Lyapunov exponents, ergodization time and out-of-time-order correlators in chaotic many-particle systems from Loschmidt echoes

We propose an experimentally realizable method to demonstrate Lyapunov instability and to estimate ergodization time in chaotic many-particle systems by monitoring equilibrium noise of virtually any observable quantity before and after time reversal of dynamics (Loschmidt echo). In the quantum regime, the quantity of interest for the method is a counterpart of out-of-time-order correlators (OTOCs). The ergodization time is defined as the characteristic time required to extract the largest Lyapunov exponent from a system’s dynamics. The proposal focuses specifically on a lattice of coupled Bose-Einstein condensates in the classical regime describable by the discrete Gross-Pitaevskii equation. We support our theoretical analysis by direct numerical simulations demonstrating that the largest Lyapunov exponent and the ergodization time can indeed be extracted from the Loschmidt echo routine.