  We extend the quantum jump method to nearly adiabatically driven open quantum
systems in a way that allows for an accurate account of the external driving in
the system-environment interaction. Using this framework, we construct the
corresponding trajectory-dependent work performed on the system and derive the
integral fluctuation theorem and the Jarzynski equality for nearly adiabatic
driving. We show that such identities hold as long as the stochastic dynamics
and work variable are consistently defined. We numerically study the emerging
work statistics for a two-level quantum system and find that the conventional
diabatic approximation is unable to capture some prominent features arising
from driving such as the continuity of the probability density of work. Our
results reveal the necessity of using accurate expressions for the
drive-dressed heat exchange in future experiments probing jump time
distributions.
