Photonic spin Hall effect using bilayer graphene in nano-optomechanical cavities

We propose a theoretical model to obtain the photonic spin Hall effect (SHE) in an optomechanical nanocavity using a graphene bilayer as the intracavity medium. In our model, the pump and probe fields coherently drive the first mirror, whereas the second mirror has mechanical oscillation due to the radiation pressure. We show that the right- and left-circular polarization components of the Gaussian probe field striking at an arbitrary incident angle become spatially separate along a direction orthogonal to the plane of incidence. Photonic SHE can be coherently controlled by adjusting the optomechanical interaction, cavity field and G-mode phonon coupling, as well as G-mode phonon and electronic state interaction. The findings of photonic SHE are equally valid for standard optomechanical systems in the absence of cavity field and G-mode phonon coupling and electronic state interaction. The cavity field and G-mode phonon coupling broadened the detuning range of the probe field to observe the dominant photonic SHE. Adding G-mode phonon and electronic state interaction generates enhanced photonic SHE at three different probe field detunings due to optomechanical-induced transparency being split into three windows. We show that asymmetric photonic SHE can be controlled through cavity field and G-mode phonon coupling and G-mode phonon and electronic state interaction when probe field detuning is non-zero. The photonic SHE in bilayer graphene integrated with an optomechanical cavity may enable further studies of spin-dependent photonic effects and quantum sensing applications.