  To optimize the performance of a heat engine in finite-time cycle, it is
important to understand the finite-time effect of thermodynamic processes.
Previously, we have shown that extra work is needed to complete a quantum
adiabatic process in finite time, and proved that the extra work follows a
\mathcal{C}/\tau^{2} scaling for long control time \tau. There the oscillating
part of the extra work is neglected due to the complex energy-level structure
of the particular quantum system. However, such oscillation of the extra work
can not be neglected in some quantum systems with simple energy-level
structure, e. g. the two-level system or the quantum harmonic oscillator. In
this paper, we build the finite-time quantum Otto engine on these simple
systems, and find that the oscillating extra work leads to a jagged edge in the
constraint relation between the output power and the efficiency. By optimizing
the control time of the quantum adiabatic processes, the oscillation in the
extra work is utilized to enhance the maximum power and the efficiency. We
further design special control schemes with the zero extra work at the specific
control time. Compared to the linear control scheme, these special control
schemes of the finite-time adiabatic process improve the maximum power and the
efficiency of the finite-time Otto engine.
