Nonlinear dynamical topological phases in a Cooper-pair box array

The topological property of a system is a static property in general. For instance, the topological edge state is observed by measuring the local density of states. In this paper we propose a system whose topological property is only revealed by dynamics. As a concrete example, we explore a nonlinear dynamical topological phase transition revealed by a quench dynamics of a Cooper-pair box array connected with capacitors, where the dynamics is driven by giving a flux to the left-end Cooper-pair box. It is described by coupled nonlinear differential equations due to the Josephson effect. It is trivial as far as the static system is concerned. However, the wave propagation induced by the quench dynamics demonstrates a rich topological phase diagram in terms of the strength of the input. It consists of topological, trivial, trapped, and dimer phases.