A general solution for the dynamics of a generalized non-degenerate optical parametric down-conversion interaction by virtue of the Lewis–Riesenfeld invariant theory

Time Evolution and Squeezing of Degenerate and Non-degenerate Coupled Parametric Down-Conversion with Driving Term

“…it returns to the solution in our previous work, [14] which corresponds to the non-degenerate parametric down-conversion with driving term. In other special cases, our solution in this paper proves to be well consistent with those results obtained by previous works.…”

“…[20] In this study, we will construct a time-dependent invariant by using a time-dependent unitary transformation for a properly initial invariant operator, then a question naturally arises how to choose the invariant operator and the unitary transformation? It is not difficult to see that on one hand, the time evolution of some parameter generation systems (such as the general non-degenerate optical parametric down-conversion [14] ) exactly preserves the coupled mode's photon number difference â † 1 â1 − â † 2 â2 ; on the other hand, the time evolution of a quantum system, due to a general quadratic parametric process with coupling of two mode (and/or single-mode) quantum fields, always approximatively preserves the energy ω 1 â †…”

“…[1−11] Recently, the single degenerate and the single non-degenerate parametric down-conversion with driving term were considered in Refs. [12] ∼ [14], respectively. With the development of the quasi-phase matching technique, several parametric processes can be realized simultaneously in a same quasiperiodic optical superlattice.…”