  We propose a scheme to investigate the topological phase transition and the
topological state transfer based on the small optomechanical lattice under the
realistic parameters regime. We find that the optomechanical lattice can be
equivalent to a topologically nontrivial Su-Schrieffer-Heeger (SSH) model via
designing the effective optomechanical coupling. Especially, the optomechanical
lattice experiences the phase transition between topologically nontrivial SSH
phase and topologically trivial SSH phase by controlling the decay of the
cavity field and the optomechanical coupling. We stress that the topological
phase transition is mainly induced by the decay of the cavity field, which is
counter-intuitive since the dissipation is usually detrimental to the system.
Also, we investigate the photonic state transfer between the two cavity fields
via the topologically protected edge channel based on the small optomechanical
lattice. We find that the quantum state transfer assisted by the topological
zero energy mode can be achieved via implying the external lasers with the
periodical driving amplitudes into the cavity fields. Our scheme provides the
fundamental and the insightful explanations toward the mapping of the photonic
topological insulator based on the micro-nano optomechanical quantum optical
platform.
