  This paper introduces a quantum heat engine model that utilizes an ultracold
atomic gas coupled with a nanomechanical mirror. The mirror's vibration induces
an opto-mechanical sideband in the control field, affecting the behavior of the
cold gas and subsequently influencing the output radiation of the engine. The
model incorporates mirror vibration while omitting cavity confinement,
establishing a bridge between a multi-level atom-laser interacting system that
plays with coherences and the mechanical vibration of the nanomechanical
mirror, which jointly function as heat engines. Three distinct heat engine
configurations are proposed: the first involves a vibration-free three-level
$\Lambda$-type system, the second introduces nanomechanical vibration to the
three-level $\Lambda$-type system, and the third constitutes a composite engine
that combines the previous setups along with nanomechanical vibration. The
spectral brightness of a three-level heat engine is diminished with mirror
vibration, whereas for a composite heat engine, there is a slight enhancement
in the brightness peak. However, the maximum brightness is attained when there
is no vibration. Comparisons between the proposed model and an ideal system are
made regarding entropy balance, adhering to the constraints of the second law
of thermodynamics. The model observed that when subjected to mirror vibration,
the proposed heat engines diverged from the characteristics expected in an
ideal heat engine.
