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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 singlebroadened cavity mode at zero temperature. By applying the hierarchy equationmethod to the Landau-Zener problem, we computationally study the survivalfidelity of adiabatic states without Born, Markov, rotating-wave or otherperturbative approximations. With this treatment it also becomes possible toinvestigate cases with very strong system-bath coupling. Different from aprevious study of infinite-time Landau-Zener processes, the fidelity of thetime-evolving state as compared with instantaneous adiabatic states showsnon-monotonic dependence on the system-bath coupling and on the sweep rate ofthe bias. We then consider the effect of applying a counter-diabatic drivingfield, which is found to be useful in improving the fidelity only forsufficiently short Landau-Zener processes. Numerically exact results show thatdifferent counter-diabatic driving fields can have much different robustnessagainst environment effects. Lastly, using a case study we discuss thepossibility of introducing a dynamical decoupling field in order to eliminatethe decoherence effect of the environment and at the same time to retain thepositive role of a counter-diabatic field. Our work indicates that finite-timeLandau-Zener processes with counter-diabatic driving offer a fruitful test bedto understand controlled adiabatic processes in open systems.

Finite-time Landau-Zener processes and counter-diabatic driving in open systems: beyond Born, Markov and Rotating-wave approximations