  Laser cooling of mesoscopic mechanical resonators is of great interest for
both fundamental studies and practical applications. We provide a general
framework to describe the cavity-assisted backaction cooling in the strong
coupling regime. By studying the cooling dynamics, we find that the temporal
evolution of mean phonon number oscillates as a function of the optomechanical
coupling strength depending on frequency mixing. The further analytical result
reveals that the optimal cooling limit is obtained when the system eigenmodes
satisfy the frequency matching condition. The reduced instantaneous-state
cooling limits with dynamic dissipative cooling approach are also presented.
Our study provides a guideline for optimizing the backaction cooling of
mesoscopic mechanical resonators in the strong coupling regime.
