  An optimized sideband cooling in the presence of initial system correlations
is investigated for a standard optomechanical system coupled to a general
mechanical non-Markovian reservoir. We study the evolution of phonon number by
incorporating the effects of initial correlations into the time-dependent
coefficients in the Heisenberg equation. We introduce the concept of cooling
rate and define an average phonon reduction function to describe the sideband
cooling effect in non-Markovian regime. Our results show that the instantaneous
phonon number can be significantly reduced by introducing either the
parametric-amplification type or the beam-splitter type initial correlations.
In addition, the ground state cooling rate can be accelerated by enhancing the
initial correlation of beam-splitter type. By optimizing the initial state of
the system and utilizing Q-modulation technology, a stable mechanical ground
state can be obtained in a very short time. Our optimized cooling protocol
provides an appealing platform for phonon manipulation and quantum information
processing in solid-state systems.
