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We derive a "classical-quantum" approximation scheme for a broad class ofbipartite quantum systems from fully quantum dynamics. In this approximation,one subsystem evolves via classical equations of motion with quantumcorrections, and the other subsystem evolves quantum mechanically withequations of motion informed by the evolving classical degrees of freedom.Using perturbation theory, we derive an estimate for the growth rate ofentanglement of the subsystems and deduce a "scrambling time" - the timerequired for the subsystems to become significantly entangled from an initialproduct state. We argue that a necessary condition for the validity of theclassical-quantum approximation is consistency of initial data with thegeneralized Bohr correspondence principle. We illustrate the general formalismby numerically studying the fully quantum, fully classical, andclassical-quantum dynamics of a system of two oscillators with nonlinearcoupling. This system exhibits parametric resonance, and we show that quantumeffects quench parametric resonance at late times. Lastly, we present a curiouslate-time scaling relation between the average value of the von Neumannentanglement of the interacting oscillator system and its total energy: $S\sim2/3 \ln E$.

Motivating semiclassical gravity: a classical-quantum approximation for bipartite quantum systems