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We present a comparative study of the control of group velocity in three-level atomic systems including Λ-, Ladder-, and V-types via spontaneously generated coherence (SGC) and relative phase of laser fields. In the steady regime, expressions of group index are derived as an analytical function of controllable parameters related to SGC and relative phase. It is shown that the group velocity can be further slowed down or speeded up under SGC in which modifications of group velocity due to the SGC in Λ- and V-systems are stronger than that in Ladder-system. In particular, when interference strength of SGC p increases from 0.7 to 1.0, the light propagation is switched from superluminal to subluminal modes (for Λ- and Ladder-systems) or from subluminal to superluminal modes (for V-system). In the presence of relative phase, the group velocity changes between subluminal and superluminal modes with a period of 2π. In resonant region, the largest positive value of group index reaches at the relative phase φ = π while the largest negative value of group index reaches at φ = 0 and 2π for the Λ- and Ladder-systems, otherwise, for the V-system the largest positive group index occurs at φ = 0 and 2π, and the largest negative group index occurs at φ = π.

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By solving the density matrix equations in the steady state, we have derived analytical expressions for the absorption, dispersion and group index of a five-level cascade-type atomic system as functions of laser intensity and frequency, spontaneously generated coherence (SGC) and relative phase of applied fields. The influences of SGC and relative phase on absorption, dispersion and group index of this system under electromagnetically induced transparency (EIT) are studied. It is shown that the three EIT windows of system become deeper and narrower as the SGC increases. These lead to an increase in the slope and the amplitude of the dispersion curves at three EIT windows. As a result, the amplitude of the group index at these three EIT windows also becomes larger when the SGC increases. In particular, the group index can easily be switched between negative and positive values i.e., the light propagation can easily be converted between superluminal to subluminal modes by adjusting the strength of the SGC, relative phase or the coupling laser intensity.

Using density-matrix theory, an analytical expression of the self-Kerr nonlinear coefficient of a three-level lambda EIT medium for a weak probe light is derived. Influences of the coupling light and Doppler broadening on the self-Kerr coefficient are investigated and compared to experimental observation with a good agreement. The self-Kerr nonlinearity is basically modified and greatly enhanced in the spectral region corresponding to EIT transparent window. Furthermore, sign, slope, and magnitude of the self-Kerr coefficient can be controlled with frequency and intensity of the coupling light and temperature. Such controllable Kerr nonlinearity can find interesting applications in optoelectronic devices working with low-light intensity.