We describe possibilities of spontaneous, degenerate four-wave mixing (FWM) processes in spin-orbit coupled Bose-Einstein condensates. Phase matching conditions (i.e., energy and momentum conservation laws) in such systems allow one to identify four different configurations characterized by involvement of distinct spinor states in which such a process can take place. We derived these conditions from first principles and then illustrated dynamics with direct numerical simulations. We found, among others, the unique configuration, where both probe waves have smaller group velocity than pump wave and proved numerically that it can be observed experimentally under proper choice of the parameters. We also reported the case when two different FWM processes can occur simultaneously. The described resonant interactions of matter waves is expected to play important role in the experiments of BEC with artificial gauge fields.

中文翻译：

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自旋轨道耦合玻色-爱因斯坦凝聚中的四波混频

我们描述了自旋轨道耦合玻色-爱因斯坦凝聚中自发、简并四波混合 (FWM) 过程的可能性。这种系统中的相位匹配条件（即，能量和动量守恒定律）允许人们识别四种不同的配置，其特征是涉及不同的旋量状态，其中可以发生这种过程。我们从第一原理推导出这些条件，然后通过直接数值模拟来说明动力学。除此之外，我们发现了独特的配置，其中两个探针波具有比泵波的较小的群体速度，并且在数字上证明它可以在实验在适当的参数下实验观察。我们还报告了两个不同的 FWM 过程可以同时发生的情况。