Goos-Hänchen shift and slow-light enhancement in a fixed cavity: Bose-Einstein condensate Bogoliubov modes as mechanical oscillators

We theoretically investigate light-matter interference phenomena in a hybrid optical cavity--Bose-Einstein condensate (BEC) system incorporating both an intracavity optical parametric amplifier (OPA) and atom-atom collisions. Starting from the linearized quantum Langevin equations, we derive the drift matrix and confirm system stability by verifying that all eigenvalue real parts remain negative. We then analyze the probe transmission spectrum under two complementary control mechanisms. First, with collisional interactions suppressed, increasing the OPA gain within the stability region transforms the standard Lorentzian cavity response into a tunable Fano-like profile, accompanied by asymmetric dispersive line shapes and pronounced group delays. Second, with the OPA turned off, including atomic collisions yields an intrinsically narrow Bogoliubov excitation that interferes with the cavity continuum, reproducing Fano resonances, ``S-shaped'' phase dispersion, slow- and fast-light windows, and positive Goos-H\"anchen shifts. By examining both absorptive (real) and dispersive (imaginary) components of the transmitted field, we establish direct links between destructive interference pathways, steep phase slopes, and enhanced optical delays. Our results demonstrate that parametric amplification and collisional nonlinearities are equally versatile tools for engineering quantum interference in cavity QED with ultracold atoms, opening prospects for reconfigurable slow-light buffers, Fano-switchable photonic routers, and atom-integrated delay lines.