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 nonlinear 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 nonperturbative effect is a distinctive feature of mesoscopic systems comprising a finite number of dimers and disappears in the thermodynamic limit which is commonly used in the description of the generalized Dicke models.

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• Cavity quantum electrodynamics
• Quantum optics with artificial atoms
• Dicke model