  Creating and manipulating quantum states of light requires nonlinear
interactions, but while nonlinear optics is inherently multi-mode, quantum
optical analyses are often done with single-mode approximations. We present a
multi-mode theory for the transformation of a quantum pulse by Hamiltonians
that are quadratic in the field creation and annihilation operators. Our theory
describes nonlinear processes, such as parametric amplification and squeezing,
as well as all linear processes, such as dispersion and beam splitting. We show
that a single input pulse feeds only two distinct output modes and, for certain
quantum states, just one. Our theory provides the quantum states in the output
modes, which are crucial for the application of pulses in quantum optics and
quantum information.
