Goos-Hanchen Shift with a rotating atomic superfluid in a ring

We investigate the Goos Hanchen shift of the transmitted probe and show that optomechanical interference in a ring Bose Einstein condensate provides a sensitive, rotation tunable beam shift response at ultralow optical powers. Without quantized circulation, conventional optomechanically induced transparency produces a strictly positive and bounded shift whose magnitude is governed primarily by cooperativity; small detuning offsets can introduce a weak, transient sign change consistent with a Fano type asymmetry, but the overall response remains limited. With circulation, the Bragg scattered mechanical side modes split, yielding a double transparency dispersion with steep dispersive flanks that strongly amplify the phase derivative and bias its sign. In this regime, the peak shift grows monotonically with control field strength, reflecting enhanced linearized coupling and increased transmission across the angular scan. At fixed power, the detuning dependence is decisive: the shift is maximized at the red sideband condition and diminishes away from resonance, tracking how effectively the scan samples the rotation split dispersive flanks. Increasing the winding number broadens the central absorption and steepens accessible phase gradients, further boosting the attainable shift. The protocol remains minimally invasive under experimentally realistic conditions, and interatomic interactions have a negligible influence on the transmission features that set the phase slope. These results identify circulation, control power, and cavity detuning as practical knobs for in situ control of the Goos Hanchen shift, enabling interferometric beam steering and phase gradient metrology in hybrid atom optomechanical platforms.