Quantum chaos, information scrambling and eigenstate thermalization in coupled harmonic oscillators

Abstract

This thesis investigates the connection between quantum chaos and the Eigenstate Thermalization Hypothesis (ETH) in a system of two and three coupled harmonic oscillators with non-linear coupling. First, we numerically calculate thermal out-oftime- ordered correlators (OTOCs) for systems of two and three coupled harmonic oscillators. Next, we identify an early-time exponential growth regime, from which we extract temperature-dependent quantum Lyapunov exponent. We analyze the diagonal and off-diagonal matrix elements of the local position operator acting on first degree of freedom e.g. the first oscillator, x21 , in the energy eigenbasis and examine their consistency with the ETH ansatz. We also probe a possible connection between early time exponential growth of OTOC and eigenstate thermalisation. By inserting the ETH ansatz into the fourpoint correlation function underlying the OTOC, we show that the smooth and gaussian random structure of chaotic eigenstates and observables within the ETH framework can consistently account for signatures of coarse-grained picture of information scrambling. This supports the connection between information scrambling and eigenstate thermalization.