Efficient Description of Parametric Amplification of Quantum Pulses

A single quantum pulse undergoing parametric amplification feeds into at most two pulses in the output. In this work, we present an efficient, analytical method for finding the quantum state of these output modes. Our method applies the amplification to the vacuum rather than to the input state, and subsequently applies a transformed version of the operator that creates the input state from vacuum. Given the input and output pulse mode functions, the method is analytical, and therefore computationally very efficient, and it can be readily generalized to multiple non-vacuum input modes. We exemplify the method by computing the output quantum states resulting from the input of a coherent, a Schrödinger cat, and a single photon input quantum state. We further employ the method to obtain the quantum state in one of the two output modes heralded upon detection of vacuum in the other, least populated, mode.