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 $\frac{T_1^{-1\mathrm{}},T_2^{-1\mathrm{}}\ll t_p^{-1\mathrm{}}\ll \Omega \mathrm{}\left(t\right)\mathrm{}}{2\pi }$ are satisfied throughout the pulse. Here, $\Omega \mathrm{}\left(t\right)={\left[{\Delta }^2\mathrm{}\right(t)+\frac{p^2{\mathcal{E}}^2\mathrm{}\left(t\right)\mathrm{}}{{\hslash }^2}]}^{\frac{1}{2}}$ is the atomic precession frequency, $\Delta \mathrm{}\left(t\right)\mathrm{}$ the detuning, $\mathcal{E}\mathrm{}\left(t\right)\mathrm{}$ the field envelope, $t_p$ the effective pulse width, and $T_1$ and $T_2$ 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 $T_1$ and $T_2$ can be comparable to or less than $t_p$. It allows the condition $T_1^{-1\mathrm{}}, T_2^{-1\mathrm{}}\ll t_p^{-1\mathrm{}}$ to be replaced by the weaker requirement $T_1^{-1\mathrm{}}$,$\frac{T_2^{-1\mathrm{}}{p}^2{\mathcal{E}}^2\mathrm{}\left(t\right)\mathrm{}}{{\hslash }^2{\Omega }^2\mathrm{}\left(t\right)\mathrm{}\ll \Omega \mathrm{}\left(t\right)\mathrm{}}$. If $T_1\gg t_p$, the atomic Bloch vector remains aligned nearly parallel to the effective driving field $\mathcal{E}\mathrm{}\left(t\right)\mathrm{}$, 0, $\frac{\hslash \Delta \mathrm{}\left(t\right)\mathrm{}}{p}$) in the rotating reference frame (as in the pure adiabatic following case); however, it decays in length if $T_2$ is comparable to $t_p$. The approximation bridges the gap between pure adiabatic following behavior for