The solution for the equations of atomic density matrix is obtained, without resorting to any approximation, in the Ξ-type of three-level system with lower-level coherence. The conditions on pumping are analyzed in the strong-coherence case for lasing without inversion and the saturation term is derived.
The LWI operation condition for an open Λ-type three level atomic system is obtained by solving the steady-state density-matrix equation with the system being driven by a strong coherent field while the inherent pumping rate and decay rate being arbitrary.It is shown that the inversions of inherent pumping rate and spontaneous decay rate replace the inversion of population,and together help to realize lasing without population inversion.Furthermore,the LWI can be realized if just either one of the two inversions is fulfilled.A schrodinger equation of three-level atomic system with appropriate conditions is analytically solved to explain the quantum interference process that leads to LWI.