  When a Brownian particle in contact with a heat bath at a constant
temperature is controlled by a time-dependent harmonic potential, its
distribution function can be rigorously derived from the Kramers equation with
the consideration of the inertial effect of the Brownian particle. Based on
this rigorous solution and the concept of shortcuts to adiabaticity, we
construct a stochastic heat engine by employing the time-dependent harmonic
potential to manipulate the Brownian particle to complete a thermodynamic
cycle. We find that the efficiency at maximum power of this stochastic heat
engine is equal to $1-\sqrt{T_c/T_h}$ where $T_c$ and $T_h$ are the
temperatures of the cold bath and the hot one in the thermodynamic cycle,
respectively.
