Quantum dynamics in the self-consistent quadratic approximation

A self-consistent quadratic theory is presented to account for nonlinear contributions in quantum dynamics. Evolution equations are shown to depend on higher-order gradients of the Hamiltonian, which are incorporated via their equations of motion or through perturbative calculations. The dynamics is proven trace-preserving, with the Hamiltonian acting as a constant of motion for initial Gaussian states. Nonlinear response functions are calculated perturbatively, and sufficient conditions are provided for the existence of their classical limit.