Spontaneous decay controlled inversionless laser in a V-type four level system

By using a semiclassical approach, we obtain 16 coupled optical Bloch equations involving density matrix elements for a V-type four level system with three closely spaced upper levels irradiated by a single mode of the electromagnetic field. The off-diagonal elements of the density matrix ρ_ij are in general complex and arise due to the interference between the probability amplitudes of the levels i and j. Through the complex off-diagonal density matrix elements, we introduce the phase angles between the levels participating in dipole allowed/forbidden transitions. Therefore, the phase angles may be regarded as the outcome of the quantum coherence. Under rotating wave approximation, we use a perturbative approach to solve these Bloch equations analytically. These solutions for density matrix elements are used to obtain a closed form analytical expression for the imaginary part of the polarization and hence the absorptive lineshape. The proper choice of the decay constants induces the necessary coherence responsible for inversionless laser.