  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.
