Transient properties in a resonant ladder-type atomic system with vacuum-induced coherence and incoherent pumping

The transient absorption and population in a resonant ladder-type atomic system with or without an incoherent pumping field are theoretically investigated. We find that the vacuum-induced coherence and the relative phase between probe and coupling fields can clearly affect the transient absorption (gain) properties, but the steady-state value of the probe absorption (gain) is remarkably dependent on the incoherent pumping. And the steady population distribution is sensitive to the incoherent pumping and insensitive to the relative phase. Additionally, it is shown that the lasing with inversion can be reached with the incoherent pumping.