  We study the possibility of achieving the Carnot efficiency in a finite-power
underdamped Brownian Carnot cycle. Recently, it was reported that the Carnot
efficiency is achievable in a general class of finite-power Carnot cycles in
the vanishing limit of the relaxation times. Thus, it may be interesting to
clarify how the efficiency and power depend on the relaxation times by using a
specific model. By evaluating the heat-leakage effect intrinsic in the
underdamped dynamics with the instantaneous adiabatic processes, we demonstrate
that the compatibility of the Carnot efficiency and finite power is achieved in
the vanishing limit of the relaxation times in the small temperature-difference
regime. Furthermore, we show that this result is consistent with a trade-off
relation between power and efficiency by explicitly deriving the relation of
our cycle in terms of the relaxation times.
