In this report, I will give three topological quantum simulations work based on synthetic lattice.

The first work is the study of the topological properties of extended SSH model - SSH4. We implement the SSH4 model in the momentum space lattice by using ultracold atoms. It is verified by experiments that the model's winding number can be obtained by the higher dimensional extension of the mean chiral displacement. The topological phase transition of the system is given experimentally, and the topological edge states are confirmed by quenching dynamics.
 
The second work is the study of the topological properties of discrete time quantum walk. The topological quantum walk in discrete time is realized by using the ultracold atom in the momentum space lattice. The topological properties of the system are proved experimentally by measuring the time average chiral displacement and the second order statistical moment. In addition, the interaction induced localization in quantum walk dynamics is also observed experimentally.

The third work is the study of non-hermitian skin effect with ultra cold atom. The non-Hermitian skin effect (NHSE), the accumulation of eigen wavefunctions at boundaries of open systems, underlies a variety of exotic properties that defy conventional wisdom. Here we report the experimental realization of a dissipative Aharonov-Bohm chain—a non-Hermitian topological model with NHSE—in the momentum space of a two-component Bose-Einstein condensate. We identify unique signatures of NHSE in the condensate dynamics, and perform Bragg spectroscopy to resolve topological edge states against a background of localized bulk states.