  We numerically investigate an on-demand single-photon source, which is
implemented with a strongly coupled atom-cavity system, proposed by Kuhn {\it
et al}., Appl. Phys. B \textbf{69}, 373 (1999). In the scheme of Kuhn {\it et
al}., a $\Lambda$-type three-level atom is captured in a single-mode optical
cavity. Considering the three atomic levels, the ground state $u$, the first
excited state $g$ accompanying the cavity mode, and the second excited state
$e$, in the $\Lambda$-configuration, we assume that a classical field and a
quantized cavity field lead to the transition between $u$ and $e$ and that
between $e$ and $g$, respectively. The classical light pulse rising
sufficiently slowly triggers an adiabatic process of the system and lets a
single photon of the cavity mode emerge. We simulate this adiabatic evolution
and transmission of the single photon through an imperfect mirror of the cavity
using the master equation. We concentrate on examining physical properties of
the efficiency of single-photon generation, the fluctuation of the duration of
the photon emission, and the time of the emission measured from a peak of the
trigger pulse. We find a function that approximates to the efficiency closely
and the upper bound of the fluctuation of the duration.
