Loschmidt Echo for edge of chaos and chaotic sea in Standard Map: Cases Study

Using the Standard Map model, this study explores the quantum Loschmidt Echo (LE) decay laws for mixed-type phase spaces, including edge of chaos and chaotic sea regimes. A universal decay law is proposed and numerically verified, characterizing the intricate dynamics of LE decay processes with a semi-classical statistical approach. Key findings include three typical decay stages and time-scale relations as well as the critical perturbations for comparing the decay extent, with critical insights into classical-quantum correspondences through the construction of classical ensemble from the uncertainty of quantum state. This work provides analytical and numerical advancements in understanding dynamical stability in quantum systems.