# What is the difference between the atomic resonance frequency and the Rabi frequency?

I've been trying to work my way through the solution to the optical bloch equations for a two-level atom system which is being driven by a laser. One thing that has been confusing me is the difference between the atomic resonance frequency $\omega_{0}$ and the Rabi frequency $\Omega$. I understand that there is a generalised Rabi frequency for this system which is related to the detuning $\delta$ between the laser frequency $\omega$ and $\omega_{0}$ $$\Omega (\delta) = \sqrt[]{ \Omega_{0}^{2} + \delta^{2} }$$ where $\Omega_{0}$ is the rabi frequency with no detuning.

So ultimately my question is: is $\omega_{0}$ equal to $\Omega_{0}$ and if not what is $\Omega_{0}$?

First, a formula. The Rabi frequency (on resonance) is given by $$\Omega = \frac{g \mathcal{E}}{\hbar}$$ where $g$ is the strength of the dipole coupling/transition matrix element, $\mathcal{E}$ is the electric field amplitude of the laser, and $\hbar$ is, well, $\hbar$.
Second, an answer. The Rabi frequency and the atomic resonance frequency are the frequencies of two different processes. The resonance frequency is the frequency of light emitted or absorbed by the two-level system, or equivalently (more focus on "resonance") the frequency which a drive must be at to excite the system. Once the system is being driven it will begin to oscillate between ground and excited states, $|g\rangle$ and $| e \rangle$. The continual change between these two states is known as Rabi oscillation or Rabi flopping, and the frequency that it occurs at is called the Rabi frequency.
Intuitively, this should square with the definition. It should make some sense that if the coupling to the laser, $g$, or the electric field amplitude $\mathcal{E}$ is very high due to laser intensity, this oscillation will occur faster. Note that the Rabi frequency is important because when we say that we apply, for instance, a $\pi$ or $\pi / 2$ pulse to change the system from one state to another, we are effectively measuring time in inverse Rabi frequencies, because we want to go through exactly one half-cycle or one quarter-cycle of the oscillation. (If you haven't encountered such pulses yet, I would expect them to be a very soon-to-be explored subject in your textbook.)
• As I understand, a two level atom under the influence of an applied electric field is put into a state of superposition i.e. $c_1|1 \rangle + c_2 |2 \rangle$, then if we consider this process as a function of time then probabilities of absorption and stimulated emission fluctuate at the frequency. In other words $c_1(t)$ and $c_2(t)$ fluctuate with time at a frequency called the Rabi frequency. Is this correct? Thanks.