  The state of an open quantum system undergoing an adiabatic process evolves
by following the instantaneous stationary state of its time-dependent
generator. This observation allows one to characterize, for a generic adiabatic
evolution, the average dynamics of the open system. However, information about
fluctuations of dynamical observables, such as the number of photons emitted or
the time-integrated stochastic entropy production in single experimental runs,
requires controlling the whole spectrum of the generator and not only the
stationary state. Here, we show how such information can be obtained in
adiabatic open quantum dynamics by exploiting tools from large deviation
theory. We prove an adiabatic theorem for deformed generators, which allows us
to encode, in a biased quantum state, the full counting statistics of generic
time-integrated dynamical observables. We further compute the probability
associated with an arbitrary "rare" time-history of the observable and derive a
dynamics which realizes it in its typical behavior. Our results provide a way
to characterize and engineer adiabatic open quantum dynamics and to control
their fluctuations.
