  We investigate Landau-Zener processes modeled by a two-level quantum system,
with its finite bias energy varied in time and in the presence of a single
broadened cavity mode at zero temperature. By applying the hierarchy equation
method to the Landau-Zener problem, we computationally study the survival
fidelity of adiabatic states without Born, Markov, rotating-wave or other
perturbative approximations. With this treatment it also becomes possible to
investigate cases with very strong system-bath coupling. Different from a
previous study of infinite-time Landau-Zener processes, the fidelity of the
time-evolving state as compared with instantaneous adiabatic states shows
non-monotonic dependence on the system-bath coupling and on the sweep rate of
the bias. We then consider the effect of applying a counter-diabatic driving
field, which is found to be useful in improving the fidelity only for
sufficiently short Landau-Zener processes. Numerically exact results show that
different counter-diabatic driving fields can have much different robustness
against environment effects. Lastly, using a case study we discuss the
possibility of introducing a dynamical decoupling field in order to eliminate
the decoherence effect of the environment and at the same time to retain the
positive role of a counter-diabatic field. Our work indicates that finite-time
Landau-Zener processes with counter-diabatic driving offer a fruitful test bed
to understand controlled adiabatic processes in open systems.
