Parametric amplification of a quantum pulse

Creating and manipulating quantum states of light requires nonlinear interactions. We present here a multimode theory for the transformation of an arbitrary quantum pulse by Hamiltonians that are quadratic in field creation and annihilation operators. We show, in particular, that any input quantum pulse will feed only one or two distinct output modes. Our result readily provides both the output modes and the quantum states after transformation of arbitrary input wave packets. While being central for applications in quantum information processing with traveling bosonic modes, e.g., in quantum networks, our theoretical method and its results are not anticipated by the conventional Bogoliubov diagonalization of the problem in many input and output eigenmodes.