Semiclassical calculations on one-, two-, three-, and four-photon absorption in truncated models of the hydrogen atom

The results of semiclassical calculations on one-, two-, three-, and four-photon absorption in truncated models of the hydrogen atom are presented. The temporal behavior near "resonance" is confirmed to be Rabi-like, with a transition probability given by $P\left(t\right)=\left\{\frac{{\gamma }^2}{[{(\frac{{\omega }_0}{N}+\delta -\omega )}^2+{\gamma }^2]}\right\}{sin}^2\left\{\right(\frac{\mathrm{Nt}}{2}\left){[{(\frac{{\omega }_0}{N}+\delta -\omega )}^2+{\gamma }^2]}^{\frac{1}{2}}\right\},$ where the resonance shift $\delta$ is proportional to the intensity of the monochromatic driving field with frequency $\omega$ and the power broadening $\gamma$ is proportional to the $\frac{N}{2}$ power of intensity. $N$ is the number of photons involved and $\mathrm{h}{\omega }_0$ is the energy separation between initial and final states. Calculated values of $\delta$ and $\gamma$ are given for several models; also, for the two-photon $1s\to 2s$ absorption, $\delta$ and $\gamma$ are given which include the contributions of all bound $p$ states.