We map the quantum problem of a free bosonic field in a space-time dependent background into a classical problem. N degrees of freedom of a real field in the quantum theory are mapped into 2*N^2 classical simple harmonic oscillators with specific initial conditions. We discuss how this classical-quantum correspondence (CQC) may be used to evaluate quantum radiation and also to analyze the backreaction of quantum fields on classical backgrounds. We also analyze the agreement between results obtained with the CQC and with a full quantum analysis. Applications include backreaction of quantum radiation during gravitational collapse as well as quantum evaporation of classical breathers.