# What is Rabi splitting? Is it related to Autler-Townes splitting?

A quantum optics text I am reading claims that the proper way to analyze a two-level atom's interaction with light is to, conceptually speaking, consider four states: $$|g, n\rangle, |e, n\rangle, |g, n+1\rangle$$, and $$|e, n-1\rangle$$, where the first letter is for the atom being in either the ground or the excited state, and the second is the number of photons in the mode needed to couple the two states of the atom. The $$|g, n+1\rangle$$ comes from a stimulated emission event, and the $$|e, n-1\rangle$$ comes from an absorption event. The text claims that, when you do this, you find that the total system has four new eigenfrequencies; it "looks" like each of the two levels of the atom has "split" in two, an effect called Rabi splitting.

I have a few questions regarding this explanation:

1. Is this a reasonable conceptual explanation of Rabi splitting?
2. Is Rabi splitting a "real" effect in the sense that, if one coupled two states in an atom (say $$|1\rangle$$ and $$|2\rangle$$) and then tried to probe one of the states by doing spectroscopy from another state (i.e. coupling $$|2\rangle$$ to some state $$|3\rangle$$), would you in fact be able to see two distinct peaks? (Ignoring all other splittings that would obviously be relevant in any real experiment.)
3. Is Rabi splitting related to Autler-Townes splitting, and if not, what is the difference?
• This stops short of addressing my question. The text I referenced above suggests that, by splitting your equation (6)'s sines and cosines into exponentials, and inserting back in the phases we got rid of in going to the rotating frame (i.e. your equation (2) inverted), you get an expression for the full wavefunction (your equation (1)) that schematically looks like four different exponential phase evolutions, which suggests four different eigenenergies, as if the two levels of the atom each "split." I am wondering more about the nature of that splitting. Aug 7 '19 at 14:23

For a two-level system with states $$|g\rangle$$ and $$|e\rangle$$, and an electromagnetic mode with Fock states $$|n\rangle$$, it does make sense to consider the four relevant levels you described. If the electromagnetic mode is resonant with the two-level system energy gap, then $$|g, n\rangle$$ is degenerate with $$|e, n-1\rangle$$, and $$|e, n\rangle$$ is degenerate with $$|g, n+1\rangle$$.

If the atom is coupled to the mode, then $$|g, n\rangle$$ is coupled to $$|e, n-1\rangle$$ through a $$(\sigma^+ \hat{a} + \sigma^- \hat{a}^\dagger)$$ type interaction. Since they are degenerate, then they diagonalize into a symmetric and anti-symmetric superposition of these states, with eigenvalues given by $$\pm E$$ relative to the original energy, where $$E$$ is the strength of the coupling. Similarly, the higher energy degenerate states $$|e, n\rangle$$ and $$|g, n+1\rangle$$ are also coupled and therefore also diagonalize into two states with eigenvalues $$\pm E'$$ relative to their original energies. (Note that in general $$E' \approx E$$ if we are considering large numbers of photons in the field, $$n \gg 1$$.)

2. Yes, if you couple $$|1\rangle$$ to $$|2\rangle$$, and then probe the transition from $$|2\rangle$$ to another state $$|3\rangle$$, then you will see two peaks at $$\pm E$$ compared to the bare transition frequency. There's another cool effect that can also emerge: you might notice that because there are two split excited states and two split lower-energy states, there are four possible transitions between these states. Two of the transitions have the original bare frequency; one transition has a lower frequency, and the last transition has a higher frequency. This can be observed and is called a 'Mollow triplet'. This seems to be a good reference for understanding Mollow triplets, and in general is well suited to your question.