# Late Time Quantum Chaos of pure states in the SYK model

## TL;DR

This paper investigates the late-time behavior of quantum chaos in the SYK model by analyzing return amplitudes, comparing numerical results with random matrix theory, and exploring effects of different Hamiltonians related to holography.

## Contribution

It provides the first detailed numerical and analytical study of return amplitudes in the SYK model, connecting quantum chaos with random matrix theory and holographic black hole models.

## Key findings

- Return amplitudes in SYK match random matrix theory predictions.
- Different Hamiltonians affect the time evolution and return amplitude patterns.
- Symplectic ensemble cases show characteristic dip, ramp, and plateau behaviors.

## Abstract

In this letter, 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 describes 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 observed an interesting pattern of the return amplitude where they show the second dip, ramp and plateau like behavior.