  In this paper, we study theoretically the bipartite and tripartite continuous
variable entanglement as well as the normal-mode splitting in a single-atom
cavity optomechanical system with intensity-dependent coupling. The system
under consideration is formed by a Fabry-Perot cavity with a thin vibrating end
mirror and a two-level atom in the Gaussian standing-wave of the cavity mode.
We first derive the general form of Hamiltonian describing the tripartite
intensity-dependent atom-field-mirror coupling due to the presence of cavity
mode structure. We then restrict our treatment to the first vibrational
sideband of the mechanical resonator and derive a novel form of tripartite
atom-field-mirror Hamiltonian. We show that when the optical cavity is
intensely driven one can generate bipartite entanglement between any pair of
the tripartite system, and that, due to entanglement sharing, the atom-mirror
entanglement is efficiently generated at the expense of optical-mechanical and
optical-atom entanglement. We also find that in such a system, when the
Lamb-Dicke parameter is large enough one can simultaneously observe the normal
mode splitting into three modes.
