Optical properties of a homogeneously broadened two-level system

For a homogeneously broadened two-level system described by the optical Bloch equations, we show that the photon-absorption cross section σ(ω,‖A${\Vert }^2$) is not equal to the particle-excitation cross section σ’(ω,‖A${\Vert }^2$), where ω is the optical frequency and ‖A${\Vert }^2$ is proportional to the intensity. We detail how conservation of energy accounts for this inequality. Analytic expressions for these cross sections for weak intensities and strong intensities (to obtain the lowest-order Bloch-Siegert shifts for σ and σ’) are derived. We define an energy-loss cross section ${\sigma }_{\mathrm{loss}(\mathrm{\omega }}$,‖A${\Vert }^2$) for the bath responsible for the homogeneous broadening, and show that it is proportional to the change of the index of refraction due to the optical transition, Δn(ω,‖A${\Vert }^2$).