Exact analytical solutions for fully quantized parametric oscillation dynamics

Abstract

In this paper, a simple method for obtaining general analytical solutions of the time evolution equations for a fully quantized parametric oscillation process is developed. Heisenberg’s equations for the signal–idler photon annihilation operators are converted into a matrix equation equivalent to a two-state Jaynes–Cummings time evolution equation which has exact analytical solutions. The mean intensity inversion for the coupled signal–idler photon pair is found to undergo fractional revivals for pump photon in a Fock state, provided both signal and idler photons are in occupied Fock states. General collapses and revivals occur for interactions with pump photon in a coherent state, but now with both or either of signal and idler photons in occupied Fock states. An interpretation of the coupled signal–idler photon pair as a circularly polarized two-state system specified by positive and negative helicity states leads to an appropriate description of photon polarization state dynamics governed by the underlying Jaynes–Cummings interaction.