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We report topological nonlinear optics with spin-orbit coupled Bose-Einsteincondensate in a cavity. The cavity is driven by a pump laser and weak probelaser which excite Bose-Einstein condensate to an intermediate storage level,where the standard Raman process engineers spin-orbit coupling. We show thatthe nonlinear photonic interactions at the transitional pathways of dressedstates result in new type of optical transparencies, which get completelyinverted with atom induced gain. These nonlinear interactions also implanttopological sort of features in probe transmission modes by inducing gaplessDirac-like cones, which become gaped in presence of Raman detuning. Thetopological features get interestingly enhanced in gain regime where thegapless topological edge-like states emerge among the probe modes, which cancause non-trivial phase transition. We show that spin-orbit coupling and Zeemanfield effects also impressively revamp fast and slow probe light. Themanipulation of dressed states for quantum nonlinear optics with topologicalcharacteristics in our findings could be a crucial step towards topologicalquantum computation.

Topological Nonlinear Optics with Spin-Orbit coupled Bose-Einstein Condensate in Cavity