# Interpretation of Rabi frequency and generalised Rabi frequency

In quantum mechanics, what is the difference, in a two level system, of the processes described by the Rabi frequency defined as $$\Omega := \frac{\langle 1| e \vec{r} \cdot \vec{E_0}| 2 \rangle}{\hbar}$$ and the generalized Rabi frequency given by $$\omega_r := \frac{1}{2}\sqrt{(\omega - \omega_0)^2 + \bigg(\frac{|V_{ab}|}{\hbar}\bigg)^2}?$$

The presence of $\omega$ in the second expression should give the game away: what you call the 'generalized' Rabi frequency is the frequency of the oscillations in population when you drive a two-level system off-resonance, i.e. at some frequency $\omega$ which does not quite match the resonance frequency $\omega_0$. At resonance, when $\omega=\omega_0,$ it is easy to see that this reduces to $$\omega_r = \frac{1}{2} \frac{|V_{ab}|}{\hbar},$$ which is exactly the same as your initial expression for $\Omega$ when you account for the different conventions and normalizations in use.
• Okay. As an example, given electric field $\vec{E} =E_0 \cos(\omega t)\hat{k}$, would I be correct in stating that the Rabi frequency is $$\Omega = \frac{\langle1| e \vec{r} \cdot \vec{E_0}|2 \rangle}{\hbar} = \frac{e \langle1|z|2 \rangle|E_0|}{\hbar},$$ where $\vec{E_0} = |E_0|\hat{k}?$ – user101311 Apr 5 '17 at 10:03
• No. That result only holds at resonance, i.e. if $\hbar\omega$ exactly matches the difference $\hbar\omega_0$ between the ground and excited-state energies. – Emilio Pisanty Apr 5 '17 at 10:06