  We consider a system comprising two groups of quantum dimers placed in a
common electromagnetic cavity, and controlled by selectively applying a static
external potential to one of the groups. We show that in the regime of deep
strong coupling to vacuum electromagnetic fluctuations, the emergent
photon-assisted interaction between the dimers leads to a strongly non-linear
quantized cross-polarization response of the first, unbiased group of dimers to
the potential applied to the second group. The total polarization shows a
series of almost ideal steps whose number and position depends on the parity of
the numbers of dimers in the groups. This non-perturbative effect is a
distinctive feature of mesoscopic systems comprising finite number of dimers
and disappears in the thermodynamic limit which is commonly used in the
desciption of the generalized Dicke models.
