Nanomechanically induced transparency in $\mathcal{PT}$-symmetric optical cavities

In this paper, we analytically present the phenomena of nanomechanically induced transparency (NMIT) and transmission rate in a parity-time-symmetric ($\mathcal{PT}$-symmetric) opto-nanomechanical system (ONMS) where a levitated dielectric nanospheres is trapped near the antinodes closest to right mirror of passive cavity which further coupled to an active cavity via hoping factor. We find that the phenomenon of NMIT may be generated from the output probe field in the presence of an effective opto-nanomechanical coupling between the cavity field and the nanosphere, whose steady-state position is influenced by the Coulomb interaction between the cavity mirror and the nanosphere. In addition, the width and height of the transparency window can be controlled through the effective optomechanical coupling, which is readily adjusted by altering changing the nanosphere's radius and the Coulomb interaction. One of the most interesting result is the transition NMIT behavior in $\mathcal{PT}$-symmetric and broken $\mathcal{PT}$-symmetric regime. We show that the presence of nanosphere in the passive cavity enhances the width and transmission rate of NMIT window in passive-passive regime and in passive-active regime, a notable decrease of sideband amplification has been observed. These results show that our scheme may find some potential applications for optical signal processing an and quantum information processing.