We propose an efficient scheme to analytically discuss the phenomena of nanomechanically induced transparency (NMIT) and transmission rate in a parity-time-symmetric ( \(\mathcal{P}\mathcal{T}\) -symmetric) optonanomechanical system (ONMS). A levitated dielectric nanosphere is trapped near the antinodes closest to the passive cavity’s right mirror, which is further coupled to an active cavity via a hoping factor. We discover that the phenomenon of NMIT may be generated from the output probe field in the presence of an effective optonanomechanical 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. This optonanomechanical coupling strength can be well modified by varying the radius of the nanosphere and the Coulomb interaction. Furthermore, optical absorption and amplification can be adjusted by manipulating the gain-to-loss ratio as well as varying the effective optonanomechanical coupling strength. The transition NMIT behavior in the \(\mathcal{P}\mathcal{T}\) -symmetric and broken \(\mathcal{P}\mathcal{T}\) -symmetric domain is among the most intriguing results. Finally, for a fixed gain-to-loss ratio, the transmission spectra can also be studied by varying the steady-state position of the nanosphere. These results show that our scheme may inspire some potential applications for optical signal processing and quantum information processing.
