an undepleted pump. However, the approach taken is quite general, such that a different parametric process or the process under different assumptions is easily derived. First the classical theory of the coupled mode equations as presented by O’Brien et al. [Phys. Rev. Lett., 113:157001] is shortly reviewed. Then a derivation of the full quantum theory is given, resulting in a Hamiltonian that describes the process of parametric amplification. We show that the coupled mode equations can be derived from this Hamiltonian in the classical limit and elaborate on the validity of the theory.