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We develop the analytic theory describing the formation and evolution ofentangled quantum states for a fermionic quantum emitter coupled to a quantizedelectromagnetic field in a nanocavity and quantized phonon or mechanicalvibrational modes. The theory is applicable to a broad range of cavity quantumoptomechanics problems and emerging research on plasmonic nanocavities coupledto single molecules and other quantum emitters. The optimal conditions for atri-state entanglement are realized near the parametric resonances in a coupledsystem. The model includes decoherence effects due to coupling of the fermion,photon, and phonon subsystems to their dissipative reservoirs within thestochastic evolution approach, which is derived from the Heisenberg-Langevinformalism. Our theory provides analytic expressions for the time evolution ofthe quantum state and observables, and the emission spectra. The limit of aclassical acoustic pumping and the interplay between parametric and standardone-photon resonances are analyzed.

Generation and dynamics of entangled fermion-photon-phonon states in nanocavities