  We propose an experimentally feasible optomechanical scheme to realize a
negative cavity photon spectral function (CPSF) which is equivalent to a
negative absorption. The system under consideration is an optomechanical system
consisting of two mechanical (phononic) modes which are linearly coupled to a
common cavity mode via the radiation pressure while parametrically driven
through the coherent time-modulation of their spring coefficients. Using the
equations of motion for the cavity retarded Green's function obtained in the
framework of the generalized linear response theory, we show that in the
red-detuned and weak-coupling regimes a frequency-dependent effective cavity
damping rate (ECDR) corresponding to a negative CPSF can be realized by
controlling the cooperativities and modulation parameters while the system
still remains in the stable regime. Nevertheless, such a negativity which acts
as an optomechanical gain never occurs in a standard (an unmodulated bare)
cavity optomechanical system. Besides, we find that the presence of two
modulated mechanical degrees of freedom provides more controllability over the
magnitude and bandwidth of the negativity of CPSF, in comparison to the setup
with a single modulated mechanical oscillator. Interestingly, the introduced
negativity may open a new platform to realize an extraordinary (modified)
optomechanically induced transparency (in which the input signal is amplified
in the output) leading to a perfect tunable optomechanical filter with
switchable bandwidth which can be used as an optical transistor.
