Photon entanglement on a chip, optical instability, and Haken–Zwanzig model

Abstract

There is a growing interest in the development of chip-scale ring resonator sources for entangled photons produced in four-wave mixing processes. Therefore, theoretical accounts that describe in particular the optical instability at which entangled photons emerge and characterize the degree of entanglement by means of suitably defined entanglement functions are needed. The present study describes a quantum-mechanical model for a class of chip devices that have been developed during the past 5 years and typically involve a Kerr nonlinearity in addition to the four-wave mixing nonlinearity. It is shown that the optical instability point can be described in terms of amplitude equations of the Haken–Zwanzig type. Using positive P representation and an entanglement function recently established in the literature, it is shown that at the instability point such devices produce entangled photons with a maximal degree of entanglement. When pumping is increased the degree of entanglement decays. This holds irrespective of the Kerr nonlinearity as long as the magnitude of the Kerr nonlinearity is comparable with the susceptibility parameter characterizing the four-wave mixing effect. The main effect of the Kerr nonlinearity is to shift the instability point to higher pumping intensities.