Loschmidt echo and Berry phase around degeneracies in nonlinear systems

We study the Loschmidt echo and Berry phase in a nonlinear system transported around a double and triple degeneracy. The nonlinearity of the system makes the Berry phase different from that in linear systems. We propose a witness of nonlinearity for the nonlinear system and show its dependence on the parameters of the system, taking the standard Landau-Zener model as an example. We calculate the Loschmidt echo(LE) or quantum fidelity in the quantum dynamics under perturbations around the degeneracy, and establish a connection between the LE and the witness of nonlinearity. These phases and Loschmidt echo can be observed with current experimental technology in a nonlinear resonator.