Comparing the winding numbers of two one-dimensional two-band topological systems by their wavefunction overlap

The measurement of topological numbers is crucial in the research of topological systems. In this article, we propose a protocol for obtaining the topological number (specifically, winding numbers in this case) of an unknown one-dimensional (1D) two-band topological system by comparing it with a known topological system. We consider two 1D two-band topological systems and their Bloch wavefunction overlap and verify a theorem. This theorem states that when the momentum varies by 2π, the number of cycles during which the magnitude of the wavefunction overlap varies from 0 to 1 and then back to 0 is equal to the absolute value of the difference between the topological numbers of these two systems. Furthermore, we propose two experimental schemes, one in a cold atom system and another one in a qubit system, which offer convenient and robust measurement methods for determining topological numbers of unknown states through quenching.