Generalized adiabatic following approximation

The response of a two-level atom to a smoothly varying near-resonant driving pulse can be described by the usual adiabatic-following approximation if the conditions T₁^-1,T₂^-1tₚ^-1Ω(t)2π are satisfied throughout the pulse. Here, Ω(t)=[Δ²(t)+p²E²(t)²]1/2 is the atomic precession frequency, Δ(t) the detuning, E(t) the field envelope, tₚ the effective pulse width, and T₁ and T₂ the atomic level and phase relaxation times, respectively. From Bloch's equations, we have developed a generalized version of this approximation applicable to cases where T₁ and T₂ can be comparable to or less than tₚ. It allows the condition T₁^-1, T₂^-1tₚ^-1 to be replaced by the weaker requirement T₁^-1,T₂^-1p²E²(t)²Ω²(t)Ω(t). If T₁tₚ, the atomic Bloch vector remains aligned nearly parallel to the effective driving field E(t), 0, Δ(t)p) in the rotating reference frame (as in the pure adiabatic following case); however, it decays in length if T₂ is comparable to tₚ. The approximation bridges the gap between pure adiabatic following behavior for tₚ^-1T₁^-1, T₂^-1 and rate-equation behavior for tₚ^-1T₂^-1≈Ω(t), thereby allowing a more complete description of phenomena such as adiabatic inversion and resonantly enhanced self-focusing and defocusing.