Hyperchaos in a Bose-Hubbard Chain with Rydberg-Dressed Interactions

Weibin Li,Rejish Nath,Gary Mccormack

doi:10.3390/photonics8120554

Abstract

We study the chaos and hyperchaos of Rydberg-dressed Bose–Einstein condensates (BECs) in a one-dimensional optical lattice. Due to the long-range, soft-core interaction between the dressed atoms, the dynamics of the BECs are described by the extended Bose-Hubbard model. In the mean-field regime, we analyze the dynamical stability of the BEC by focusing on the ground state and localized state configurations. Lyapunov exponents of the two configurations are calculated by varying the soft-core interaction strength, potential bias, and length of the lattice. Both configurations can have multiple positive Lyapunov exponents, exhibiting hyperchaotic dynamics. We show the dependence of the number of the positive Lyapunov exponents and the largest Lyapunov exponent on the length of the optical lattice. The largest Lyapunov exponent is directly proportional to areas of phase space encompassed by the associated Poincaré sections. We demonstrate that linear and hysteresis quenches of the lattice potential and the dressed interaction lead to distinct dynamics due to the chaos and hyperchaos. Our work is relevant to current research on chaos as well as collective and emergent nonlinear dynamics of BECs with long-range interactions.

Highlights

Over the past two decades, Bose–Einstein condensates (BECs) of ultracold atomic gases have become an ideal system for studying both quantum and nonlinear dynamics due to the high controllability of the two-body interactions [1], trapping potentials [2], and spatial dimensions [3,4], along with long coherence times
We explore the dynamical stability of the ground state and localized state, where the dependence of the largest and the total number of positive Lyapunov exponents [72,73] on the dressed interaction and system size are explored
We will discuss how the maximal and total number of Lyapunov exponents depend on the system size and initial state, focusing on parameter regimes where the nonlinear interaction cannot be neglected, i.e., chaos and hyperchaos are expected in the dynamics

Summary

Introduction

Over the past two decades, Bose–Einstein condensates (BECs) of ultracold atomic gases have become an ideal system for studying both quantum and nonlinear dynamics due to the high controllability of the two-body interactions [1], trapping potentials [2], and spatial dimensions [3,4], along with long coherence times. We explore the dynamical stability of the ground state and localized state, where the dependence of the largest and the total number of positive Lyapunov exponents [72,73] on the dressed interaction and system size are explored. We explore the static (eigenenergies and Bogoliubov spectra) and dynamical properties (Lyapunov exponents) of the ground state and localized state configurations in Sections 3 and 4, respectively. Dynamics driven by both the linear and hysteresis quenching parameters are explored with different initial states.

Nonlinear Eigenenergies and Bogoliubov Spectra

Poincaré Sections and Lyapunov Exponents

Quenching Schemes

Quench Dynamics

Discussion

Conclusions and Outlook