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Is there an intuitive explanation of why the Rabi oscillations with frequency $\Omega$ occur in a two-level systemand why they get faster when the transition dipole moment $M$ ($M\propto V$) gets larger or/and when the frequency of the light $\omega$ is not near the transition frequency $\omega_0$? $$\Omega=\sqrt{V^2+(\omega_0-\omega)^2}$$

Buzz
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Very unlikely. Rabi's oscillations are a specific result based on Schroedinger's time-dependent equation, which is anything but intuitive.