There are two distinct frequency regimes for nonlinear optics in semiconductors which correspond to real and virtual excitation. Real excitations usually result in a reduction of the refractive index at frequencies of interest. In contrast, by exciting optical solids at frequencies much less than the gap, a considerably smaller, but faster, positive nonlinear refractive index n2 due to bound electronic effects are observed.
Why do real excitations result in a reduction of the refractive index while virtual excitations result in an increase? What is the fundamental mechanism behind it?
I think you are conflating two different things: the state of the material before the light arrives, and the states involved in how the light interacts with the material.
One way to think of light propagating in a material is a series of absorptions and re-emissions. The light interacting with an electron causes the electron to transition to a higher energy state. However, depending on the material and the photon energy, that energy state may not be an allowed state of the material. This is the virtual state you are talking about. It isn’t a state of the material before the light arrives. It is a state created by the interaction with the light itself. Because it isn‘t allowed, this absorption of the light, this electronic transition cannot actually occur, BUT it can sort of occur briefly. In quantum mechanics energy and time are conjugate variables that are governed by the Heisenberg uncertainty principle. For a brief enough time the energy of the transition is not well defined and it is a quantum mechanical possibility until the Heisenberg uncertainty resolves the energy to a degree that the transition is clearly not allowed. Here is the trick: the closer in energy the disallowed virtual transition is to a real allowed transition the longer it takes for the uncertainty of the energy to become precise enough to prevent the transition. This is why the index of refraction increases as the photon energy approaches an absorption edge.
Now, actual excitations are a different matter. The idea here is that they are something that has already happened before the light arrives. (or if the light is causing the transitions, let’s say “the next bit of light” arrives) the light encounters a different material. Suppose in the ground state A the virtual transition for a particular wavelength of light produces an energy that is very close to the energy of a real allowed state B. Encountering a material where essentially all of the electrons are in the ground state, the light experiences a high index of refraction. But now suppose many of the electrons are excited into the higher state B. Now the virtual transition is not from the lower state A. The virtual absorption is now a photon energy above B. Is there some other state C above B? Maybe, maybe not. Is that virtual absorption high index? It depends on the state structure above B, but one thing you can say for sure, it’s not the same as when the photon encountered electrons in the ground state.
there is a direct relationship between the refractive index and the energy gap of the semiconductor
you can check the following: https://www.sciencedirect.com/science/article/abs/pii/0020089181900336
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