We investigate topological nonlinear optics with spin-orbit coupled Bose-Einstein condensate in a cavity. The cavity is driven by a pump laser and a weak probe laser. Both lasers excite Bose-Einstein condensate, in the presence of standard Raman process for spin-orbit coupling, to an intermediate storage level. We theoretically show that the quantum interference at the transitional pathways of dressed atomic states results in different types of optical transparencies, which get completely inverted in atomic damping induced gain regime. The synthetic pseudo-spin states also implant different phases in the probe field forcing modes in probe transparencies to form gapless Dirac cones, which become gapped in presence of Raman detuning. These features get interestingly enhanced in gain regime where the amplified part of probe transparencies appear as gapless topological edge-like states between the probe bulk modes and cause non-trivial phase transition. We illustrate that the nonlinear interactions of the pseudo-spin states also enhance the slow light features in probe transmission. The manipulation of dressed states for topological optical transparencies in our findings could be a crucial step towards topological photonics and their application in quantum computation.

我们研究了腔内自旋轨道耦合玻色-爱因斯坦凝聚体的拓扑非线性光学。该腔由泵浦激光器和弱探测激光器驱动。在标准拉曼过程用于自旋轨道耦合的情况下，两种激光器都将玻色-爱因斯坦凝聚物激发到中间存储水平。我们从理论上表明，修饰原子态的过渡路径上的量子干涉导致不同类型的光学透明度，这些光学透明度在原子阻尼诱导的增益状态下完全反转。合成的伪自旋态还在探针透明体中的探针场强制模式中植入不同的相位，以形成无间隙的狄拉克锥，在拉曼失谐的存在下，狄拉克锥变得有间隙。这些特征在增益状态下得到了有趣的增强，其中探针透明度的放大部分表现为探针体模式之间的无间隙拓扑边缘状状态，并导致非平凡的相变。我们说明伪自旋态的非线性相互作用也增强了探测传输中的慢光特征。在我们的研究结果中，对拓扑光学透明度的修饰状态进行操作可能是迈向拓扑光子学及其在量子计算中应用的关键一步。