What is the role of "vacuum" in vacuum Rabi splitting (VRS)? Also, does VRS arise because of quantized nature of light? If not, what is a semiclassical explanation for VRS?
Vacuum Rabi splitting is about an initially excited atom in a cavity which spontaneously emits a photon into the cavity. The photon's wave-function propogates to the end mirror(s) is reflected and then comes back and interacts back on the atom. Theoretically there are two parts of this process that requires (at least in principle) a quantized electric field.
An atom in an excited eigenstate is a stationary state (i.e. doesn't change in time). You only get spontaneous emission if the atom is coupled to a quantized EM field. Sometimes this is described with the picture that 'vacuum fluctuations cause the spontaneous decay.' A more complete description is that the energy states of the atom alone are not eigenstates of a quantum atom+field, and thus an excited atom won't stay that way.
If you describe the atom quantum mechanically (i.e. correctly), then a single excited atom will only emit a single quanta of energy or single photon, which is a non-classical state and thus requires (at least superficially) a quantum field description.
Now if you ignore these two essentially quantum aspects of the EM field, the rest can be described semi-classically. Specifically the propagation of the single photon wave-packet that is emitted by the atom can be fully described by classical EM theory (Maxwell's equations).
Taken from Nature Physics 2, 81 - 90 (2006):
If the coupling strength between two same-energy oscillators exceeds the mean of their decay rates, then the coupled system has two eigenenergies — that is, their states split. VRS occurs when one of the oscillators consists of a two-level atom or QD, and the other is that of a small-volume high-quality (high-Q) cavity. The name comes about because the VRS is numerically equal to twice the product of the transition dipole moment and the vacuum field arising from the root-mean-square of the vacuum fluctuations (zero-point energy of half a photon energy in the cavity mode). The term VRS is also used when the matter involves more than a single oscillator; for example, there is VRS due to many atoms or to a quantum well (QW). A simple way to understand VRS is to insert the QD or QW absorption and refractive index into the formula for the transmission of a Fabry–Perot interferometer; the transmission becomes double-peaked if the absorption is strong and narrow enough.