Quantum difference parametric amplification and oscillation

A recent article [W.C.W. Huang and H. Batelaan, arXiv:1708.0057v1] analysed the dualism between optical and difference parametric amplification, performing a classical analysis of a system where two electromagnetic fields are produced by another of a frequency which is the difference of the frequency of the other two. The authors claimed that this process would not violate energy conservation at the classical level, but that a quantum description would necessarily require a non-Hermitian Hamiltonian and therefore would not exist. In this work we show that the process can proceed quantum mechanically if described by the correct Hamiltonian, that energy conservation is not violated, and that fields are produced with interesting quantum statistics. Furthermore, we show that the process can be thought of as different types of already known three-wave mixing processes, with the actual type depending on either initial conditions or personal preference.