  In recent years, there has been increasing interest in studying the quantum
characteristics in plasmonic metamaterials. By using the Hamiltonian combined
with second quantization, we have investigated the basic excitation of the
coupled metamaterials and presented a quantum description of them. The
interaction between the excitation of the coupled metamaterial and the exciton
in the quantum dot material has been further considered with the aid of the
interaction Hamiltonian. In such a damping system, a set of Langevin equations
has been used to deal with the quantum motion of the system. And the stimulated
emission has been found, which can be used to form a nano-laser. The quantum
description of coupled metamaterial and the interaction system could work as
the fundamental method in paving the way for studies on quantum effects in
coupled metamaterials.
