  This paper presents a algebraic analysis of electromagnetic chirality-induced
negative refractive index in a four-level atomic medium. According to analyze
mathematically its argument of the complex refractive index for one circular
polarization, it found that the negative refractive index without
simultaneously negative permittivity and permeability can be obtained when the
argument is in the second quadrant of the cartesian coordinate system, and that
the probe field coupling to two equal transition frequencies in the atomic
level doesn't require. This undoubtedly reduced stringent conditions to
negative refractive index by quantum optics. As an application, our scheme may
possibly give a novel approach to obtain negative refractive index by
electromagnetic chirality-inducing.
