Adiabatic following in two-photon transition

The coherent interaction of two smoothly varying, near-resonant, two-photon pulses with a three-level system can be described by "two-photon damped Bloch equations" which are analogous to those for a one-photon transition in a two-level system except for the presence of a two-photon coupling and a frequency shift. These equations are solved for the cases γ₁, γ₂Ω, γ₁=γ₂, and γ₂k²ε⁴Ω², γ₁Ω, where γ₁ and γ₂ are the atomic energy and phase relaxation widths, respectively, and Ω is the Rabi frequency. The leading contribution to the refractive index is intensity dependent, caused by the level shifts inherent in multiphoton processes; it includes a relaxation dependent part which is important at times shorter than γ₁^-1. The second-order contributions depend on the square of the intensity and the time-integrated square of the intensity. The latter contribution, which is relaxation dependent, causes line asymmetry at the long-wavelength wing; it consists of a term proportional to γ₂-γ₁ and only important at early times and a term proportional to 2γ₂-γ₁.