  Simultaneous ground-state cooling of two levitated nanoparticles is a crucial
prerequisite for investigation of macroscopic quantum effects such as quantum
entanglement and quantum correlation involving translational motion of
particles. Here we consider a coupled cavity-levitated-particle system and
present a detailed derivation of its Hamiltonian. We find that the
$y$-direction motions of the two particles are decoupled from the cavity field
and both the $x$- and $z$-direction motions, and that the $z$-direction motions
can be further decoupled from the cavity field and the $x$-direction motions by
choosing proper locations of the particles. We study the simultaneous cooling
of these mechanical modes in both the three-mode and five-mode cavity-levitated
optomechanical models. It is found that there exists the dark-mode effect when
the two tweezers have the same powers, which suppress the simultaneous
ground-state cooling. Nevertheless, the simultaneous ground-state cooling of
these modes can be realized by breaking the dark-mode effect under proper
parameters. Our system provides a versatile platform to study quantum effects
and applications in cavity-levitated optomechanical systems.
