Direct Measurement of Topological Number by Quench Dynamics

Abstract: The measurement of topological number is crucial in the research of topological systems. Recently, the relations between the topological number and the dynamics are built. But a direct method to read out the topological number via the dynamics is still lacking. In this work, we propose a new dynamical protocol to directly measure the topological number of an unknown system. Different from common quench operations, we change the Hamiltonian of the unknown system to another one with known topological properties. After the quench, different initial states result in different particle number distributions on the post-quench final Bloch bands. Such distributions depend on the wavefunction overlap between the initial Bloch state and the final Bloch state, which is a complex number depending on the momentum. We prove a theorem that when the momentum varies by $2\pi$, the phase of the wavefunction overlap change by $\Delta n\pi$ where $\Delta n$ is the topological number difference between the initial Bloch band and the final Bloch band. Based on this and the known topological number of the final Bloch band, we can directly deduce the topological number of the initial state from the particle number distribution and need not track the evolution of the system nor measure the spin texture. Two experimental schemes are also proposed as well. These schemes provide a convenient and robust measurement method and also deepens the understanding of the relation between topology and dynamics.