Averaging method and coherence applied to Rabi oscillations in a two-level system

We study Rabi oscillations in a two-level system within the semiclassical approximation as an archetype test field of the Averaging Method (AM). The population transfer between the two levels is approached within the first and the second order AM. We systematically compare AM predictions with the rotating wave approximation (RWA) and with the complete numerical solution utilizing standard algorithms (NRWA). We study both the resonance ($\Delta = 0$) and out-of-resonance ($\Delta \ne 0$) cases, where $\Delta = \omega-\Omega$, and $\hbar \Omega = E_2-E_1$ is the two-level energetic separation, while $\omega$ is the (cyclic) frequency of the electromagnetic field. We introduce three types of dimensionless factors $\epsilon$, i.e., $\Omega_{\textrm{R}}/\Delta$, $\Omega_{\textrm{R}}/\Sigma$, and $\Omega_{\textrm{R}}/\omega$, where $\Omega_{\textrm{R}}$ is the Rabi (cyclic) frequency and $\Sigma = \omega + \Omega$ and explore the range of $\epsilon$ where the AM results are equivalent to NRWA. Finally, by allowing for a phase difference in the initial electron wave functions, we explore the prospects coherence can offer. We illustrate that even with equal initial probabilities at the two levels, but with phase difference, strong oscillations can be generated and manipulated.