Many molecular, quantum-dot, and optomechanical nanocavity-QED systems demonstrate strong nonlinear interactions between electrons, photons, and phonon (vibrational) modes. We show that such systems can be described by a universal model in the vicinity of the nonlinear resonance involving all three degrees of freedom. We solve for the ultrafast nonperturbative quantum dynamics in the strong coupling regime, taking into account quantization, dissipation, and fluctuations of all fields. We find analytic solutions for quantum states which demonstrate tripartite quantum entanglement. We show how the strong coupling at the nonlinear resonance modifies photon emission and vibrational spectra.
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