A Keldysh formalism approach to superradiance of two-level system

We use a non-equilibrium Keldysh formalism to develop a method of describing superradiance. A time evolution operator on the Schwinger-Keldysh contour is written in terms of the field and atomic operators. The quantized field’s degrees of freedom are formally eliminated to obtain an effective two-atom master equation. Using non-equilibrium Green’s function method and Dyson equation formalism, we derive a self-consistent expression for superradiant decay rates. We consider a homogeneous gas of initially inverted two-level atoms, and numerical results are obtained by solving the closed form of its equation of motion. The effect of optical depth on super- and sub-radiance is also discussed.