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The first exact quantum simulation of a real molecular system (HD$^+$) understrong ro-vibrational coupling to a quantized optical cavity mode in thermalequilibrium is presented. Theoretical challenges in describing strongly coupledsystems of mixed quantum statistics (Bosons and Fermions) are discussed andcircumvented by the specific choice of our molecular system. Our exactsimulations reveal the absence of a zero temperature for the strongly coupledmatter and light subsystems, due to cavity induced non-equilibrium conditions.Furthermore, we explore the temperature dependency of light-matter quantumentanglement, which emerges for the groundstate, but is quickly lost already inthe deep cryogenic regime, opposing predictions from phenomenological models(Jaynes-Cummings). Distillable molecular light-matter entanglement ofro-vibrational states may open interesting perspectives for quantumtechnological applications. Moreover, we find that the dynamics (fluctuations)of matter remains modified by the quantum nature of the thermal and vacuumfield fluctuations for significant temperatures, e.g. at ambient conditions.These observations (loss of entanglement and coupling to quantum fluctuations)has far reaching consequences for the understanding and control of polaritonicchemistry and materials science, since a semi-classical theoretical descriptionof light-matter interaction becomes feasible, but the typical canonicalequilibrium assumption for the nuclear dynamics remains broken. This opens thedoor for quantum fluctuations induced stochastic resonance phenomena undervibrational strong coupling. A plausible theoretical mechanism to explain theexperimentally observed resonance phenomena in absence of periodic driving,which have not yet been understood.

Exact Solution for A Real Polaritonic System Under Vibrational Strong  Coupling in Thermodynamic Equilibrium: Absence of Zero Temperature and Loss  of Light-Matter Entanglement