Quantum evolution of classical nonlinear eigenmode in parametric interaction and realization in traps with ions

We study the quantum evolution of Kaplan's nonlinear eigenmode for the degenerate parametric oscillator. We calculate the energy exchange between the subharmonic and fundamental modes under conditions where the classical solution disallows any energy exchange between them. This energy exchange is strictly quantum in nature, i.e., it has no classical analog, and exhibits fractional revivals as well as large quantum fluctuations. We demonstrate how the quantum regime of nonlinear optical interactions can be studied in ion traps.