Understanding Vacuum Rabi Splitting in QED Cavities

[afrano]

I am studying a little about QED in a cavity. I don't really understand Vacuum Rabi splitting. The way I see it, it is the effect on a cavity's transmission spectrum of an atom (or quantum dot) inside the cavity. Well, this leads to two questions:
1) why is the cavity's transmission spectrum a "Lorenztian" bell, or something of the sort?

2)Why does the quantum dot have only one discrete absorption peak? It is made out of a solid (semicond.), so it must have a band structure, and thus a continuous spectrum for energies above the energy band width. So why is it discrete? Why is it only one sharp line?

Thanks.

 

[Cthugha]

Well, you will have to consider the QD and the cavity as a joint system in order to understand Rabi splitting.

1) A resonant cavity will form standing waves for some special wavelength of the light due to its geometry. Now the situation is quite similar to usual atomic transitions or such. A state with longer lifetime corresponds to a more precisely defined transition energy while a shorter lifetime corresponds to a broader spectral shape. Usually these shapes are Lorentzians. In cavities it is more or less the same. You have one central wavelength and the longer the average lifetime of a photon inside the cavity is (this corresponds to a higher Q-factor or higher reflectivity of the mirrors) the narrower the cavity mode energy will be. So this shape is a Lorentzian like in the atomic case.

2) Quantum dots are the semiconductor analogon to a 3D potential trap. So go back and solve the basic quantum mechanics exercise of a particle in a box leading to discrete energies and you will be close to what a QD actually is. However one needs to remember that the actual particles inside a QD are excitons - bound electron-hole pairs. In bulk materials excitons manifest as absorption already below the band gap. This happens because the bound state is energetically lower than free electrons and holes. In low dimensional systems this effect is even stronger due to the strong confinement and the increased Coulomb interaction. So QD emission will be from below the material band gap. You can of course also find a continuous spectrum due to emission the material itself or from an electron-hole plasma, but this will happen at higher energies.

Now for the effect of Rabi splitting you need to bring the cavity mode and a discrete QD state into resonance. The effect of Rabi splitting itself is quite analogous to an effect, which can already be seen in classical mechanics. If you have two coupled oscillators in resonance, you will find out that the coupled oscillator system will now show two normal modes with different eigenfrequencies - usually moving in phase and out of phase of each other. Now in the Rabi splitting case it is more or less the same. If the coupling between the QD mode and the cavity mode is strong enough (the dipole moment of the transition exceeds the cavity decay rate) you will have coherent absorption and reemission of photons by the excitons. So the energy oscillates back and forth between photons and excitons. And due to the strong coupling systems you will find two new eigenmodes - just like in the classical case - of the new system, which is usually described as a new quasiparticle, the polariton.

 

[Minich]

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