# Extracting Lyapunov exponents from the echo dynamics of Bose-Einstein condensates on a lattice

## TL;DR

This paper proposes an experimentally feasible method to measure the largest Lyapunov exponent in a chaotic Bose-Einstein condensate lattice system using Loschmidt echo, supported by numerical simulations.

## Contribution

It introduces a novel approach to extract Lyapunov exponents from Bose-Einstein condensates via time-reversal dynamics and noise analysis, bridging theory and experiment.

## Key findings

- Numerical simulations confirm the feasibility of extracting Lyapunov exponents.
- The method can be implemented with current experimental techniques.
- Parameter ranges for successful measurement are discussed.

## Abstract

We propose theoretically an experimentally realizable method to demonstrate the Lyapunov instability and to extract the value of the largest Lyapunov exponent for a chaotic many-particle interacting system. The proposal focuses specifically on a lattice of coupled Bose-Einstein condensates in the classical regime describable by the discrete Gross-Pitaevskii equation. We suggest to use imperfect time-reversal of system's dynamics known as Loschmidt echo, which can be realized experimentally by reversing the sign of the Hamiltonian of the system. The routine involves tracking and then subtracting the noise of virtually any observable quantity before and after the time-reversal. We support the theoretical analysis by direct numerical simulations demonstrating that the largest Lyapunov exponent can indeed be extracted from the Loschmidt echo routine. We also discuss possible values of experimental parameters required for implementing this proposal.