Topological photonics mediated Hamiltonian chaos in cavity QED with spin–orbit coupled Bose–Einstein condensate

We study topological-photonics-mediated Hamiltonian chaos in a cavity QED system containing a spin–orbit coupled Bose–Einstein condensate driven by an amplitude-modulated pump. Starting from the quantum Langevin description, we derive a closed set of semiclassical quadrature equations for the cavity field and the two atomic pseudospin modes, where the modulation enters as a controlled time-periodic perturbation. We analyze the long-time dynamics by combining time-series signatures, phase-space trajectories for both pseudospin branches, and stroboscopic Poincaré sections, and we quantify dynamical instability using the largest Lyapunov exponent together with the associated Kolmogorov–Sinai entropy. In the trivial ordering κ>γ, increasing the normalized modulation amplitude produces a clear transition from regular motion to chaotic phase-space spreading. When the system is tuned to the topological-photonic side, identified by γ>κ, the same modulation reshapes the route to chaos by reorganizing the phase-space transport and the instability landscape, rather than shifting it uniformly. We further show that Raman detuning controls the symmetry of the SOC phase: δ=0 yields a more balanced response, while δ≠0 induces an asymmetric phase and a detuning-selective chaotic window. This work demonstrates that topological photonics provides a practical handle to control amplitude-modulated chaos in hybrid light–matter systems, which is important for future design of driven nonlinear photonic devices and synthetic-matter platforms.