  We derive a "classical-quantum" approximation scheme for a broad class of
bipartite quantum systems from fully quantum dynamics. In this approximation,
one subsystem evolves via classical equations of motion with quantum
corrections, and the other subsystem evolves quantum mechanically with
equations of motion informed by the evolving classical degrees of freedom.
Using perturbation theory, we derive an estimate for the growth rate of
entanglement of the subsystems and deduce a "scrambling time" - the time
required for the subsystems to become significantly entangled from an initial
product state. We argue that a necessary condition for the validity of the
classical-quantum approximation is consistency of initial data with the
generalized Bohr correspondence principle. We illustrate the general formalism
by numerically studying the fully quantum, fully classical, and
classical-quantum dynamics of a system of two oscillators with nonlinear
coupling. This system exhibits parametric resonance, and we show that quantum
effects quench parametric resonance at late times. Lastly, we present a curious
late-time scaling relation between the average value of the von Neumann
entanglement of the interacting oscillator system and its total energy: $S\sim
2/3 \ln E$.
