Parametric resonant enhancement of motional entanglement under optimal control: an analytical study

We study theoretically continuous-variable entanglement between the motional degrees of freedom of optically trapped massive particles coupled via the Coulomb interaction, in the presence of a feedback control scheme. We perform a detailed analysis of the parametric resonance induced by temporal modulation of the coupling strength, based on the system's coupled nonlinear, nonhomogeneous dynamical equations. Our model accurately reproduces the numerical findings and provides closed-form expressions for the entanglement degree. We demonstrate that a stationary nonequilibrium entangled state is realized as a result of the competition between parametric gain and decoherence.