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We consider a system comprising two groups of quantum dimers placed in acommon electromagnetic cavity, and controlled by selectively applying a staticexternal potential to one of the groups. We show that in the regime of deepstrong coupling to vacuum electromagnetic fluctuations, the emergentphoton-assisted interaction between the dimers leads to a strongly non-linearquantized cross-polarization response of the first, unbiased group of dimers tothe potential applied to the second group. The total polarization shows aseries of almost ideal steps whose number and position depends on the parity ofthe numbers of dimers in the groups. This non-perturbative effect is adistinctive feature of mesoscopic systems comprising finite number of dimersand disappears in the thermodynamic limit which is commonly used in thedesciption of the generalized Dicke models.

Non-perturbative effects of deep-strong light-matter interaction in a mesoscopic cavity-QED system