Simulating and studying the topological properties of the photon-phonon coupled modes in a one-dimensional superconducting circuit lattice

We theoretically study the topological properties in a one-dimensional superconducting circuit lattice, where each unit cell is composed of a superconducting microwave cavity, a nanomechanical resonator (NAMR) and a superconducting qubit. By using diagonalization and dimensional reduction methods, the quantum spin Hall system of the photon-phonon coupled modes can be achieved. We analyze the energy spectrum and edge states of different lattice sizes. It is found that the number of lattices affects the topological characteristics of the system, and the inversion of the edge state distribution can be realized. In addition, considering the effects of defects, dissipation and disorder on the system, we find that they will respectively expand the region of the edge state, change the maximum value of the energy spectrum and cause small fluctuation of the energy spectrum. But the edge state is still very stable, which is due to the topological protection of the edge state. This scheme opens up a new way to study the application of topological matter in quantum information processing, meanwhile, the research results can be used to design novel quantum devices.