Photon Transport in a Gas of Two-Level Atoms: Unveiling Quantum Light Creation

We present a theoretical analysis of nearly monochromatic light propagation through a gas of two-level atoms using the Heisenberg-Langevin equation method. Our focus is on the evolution of the photon annihilation operator and its impact on the second-order correlation function, $g^{(2)}(\tau )$, with particular emphasis on photon antibunching behavior. The model accounts for both open and closed atomic system approximations, including Doppler broadening and the influence of pump field detuning. We derive expressions that reproduce known results from scattering theory and extend the analysis to complex systems, such as inhomogeneously broadened media. The theoretical predictions are compared with experimental data from a waveguide QED platform, which show good agreement and thereby demonstrate the power of our approach. Future work will explore extensions to even more complex systems and other quantum light characteristics for practical applications.