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.

对于四波混频过程中产生的纠缠光子的芯片级环形谐振器源的开发越来越引起人们的兴趣。因此，需要理论上的描述，该描述尤其描述了纠缠的光子出现时的光学不稳定性，并通过适当定义的纠缠函数来表征纠缠的程度。本研究描述了过去五年来开发的一类芯片设备的量子力学模型，除了四波混合非线性外，还通常涉及Kerr非线性。结果表明，光学不稳定点可以用Haken-Zwanzig类型的振幅方程来描述。使用文献中最近建立的正P表示和纠缠函数，结果表明，在不稳定性点上，这种装置产生具有最大纠缠度的纠缠光子。当泵浦增加时，纠缠度降低。这与Kerr非线性无关，只要Kerr非线性的大小与表征四波混频效果的磁化率参数相当即可。Kerr非线性的主要作用是将不稳定性点转移到较高的泵浦强度。