  In this work we study the effects of nonadiabatic external driving on the
thermodynamics of an electronic system coupled to two electronic leads and to a
phonon mode, with and without damping. In the limit of slow driving, we
establish nonadiabatic corrections to quantum thermodynamic quantities. In
particular, we study the first-order correction to the electronic population,
charge-current, and vibrational excitation using a perturbative expansion, and
compare the results to the numerically exact hierarchical equations of motion
(HEOM) approach. Furthermore, the HEOM analysis spans both the weak and strong
system-bath coupling regime and the slow and fast driving limits. We show that
the electronic friction and the nonadiabatic corrections to the charge-current
provide a clear indicator for the Franck-Condon effect and for non-resonant
tunneling processes. We also discuss the validity of the approximate quantum
master equation approach and the benefits of using HEOM to study quantum
thermodynamics out of equilibrium.
