quantum systems with many-body interactions subject to slow time-dependent
driving, and show that fluctuation relations of the charge current can in
general not be formulated without taking into account the total energy current
put into the system through the pumping process. Moreover, we show that this
correction due to the input energy is nonzero only when Coulomb-interactions
are present. Thus, fluctuation response relations offer an until now unrevealed
opportunity to probe many-body correlations in quantum systems. We demonstrate
our general findings at the concrete example of a single-level quantum dot
model, and propose a scheme to measure the interaction-induced discrepancies
from the stationary case.
