  Here we present details on how the cooling effects of an opto-mechanical
system are affected beyond the secular approximation. To this end, a laser
driven two-level quantum dot (QD) embed- ded in a phononic nano-cavity is
investigated for moderately strong QD-phonon couplings regimes. For these
regimes, the use of a secular approximation within the QD-phonon interaction
terms is no longer justified as the rapidly oscillating terms cannot be
neglected from the system dynamics. Therefore, one shows that although being
small, their contribution plays an important role when quantum cooling is
achieved. The main contribution of the fast oscillating terms is analytically
estimated and one compares how the quantum cooling dynamics change within or
beyond the sec- ular approximation. The behavior of the quantum cooling effect
is investigated in the steady-state regime via the phonon field statistics.
