  We examine conditions under which an open quantum system composed of a driven
degenerated parametric oscillator cavity and a driven two-level atom coupled to
a waveguide could decay to a pure dark state rather than the expected mixed
state. The calculations are carried out analytically in a low dimensional
Hilbert space truncated at the double-excitation states of the combined system.
The validity of the truncation is confirmed by the exact numerical analysis. It
is found that one way to produce the pure state is to chirally couple the
cavity and the atom to the waveguide. Another way to produce the pure state is
to drive the cavity and the atom with unequal detunings. In both cases, if the
driving fields are weak, the produced state is a coherent superposition of only
the single-excitation and ground states of the combined system. In addition, we
have found a direct correspondence between the generation of the dark state and
the photon blockade effect. In other words, the generation of the dark state
acts as a blockade to the number of photons so that only a single photon can be
present in the cavity. We investigate the normalized second-order correlation
function of the cavity field and find that the conditions under which the
correlation function vanishes coincide with the conditions for the creation of
the pure dark state. This system is, therefore, suggested as an alternative
scheme for the generation of single-photon states.
