Late time quantum chaos of pure states in random matrices and in the Sachdev-Ye-Kitaev model

We study the return amplitude, which is the overlap between the initial state and the time-evolved state, in the Sachdev-Ye-Kitaev (SYK) model. Initial states are taken to be product states in a spin basis. We numerically study the return amplitude by exactly diagonalizing the Hamiltonian. We also derive the analytic expression for the return amplitude in random matrix theory. The SYK results agree with the random matrix expectation. We also study the time evolution under the different Hamiltonian that is originally proposed to describe the traversable wormholes in projected black holes in the context of holography. The time evolution now depends on the choice of initial product states. The results are again explained by random matrix theory. In the symplectic ensemble cases, we observe an interesting pattern of the return amplitude in which they show the second dip, ramp, and plateaulike behavior.