Many (all?) materials are polarizable at some level; meaning that when we apply an electric field across the material, there is an induced dipole moment $$ \vec{p}=\alpha(\omega)\vec{E}, $$ where $\alpha(\omega)$ is the polarizability of the material at frequency $\omega$, $\vec{E}$ is the applied electric field, and $\vec{p}$ is the induced polarization. If we assume that the electrons in the material are bound to their nuclei as simple harmonic oscillators (the standard physics assumption), then we find that the polarizability of the material is given by $$ \alpha(\omega)=\frac{e^2/m}{\omega_0^2-\omega^2}, $$ where $\omega_0$ contains the information about the stiffness of the binding. This leads to interesting optical phenomena in materials, one example of which is that it can be related to the index of refraction by $$ n^2(\omega)=1+\frac{N}{\epsilon_0}\alpha(\omega), $$ where $N$ is the number density of the atoms and $\epsilon_0$ is the permittivity of free space. All of this is just background information.
My actual questions are: What are the experimental limits on the polarizibility of the vacuum, and which experiments set these limits? Is anyone actively trying to improve the limit? It is obviously expected to be identically zero, but we have been surprised by measurements in the past.